What is the integral of 1/(sqrt((1+x^2)*ln(x+\sqrt{1+x^2)))} ?

The integral of 1/(sqrt((1+x^2)*ln(x+\sqrt{1+x^2)))} is 2sqrt(ln(x+\sqrt{1+x^2))}+C

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integral of 1/(sqrt((1+x^2)*ln(x+\sqrt{1+x^2)))}

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

2 ln (x + 1 + x^{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ln ( u )} d u

= \int \frac{1}{u ln ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

= \int \frac{1}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= 2 v^{\frac{1}{2}}

Substitute back

= 2 ln^{\frac{1}{2}} (x + 1 + x^{2})

Apply radical rule:

a^{\frac{1}{n}} = n a

= 2 ln (x + 1 + x^{2})

Add a constant to the solution

= 2 ln (x + 1 + x^{2}) + C

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What is the integral of 1/(sqrt((1+x^2)*ln(x+\sqrt{1+x^2)))} ?
The integral of 1/(sqrt((1+x^2)*ln(x+\sqrt{1+x^2)))} is 2sqrt(ln(x+\sqrt{1+x^2))}+C