Integrand size = 8, antiderivative size = 147 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=-\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{a x}-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{\sqrt {a}} \]
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Time = 0.05 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6465, 30, 265, 331, 313, 227, 1213, 435} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=-\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{a x}+\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{\sqrt {a}}-\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x} \]
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Rule 30
Rule 227
Rule 265
Rule 313
Rule 331
Rule 435
Rule 1213
Rule 6465
Rubi steps \begin{align*} \text {integral}& = e^{\text {sech}^{-1}\left (a x^2\right )} x+\frac {2 \int \frac {1}{x^2} \, dx}{a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^2 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a} \\ & = -\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^4}} \, dx}{a} \\ & = -\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{a x}-\left (2 a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^4}} \, dx \\ & = -\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{a x}+\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx-\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1+a x^2}{\sqrt {1-a^2 x^4}} \, dx \\ & = -\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{a x}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{\sqrt {a}}-\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1+a x^2}}{\sqrt {1-a x^2}} \, dx \\ & = -\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{a x}-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{\sqrt {a}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=-\frac {1}{a x}+\left (-\frac {1}{a x}-x\right ) \sqrt {\frac {1-a x^2}{1+a x^2}}-\frac {2 i \sqrt {\frac {1-a x^2}{1+a x^2}} \sqrt {1-a^2 x^4} \left (E\left (\left .i \text {arcsinh}\left (\sqrt {-a} x\right )\right |-1\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-a} x\right ),-1\right )\right )}{\sqrt {-a} \left (-1+a x^2\right )} \]
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Time = 0.77 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {1}{a x}-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (x^{4} a^{2}+2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \operatorname {EllipticF}\left (x \sqrt {a}, i\right ) \sqrt {a}-2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \operatorname {EllipticE}\left (x \sqrt {a}, i\right ) \sqrt {a}-1\right )}{x^{4} a^{2}-1}\) | \(132\) |
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=\int { \sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}} \,d x } \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=\frac {\int \frac {1}{x^{2}}\, dx + \int a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=\int { \sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}} \,d x } \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=\int { \sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}} \,d x } \]
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Timed out. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=\int \sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2} \,d x \]
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