Optimal. Leaf size=99 \[ \frac {\sqrt {\frac {\pi }{2}} e^{-2 a} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^{2 a} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )}{32 b^{3/2}}+\frac {x \sinh \left (2 a+2 b x^2\right )}{8 b}-\frac {x^3}{6} \]
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Rubi [A] time = 0.10, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5340, 5325, 5298, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e^{-2 a} \text {Erf}\left (\sqrt {2} \sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^{2 a} \text {Erfi}\left (\sqrt {2} \sqrt {b} x\right )}{32 b^{3/2}}+\frac {x \sinh \left (2 a+2 b x^2\right )}{8 b}-\frac {x^3}{6} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5325
Rule 5340
Rubi steps
\begin {align*} \int x^2 \sinh ^2\left (a+b x^2\right ) \, dx &=\int \left (-\frac {x^2}{2}+\frac {1}{2} x^2 \cosh \left (2 a+2 b x^2\right )\right ) \, dx\\ &=-\frac {x^3}{6}+\frac {1}{2} \int x^2 \cosh \left (2 a+2 b x^2\right ) \, dx\\ &=-\frac {x^3}{6}+\frac {x \sinh \left (2 a+2 b x^2\right )}{8 b}-\frac {\int \sinh \left (2 a+2 b x^2\right ) \, dx}{8 b}\\ &=-\frac {x^3}{6}+\frac {x \sinh \left (2 a+2 b x^2\right )}{8 b}+\frac {\int e^{-2 a-2 b x^2} \, dx}{16 b}-\frac {\int e^{2 a+2 b x^2} \, dx}{16 b}\\ &=-\frac {x^3}{6}+\frac {e^{-2 a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )}{32 b^{3/2}}-\frac {e^{2 a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )}{32 b^{3/2}}+\frac {x \sinh \left (2 a+2 b x^2\right )}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 101, normalized size = 1.02 \[ \frac {3 \sqrt {2 \pi } (\cosh (2 a)-\sinh (2 a)) \text {erf}\left (\sqrt {2} \sqrt {b} x\right )-3 \sqrt {2 \pi } (\sinh (2 a)+\cosh (2 a)) \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )+8 \sqrt {b} x \left (3 \sinh \left (2 \left (a+b x^2\right )\right )-4 b x^2\right )}{192 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 427, normalized size = 4.31 \[ -\frac {32 \, b^{2} x^{3} \cosh \left (b x^{2} + a\right )^{2} - 12 \, b x \cosh \left (b x^{2} + a\right )^{4} - 48 \, b x \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{3} - 12 \, b x \sinh \left (b x^{2} + a\right )^{4} - 3 \, \sqrt {2} \sqrt {\pi } {\left (\cosh \left (b x^{2} + a\right )^{2} \cosh \left (2 \, a\right ) + {\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + \cosh \left (b x^{2} + a\right )^{2} \sinh \left (2 \, a\right ) + 2 \, {\left (\cosh \left (b x^{2} + a\right ) \cosh \left (2 \, a\right ) + \cosh \left (b x^{2} + a\right ) \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {2} \sqrt {-b} x\right ) - 3 \, \sqrt {2} \sqrt {\pi } {\left (\cosh \left (b x^{2} + a\right )^{2} \cosh \left (2 \, a\right ) + {\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} - \cosh \left (b x^{2} + a\right )^{2} \sinh \left (2 \, a\right ) + 2 \, {\left (\cosh \left (b x^{2} + a\right ) \cosh \left (2 \, a\right ) - \cosh \left (b x^{2} + a\right ) \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {2} \sqrt {b} x\right ) + 8 \, {\left (4 \, b^{2} x^{3} - 9 \, b x \cosh \left (b x^{2} + a\right )^{2}\right )} \sinh \left (b x^{2} + a\right )^{2} + 12 \, b x + 16 \, {\left (4 \, b^{2} x^{3} \cosh \left (b x^{2} + a\right ) - 3 \, b x \cosh \left (b x^{2} + a\right )^{3}\right )} \sinh \left (b x^{2} + a\right )}{192 \, {\left (b^{2} \cosh \left (b x^{2} + a\right )^{2} + 2 \, b^{2} \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + b^{2} \sinh \left (b x^{2} + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 97, normalized size = 0.98 \[ -\frac {1}{6} \, x^{3} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} \sqrt {-b} x\right ) e^{\left (2 \, a\right )}}{64 \, \sqrt {-b} b} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} \sqrt {b} x\right ) e^{\left (-2 \, a\right )}}{64 \, b^{\frac {3}{2}}} + \frac {x e^{\left (2 \, b x^{2} + 2 \, a\right )}}{16 \, b} - \frac {x e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 90, normalized size = 0.91 \[ -\frac {x^{3}}{6}-\frac {{\mathrm e}^{-2 a} x \,{\mathrm e}^{-2 b \,x^{2}}}{16 b}+\frac {{\mathrm e}^{-2 a} \sqrt {\pi }\, \sqrt {2}\, \erf \left (x \sqrt {2}\, \sqrt {b}\right )}{64 b^{\frac {3}{2}}}+\frac {{\mathrm e}^{2 a} x \,{\mathrm e}^{2 b \,x^{2}}}{16 b}-\frac {{\mathrm e}^{2 a} \sqrt {\pi }\, \erf \left (\sqrt {-2 b}\, x \right )}{32 b \sqrt {-2 b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 95, normalized size = 0.96 \[ -\frac {1}{6} \, x^{3} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-b} x\right ) e^{\left (2 \, a\right )}}{64 \, \sqrt {-b} b} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {b} x\right ) e^{\left (-2 \, a\right )}}{64 \, b^{\frac {3}{2}}} + \frac {x e^{\left (2 \, b x^{2} + 2 \, a\right )}}{16 \, b} - \frac {x e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{16 \, b} \]
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 \[ \int x^2\,{\mathrm {sinh}\left (b\,x^2+a\right )}^2 \,d x \]
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 \[ \int x^{2} \sinh ^{2}{\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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