Integrand size = 10, antiderivative size = 64 \[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\frac {1}{3} x^3 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2 n},\frac {3 (1+n)}{2 n},-a^2 x^{2 n}\right )}{3 (3+n)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5875, 12, 371} \[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\frac {1}{3} x^3 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{n+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+3}{2 n},\frac {3 (n+1)}{2 n},-a^2 x^{2 n}\right )}{3 (n+3)} \]
[In]
[Out]
Rule 12
Rule 371
Rule 5875
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arcsinh}\left (a x^n\right )-\frac {1}{3} \int \frac {a n x^{2+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = \frac {1}{3} x^3 \text {arcsinh}\left (a x^n\right )-\frac {1}{3} (a n) \int \frac {x^{2+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = \frac {1}{3} x^3 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2 n},\frac {3 (1+n)}{2 n},-a^2 x^{2 n}\right )}{3 (3+n)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\frac {1}{3} x^3 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2 n},1+\frac {3+n}{2 n},-a^2 x^{2 n}\right )}{3 (3+n)} \]
[In]
[Out]
\[\int x^{2} \operatorname {arcsinh}\left (a \,x^{n}\right )d x\]
[In]
[Out]
Exception generated. \[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\int x^{2} \operatorname {asinh}{\left (a x^{n} \right )}\, dx \]
[In]
[Out]
\[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\int { x^{2} \operatorname {arsinh}\left (a x^{n}\right ) \,d x } \]
[In]
[Out]
\[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\int { x^{2} \operatorname {arsinh}\left (a x^{n}\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \text {arcsinh}\left (a x^n\right ) \, dx=\int x^2\,\mathrm {asinh}\left (a\,x^n\right ) \,d x \]
[In]
[Out]