Integrand size = 12, antiderivative size = 133 \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {p x^{1+m-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m-p}{2 p},\frac {1+m+p}{2 p},a^2 x^{2 p}\right )}{a (1+m) (1+m-p)} \]
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Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6470, 30, 265, 371} \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} x^{m-p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m-p+1}{2 p},\frac {m+p+1}{2 p},a^2 x^{2 p}\right )}{a (m+1) (m-p+1)}+\frac {p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1} \]
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Rule 30
Rule 265
Rule 371
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p \int x^{m-p} \, dx}{a (1+m)}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{m-p}}{\sqrt {1-a x^p} \sqrt {1+a x^p}} \, dx}{a (1+m)} \\ & = \frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{m-p}}{\sqrt {1-a^2 x^{2 p}}} \, dx}{a (1+m)} \\ & = \frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {p x^{1+m-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m-p}{2 p},\frac {1+m+p}{2 p},a^2 x^{2 p}\right )}{a (1+m) (1+m-p)} \\ \end{align*}
Time = 4.33 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.40 \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {2^{\frac {1+m}{p}} e^{\text {sech}^{-1}\left (a x^p\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^p\right )}}\right )^{\frac {1+m}{p}} x^{1+m} \left (a x^p\right )^{-\frac {1+m}{p}} \left (-e^{2 \text {sech}^{-1}\left (a x^p\right )} (1+m+p) \operatorname {Hypergeometric2F1}\left (1,-\frac {1+m-3 p}{2 p},\frac {1+m+5 p}{2 p},-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )+(1+m+3 p) \operatorname {Hypergeometric2F1}\left (1,1-\frac {1+m+p}{2 p},\frac {1+m+3 p}{2 p},-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )\right )}{(1+m+p) (1+m+3 p)} \]
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\[\int \left (\frac {x^{-p}}{a}+\sqrt {\frac {x^{-p}}{a}-1}\, \sqrt {\frac {x^{-p}}{a}+1}\right ) x^{m}d x\]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\text {Exception raised: TypeError} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {\int x^{m} x^{- p}\, dx + \int a x^{m} \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}\, dx}{a} \]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\text {Exception raised: ValueError} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}\right )} \,d x } \]
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Timed out. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\int x^m\,\left (\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}\right ) \,d x \]
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