Integrand size = 12, antiderivative size = 63 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {x^2}{4 a}+\frac {1}{4} e^{\text {sech}^{-1}\left (a x^2\right )} x^4+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \arcsin \left (a x^2\right )}{4 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6470, 30, 265, 281, 222} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \arcsin \left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}\left (a x^2\right )} \]
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Rule 30
Rule 222
Rule 265
Rule 281
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} e^{\text {sech}^{-1}\left (a x^2\right )} x^4+\frac {\int x \, dx}{2 a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{2 a} \\ & = \frac {x^2}{4 a}+\frac {1}{4} e^{\text {sech}^{-1}\left (a x^2\right )} x^4+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x}{\sqrt {1-a^2 x^4}} \, dx}{2 a} \\ & = \frac {x^2}{4 a}+\frac {1}{4} e^{\text {sech}^{-1}\left (a x^2\right )} x^4+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )}{4 a} \\ & = \frac {x^2}{4 a}+\frac {1}{4} e^{\text {sech}^{-1}\left (a x^2\right )} x^4+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \arcsin \left (a x^2\right )}{4 a^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.46 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {2 a x^2+a \sqrt {\frac {1-a x^2}{1+a x^2}} \left (x^2+a x^4\right )+i \log \left (-2 i a x^2+2 \sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )\right )}{4 a^2} \]
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Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.78
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (x^{2} \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}\, a^{2}+\arctan \left (\frac {x^{2}}{\sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}}\right )\right )}{4 \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}\, a^{2}}+\frac {x^{2}}{2 a}\) | \(112\) |
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none
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {a^{2} x^{4} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 2 \, a x^{2} - 2 \, \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right )}{4 \, a^{2}} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {\int x\, dx + \int a x^{3} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\int { x^{3} {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (56) = 112\).
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.10 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {2 \, a^{2} x^{2} + 4 \, a \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) + 2 \, \sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} + 2 \, a - \frac {2 \, a^{2} \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) - \sqrt {a^{2} x^{2} + a} {\left (a^{2} x^{2} - 2 \, a\right )} \sqrt {-a^{2} x^{2} + a}}{a}}{4 \, a^{3}} \]
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Time = 10.88 (sec) , antiderivative size = 306, normalized size of antiderivative = 4.86 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx=\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{4\,a^2}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{4\,a^2}+\frac {\frac {1{}\mathrm {i}}{64\,a^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{64\,a^2\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^6}}+\frac {x^2}{2\,a}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{64\,a^2\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2} \]
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