Optimal. Leaf size=101 \[ -\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )+\frac {\left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{9 a^{3/2} \sqrt {1+a^2 x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5875, 12, 327,
226} \begin {gather*} -\frac {2 x \sqrt {a^2 x^4+1}}{9 a}+\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{9 a^{3/2} \sqrt {a^2 x^4+1}}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 226
Rule 327
Rule 5875
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )-\frac {1}{3} \int \frac {2 a x^4}{\sqrt {1+a^2 x^4}} \, dx\\ &=\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )-\frac {1}{3} (2 a) \int \frac {x^4}{\sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx}{9 a}\\ &=-\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )+\frac {\left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{9 a^{3/2} \sqrt {1+a^2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 75, normalized size = 0.74 \begin {gather*} \frac {1}{9} \left (-\frac {2 \left (x+a^2 x^5\right )}{a \sqrt {1+a^2 x^4}}+3 x^3 \sinh ^{-1}\left (a x^2\right )-\frac {2 \sqrt {i a} F\left (\left .i \sinh ^{-1}\left (\sqrt {i a} x\right )\right |-1\right )}{a^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.22, size = 89, normalized size = 0.88
method | result | size |
default | \(\frac {x^{3} \arcsinh \left (a \,x^{2}\right )}{3}-\frac {2 a \left (\frac {x \sqrt {a^{2} x^{4}+1}}{3 a^{2}}-\frac {\sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \EllipticF \left (x \sqrt {i a}, i\right )}{3 a^{2} \sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3}\) | \(89\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 67, normalized size = 0.66 \begin {gather*} \frac {3 \, a x^{3} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) + 2 \, a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}}}{x}, -1\right ) - 2 \, \sqrt {a^{2} x^{4} + 1} x}{9 \, a} \end {gather*}
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 \begin {gather*} \int x^{2} \operatorname {asinh}{\left (a x^{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,\mathrm {asinh}\left (a\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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