What is the integral of (e^{3x})/((1+e^x)^3) ?

The integral of (e^{3x})/((1+e^x)^3) is

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integral of (e^{3x})/((1+e^x)^3)

\int \frac{e ^{3 x}}{( 1 + e ^{x} ) ^{3}} d x

\frac{e ^{3 x}}{( 1 + e ^{x} ) ^{3}} + ln ∣ e^{x} + 1 ∣ + \frac{3}{e ^{x} + 1} - \frac{3}{2 ( e ^{x} + 1 ) ^{2}} + \frac{1}{3 ( e ^{x} + 1 ) ^{3}} + \frac{2 e ^{2 x}}{3 ( 1 + e ^{x} ) ^{3}} - \frac{2 ( - 2 e ^{x} - 1 )}{3 ( e ^{x} + 1 ) ^{2}} + C

Solution steps

\int \frac{e ^{3 x}}{( 1 + e ^{x} ) ^{3}} d x

Apply u-substitution

: \int \frac{u ^{2}}{( 1 + u ) ^{3}} d u

= \int \frac{u ^{2}}{( 1 + u ) ^{3}} d u

Apply Integration By Parts

: \frac{u ^{3}}{( 1 + u ) ^{3}} - \int \frac{u ^{2} ( - u + 2 )}{( 1 + u ) ^{4}} d u

= \frac{u ^{3}}{( 1 + u ) ^{3}} - \int \frac{u ^{2} ( - u + 2 )}{( 1 + u ) ^{4}} d u

\int \frac{u ^{2} ( - u + 2 )}{( 1 + u ) ^{4}} d u = - ln ∣ u + 1 ∣ - \frac{3}{u + 1} + \frac{3}{2 ( u + 1 ) ^{2}} - \frac{1}{3 ( u + 1 ) ^{3}} - \frac{2 u ^{2}}{3 ( 1 + u ) ^{3}} + \frac{2 ( - 2 u - 1 )}{3 ( u + 1 ) ^{2}}

= \frac{u ^{3}}{( 1 + u ) ^{3}} - (- ln ∣ u + 1 ∣ - \frac{3}{u + 1} + \frac{3}{2 ( u + 1 ) ^{2}} - \frac{1}{3 ( u + 1 ) ^{3}} - \frac{2 u ^{2}}{3 ( 1 + u ) ^{3}} + \frac{2 ( - 2 u - 1 )}{3 ( u + 1 ) ^{2}})

Substitute back

u = e^{x}

= \frac{( e ^{x} ) ^{3}}{( 1 + e ^{x} ) ^{3}} - (- ln ∣ e^{x} + 1 ∣ - \frac{3}{e ^{x} + 1} + \frac{3}{2 ( e ^{x} + 1 ) ^{2}} - \frac{1}{3 ( e ^{x} + 1 ) ^{3}} - \frac{2 ( e ^{x} ) ^{2}}{3 ( 1 + e ^{x} ) ^{3}} + \frac{2 ( - 2 e ^{x} - 1 )}{3 ( e ^{x} + 1 ) ^{2}})

Simplify

\frac{( e ^{x} ) ^{3}}{( 1 + e ^{x} ) ^{3}} - (- ln ∣ e^{x} + 1 ∣ - \frac{3}{e ^{x} + 1} + \frac{3}{2 ( e ^{x} + 1 ) ^{2}} - \frac{1}{3 ( e ^{x} + 1 ) ^{3}} - \frac{2 ( e ^{x} ) ^{2}}{3 ( 1 + e ^{x} ) ^{3}} + \frac{2 ( - 2 e ^{x} - 1 )}{3 ( e ^{x} + 1 ) ^{2}}) : \frac{e ^{3 x}}{( 1 + e ^{x} ) ^{3}} + ln ∣ e^{x} + 1 ∣ + \frac{3}{e ^{x} + 1} - \frac{3}{2 ( e ^{x} + 1 ) ^{2}} + \frac{1}{3 ( e ^{x} + 1 ) ^{3}} + \frac{2 e ^{2 x}}{3 ( 1 + e ^{x} ) ^{3}} - \frac{2 ( - 2 e ^{x} - 1 )}{3 ( e ^{x} + 1 ) ^{2}}

= \frac{e ^{3 x}}{( 1 + e ^{x} ) ^{3}} + ln ∣ e^{x} + 1 ∣ + \frac{3}{e ^{x} + 1} - \frac{3}{2 ( e ^{x} + 1 ) ^{2}} + \frac{1}{3 ( e ^{x} + 1 ) ^{3}} + \frac{2 e ^{2 x}}{3 ( 1 + e ^{x} ) ^{3}} - \frac{2 ( - 2 e ^{x} - 1 )}{3 ( e ^{x} + 1 ) ^{2}}

Add a constant to the solution

= \frac{e ^{3 x}}{( 1 + e ^{x} ) ^{3}} + ln ∣ e^{x} + 1 ∣ + \frac{3}{e ^{x} + 1} - \frac{3}{2 ( e ^{x} + 1 ) ^{2}} + \frac{1}{3 ( e ^{x} + 1 ) ^{3}} + \frac{2 e ^{2 x}}{3 ( 1 + e ^{x} ) ^{3}} - \frac{2 ( - 2 e ^{x} - 1 )}{3 ( e ^{x} + 1 ) ^{2}} + C

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What is the integral of (e^{3x})/((1+e^x)^3) ?
The integral of (e^{3x})/((1+e^x)^3) is