Optimal. Leaf size=67 \[ \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{3 a^{3/2}}+\frac {1}{3} x^3 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {2 x}{3 a} \]
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Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6335, 8, 248, 221} \[ \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{3 a^{3/2}}+\frac {1}{3} x^3 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {2 x}{3 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 221
Rule 248
Rule 6335
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx &=\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \int 1 \, dx}{3 a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{3 a}\\ &=\frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\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}{3 a}\\ &=\frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{3 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 116, normalized size = 1.73 \[ -\frac {2 i \sqrt {\frac {1-a x^2}{a x^2+1}} \sqrt {1-a^2 x^4} F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )}{3 (-a)^{3/2} \left (a x^2-1\right )}+\frac {\sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^3+x\right )}{3 a}+\frac {x}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 102, normalized size = 1.52 \[ \frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (x^{5} a^{\frac {5}{2}}-2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}-x \sqrt {a}\right )}{3 \left (a^{2} x^{4}-1\right ) \sqrt {a}}+\frac {x}{a} \]
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 \[ \frac {x}{a} + \frac {\int \sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}\,{d x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int 1\, dx + \int a x^{2} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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