Integrand size = 12, antiderivative size = 112 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{5 a^{5/2}}-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{5 a^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6470, 30, 265, 313, 227, 1213, 435} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=-\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{5 a^{5/2}}+\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac {2 x^3}{15 a}+\frac {1}{5} x^5 e^{\text {sech}^{-1}\left (a x^2\right )} \]
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Rule 30
Rule 227
Rule 265
Rule 313
Rule 435
Rule 1213
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {2 \int x^2 \, dx}{5 a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{5 a} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^4}} \, dx}{5 a} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5-\frac {\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}{5 a^2}+\frac {\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}{5 a^2} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5-\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 )}{5 a^{5/2}}+\frac {\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}{5 a^2} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\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 )}{5 a^{5/2}}-\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 )}{5 a^{5/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.25 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\frac {1}{15} \left (\frac {5 x^3}{a}+\frac {3 \sqrt {\frac {1-a x^2}{1+a x^2}} \left (x^3+a x^5\right )}{a}+\frac {6 i \sqrt {\frac {1-a x^2}{1+a x^2}} \sqrt {1-a^2 x^4} \left (E\left (\left .i \text {arcsinh}\left (\sqrt {-a} x\right )\right |-1\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-a} x\right ),-1\right )\right )}{(-a)^{5/2} \left (-1+a x^2\right )}\right ) \]
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Time = 0.66 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{\frac {7}{2}} x^{7}-x^{3} a^{\frac {3}{2}}+2 \operatorname {EllipticF}\left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}-2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \operatorname {EllipticE}\left (x \sqrt {a}, i\right )\right )}{5 \left (x^{4} a^{2}-1\right ) a^{\frac {3}{2}}}+\frac {x^{3}}{3 a}\) | \(136\) |
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Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\frac {5 \, a^{2} x^{3} + 3 \, {\left (a^{3} x^{5} - 2 \, a x\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - \frac {6 i \, E(\arcsin \left (\frac {1}{\sqrt {a} x}\right )\,|\,-1)}{\sqrt {a}} + \frac {6 i \, F(\arcsin \left (\frac {1}{\sqrt {a} x}\right )\,|\,-1)}{\sqrt {a}}}{15 \, a^{3}} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\frac {\int x^{2}\, dx + \int a x^{4} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\int { x^{4} {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx=\int x^4\,\left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \]
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