Integrand size = 10, antiderivative size = 65 \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=-\frac {\text {arcsinh}\left (a x^n\right )}{x}-\frac {a n x^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1-n}{2 n},\frac {1}{2} \left (3-\frac {1}{n}\right ),-a^2 x^{2 n}\right )}{1-n} \]
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Time = 0.02 (sec) , antiderivative size = 65, 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} \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=-\frac {a n x^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1-n}{2 n},\frac {1}{2} \left (3-\frac {1}{n}\right ),-a^2 x^{2 n}\right )}{1-n}-\frac {\text {arcsinh}\left (a x^n\right )}{x} \]
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Rule 12
Rule 371
Rule 5875
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}\left (a x^n\right )}{x}+\int \frac {a n x^{-2+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^n\right )}{x}+(a n) \int \frac {x^{-2+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = -\frac {\text {arcsinh}\left (a x^n\right )}{x}-\frac {a n x^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1-n}{2 n},\frac {1}{2} \left (3-\frac {1}{n}\right ),-a^2 x^{2 n}\right )}{1-n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=-\frac {\text {arcsinh}\left (a x^n\right )}{x}+\frac {a n x^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {-1+n}{2 n},1+\frac {-1+n}{2 n},-a^2 x^{2 n}\right )}{-1+n} \]
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\[\int \frac {\operatorname {arcsinh}\left (a \,x^{n}\right )}{x^{2}}d x\]
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Exception generated. \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{n} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x^2} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x^2} \,d x \]
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