Optimal. Leaf size=63 \[ \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sin ^{-1}\left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6335, 30, 259, 275, 216} \[ \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sin ^{-1}\left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}\left (a x^2\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 216
Rule 259
Rule 275
Rule 6335
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^3 \, dx &=\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 ) \operatorname {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} \sin ^{-1}\left (a x^2\right )}{4 a^2}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 92, normalized size = 1.46 \[ \frac {2 a x^2+i \log \left (2 \sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right )-2 i a x^2\right )+a \sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^4+x^2\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.76, size = 102, normalized size = 1.62 \[ \frac {a^{2} x^{4} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 2 \, a x^{2} - 2 \, \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 132, normalized size = 2.10 \[ \frac {2 \, a^{2} x^{2} + 4 \, a \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) + 2 \, \sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} + 2 \, a - \frac {2 \, a^{2} \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) - \sqrt {a^{2} x^{2} + a} {\left (a^{2} x^{2} - 2 \, a\right )} \sqrt {-a^{2} x^{2} + a}}{a}}{4 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 112, normalized size = 1.78 \[ \frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (x^{2} \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, a^{2}+\arctan \left (\frac {x^{2}}{\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}\right )\right )}{4 \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, a^{2}}+\frac {x^{2}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x^{2}}{2 \, a} + \frac {\frac {1}{4} \, \sqrt {-a^{2} x^{4} + 1} x^{2} + \frac {\arcsin \left (a x^{2}\right )}{4 \, a}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.14, size = 306, normalized size = 4.86 \[ \frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{4\,a^2}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{4\,a^2}+\frac {\frac {1{}\mathrm {i}}{64\,a^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,a^2\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{64\,a^2\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^6}}+\frac {x^2}{2\,a}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{64\,a^2\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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