Integrand size = 12, antiderivative size = 107 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=-\frac {2 x^{-1+m}}{a \left (1-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}-\frac {2 x^{-1+m} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1+m),\frac {3+m}{4},a^2 x^4\right )}{a \left (1-m^2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 107, 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} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=-\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} x^{m-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m-1}{4},\frac {m+3}{4},a^2 x^4\right )}{a \left (1-m^2\right )}-\frac {2 x^{m-1}}{a \left (1-m^2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^2\right )}}{m+1} \]
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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^2\right )} x^{1+m}}{1+m}+\frac {2 \int x^{-2+m} \, dx}{a (1+m)}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^{-2+m}}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a (1+m)} \\ & = -\frac {2 x^{-1+m}}{a \left (1-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^{-2+m}}{\sqrt {1-a^2 x^4}} \, dx}{a (1+m)} \\ & = -\frac {2 x^{-1+m}}{a \left (1-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}-\frac {2 x^{-1+m} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1+m),\frac {3+m}{4},a^2 x^4\right )}{a \left (1-m^2\right )} \\ \end{align*}
Time = 2.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.49 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=\frac {2^{\frac {1+m}{2}} e^{\text {sech}^{-1}\left (a x^2\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^2\right )}}\right )^{\frac {1+m}{2}} x^{1+m} \left (a x^2\right )^{\frac {1}{2} (-1-m)} \left ((7+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{4},\frac {7+m}{4},-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )-e^{2 \text {sech}^{-1}\left (a x^2\right )} (3+m) \operatorname {Hypergeometric2F1}\left (1,\frac {5-m}{4},\frac {11+m}{4},-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )\right )}{(3+m) (7+m)} \]
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\[\int \left (\frac {1}{a \,x^{2}}+\sqrt {\frac {1}{a \,x^{2}}-1}\, \sqrt {\frac {1}{a \,x^{2}}+1}\right ) x^{m}d x\]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=\frac {\int \frac {x^{m}}{x^{2}}\, dx + \int a x^{m} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx=\int x^m\,\left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \]
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