Integrand size = 12, antiderivative size = 109 \[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=-\frac {3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}-\frac {3 x^{-2+m} \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-2+m),\frac {4+m}{6},a^2 x^6\right )}{a \left (2+m-m^2\right )} \]
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Time = 0.05 (sec) , antiderivative size = 109, 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} \[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=-\frac {3 \sqrt {\frac {1}{a x^3+1}} \sqrt {a x^3+1} x^{m-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m-2}{6},\frac {m+4}{6},a^2 x^6\right )}{a \left (-m^2+m+2\right )}-\frac {3 x^{m-2}}{a \left (-m^2+m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^3\right )}}{m+1} \]
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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^3\right )} x^{1+m}}{1+m}+\frac {3 \int x^{-3+m} \, dx}{a (1+m)}+\frac {\left (3 \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3}\right ) \int \frac {x^{-3+m}}{\sqrt {1-a x^3} \sqrt {1+a x^3}} \, dx}{a (1+m)} \\ & = -\frac {3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}+\frac {\left (3 \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3}\right ) \int \frac {x^{-3+m}}{\sqrt {1-a^2 x^6}} \, dx}{a (1+m)} \\ & = -\frac {3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}-\frac {3 x^{-2+m} \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-2+m),\frac {4+m}{6},a^2 x^6\right )}{a \left (2+m-m^2\right )} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.72 \[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=\frac {2^{\frac {1+m}{3}} e^{\text {sech}^{-1}\left (a x^3\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^3\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^3\right )}}\right )^{\frac {1+m}{3}} \left (1+e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )^{\frac {1+m}{3}} x^{1+m} \left (a x^3\right )^{\frac {1}{3} (-1-m)} \left ((10+m) \operatorname {Hypergeometric2F1}\left (\frac {4+m}{6},\frac {4+m}{3},\frac {10+m}{6},-e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )-e^{2 \text {sech}^{-1}\left (a x^3\right )} (4+m) \operatorname {Hypergeometric2F1}\left (\frac {4+m}{3},\frac {10+m}{6},\frac {16+m}{6},-e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )\right )}{(4+m) (10+m)} \]
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\[\int \left (\frac {1}{a \,x^{3}}+\sqrt {\frac {1}{a \,x^{3}}-1}\, \sqrt {\frac {1}{a \,x^{3}}+1}\right ) x^{m}d x\]
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\[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {1}{a x^{3}} + 1} \sqrt {\frac {1}{a x^{3}} - 1} + \frac {1}{a x^{3}}\right )} \,d x } \]
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\[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=\frac {\int \frac {x^{m}}{x^{3}}\, dx + \int a x^{m} \sqrt {-1 + \frac {1}{a x^{3}}} \sqrt {1 + \frac {1}{a x^{3}}}\, dx}{a} \]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=\text {Exception raised: ValueError} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {1}{a x^{3}} + 1} \sqrt {\frac {1}{a x^{3}} - 1} + \frac {1}{a x^{3}}\right )} \,d x } \]
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Timed out. \[ \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx=\int x^m\,\left (\sqrt {\frac {1}{a\,x^3}-1}\,\sqrt {\frac {1}{a\,x^3}+1}+\frac {1}{a\,x^3}\right ) \,d x \]
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