What is the integral of 1/((1+x^2)^4) ?

The integral of 1/((1+x^2)^4) is

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integral of 1/((1+x^2)^4)

\int \frac{1}{( 1 + x ^{2} ) ^{4}} d x

\frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} - \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1} + C

Solution steps

\int \frac{1}{( 1 + x ^{2} ) ^{4}} d x

Apply Trigonometric Substitution

: \int - \frac{1}{csc ^{8} ( u ) sin ^{2} ( u )} d u

= \int - \frac{1}{csc ^{8} ( u ) sin ^{2} ( u )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{csc ^{8} ( u ) sin ^{2} ( u )} d u

Rewrite using trig identities

= - \int sin^{6} (u) d u

Apply Integral Reduction

= - (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} \cdot \int sin^{4} (u) d u)

\int sin^{4} (u) d u = - \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))

= - (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

Substitute back

u = arccot (x)

= - (- \frac{cos ( arccot ( x ) ) sin ^{5} ( arccot ( x ) )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (arccot (x)) cos (arccot (x)) + \frac{3}{8} (arccot (x) - \frac{1}{2} sin (2 arccot (x)))))

Simplify

- (- \frac{cos ( arccot ( x ) ) sin ^{5} ( arccot ( x ) )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (arccot (x)) cos (arccot (x)) + \frac{3}{8} (arccot (x) - \frac{1}{2} sin (2 arccot (x))))) : \frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} - \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1}

= \frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} - \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1}

Add a constant to the solution

= \frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} - \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1} + C

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f (x) = x^{2}