Integrand size = 10, antiderivative size = 68 \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=-\frac {\text {arcsinh}\left (a x^n\right )}{2 x^2}-\frac {a n x^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right ),\frac {1}{2} \left (3-\frac {2}{n}\right ),-a^2 x^{2 n}\right )}{2 (2-n)} \]
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Time = 0.03 (sec) , antiderivative size = 68, 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} \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=-\frac {a n x^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right ),\frac {1}{2} \left (3-\frac {2}{n}\right ),-a^2 x^{2 n}\right )}{2 (2-n)}-\frac {\text {arcsinh}\left (a x^n\right )}{2 x^2} \]
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Rule 12
Rule 371
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (a x^n\right )}{2 x^2}+\frac {1}{2} \int \frac {a n x^{-3+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^n\right )}{2 x^2}+\frac {1}{2} (a n) \int \frac {x^{-3+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^n\right )}{2 x^2}-\frac {a n x^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right ),\frac {1}{2} \left (3-\frac {2}{n}\right ),-a^2 x^{2 n}\right )}{2 (2-n)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=\frac {-\left ((-2+n) \text {arcsinh}\left (a x^n\right )\right )+a n x^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {1}{n},\frac {3}{2}-\frac {1}{n},-a^2 x^{2 n}\right )}{2 (-2+n) x^2} \]
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\[\int \frac {\operatorname {arcsinh}\left (a \,x^{n}\right )}{x^{3}}d x\]
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Exception generated. \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{n} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x^3} \,d x \]
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