Optimal. Leaf size=69 \[ \frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{8 b^{3/2}}+\frac {x \sinh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 69, 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 = {5325, 5298, 2204, 2205} \[ \frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x\right )}{8 b^{3/2}}+\frac {x \sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5325
Rubi steps
\begin {align*} \int x^2 \cosh \left (a+b x^2\right ) \, dx &=\frac {x \sinh \left (a+b x^2\right )}{2 b}-\frac {\int \sinh \left (a+b x^2\right ) \, dx}{2 b}\\ &=\frac {x \sinh \left (a+b x^2\right )}{2 b}+\frac {\int e^{-a-b x^2} \, dx}{4 b}-\frac {\int e^{a+b x^2} \, dx}{4 b}\\ &=\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{8 b^{3/2}}-\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{8 b^{3/2}}+\frac {x \sinh \left (a+b x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 67, normalized size = 0.97 \[ \frac {\sqrt {\pi } (\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x\right )-\sqrt {\pi } (\sinh (a)+\cosh (a)) \text {erfi}\left (\sqrt {b} x\right )+4 \sqrt {b} x \sinh \left (a+b x^2\right )}{8 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 189, normalized size = 2.74 \[ \frac {2 \, b x \cosh \left (b x^{2} + a\right )^{2} + 4 \, b x \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + 2 \, b x \sinh \left (b x^{2} + a\right )^{2} + \sqrt {\pi } {\left (\cosh \left (b x^{2} + a\right ) \cosh \relax (a) + {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \sinh \left (b x^{2} + a\right ) + \cosh \left (b x^{2} + a\right ) \sinh \relax (a)\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {-b} x\right ) + \sqrt {\pi } {\left (\cosh \left (b x^{2} + a\right ) \cosh \relax (a) + {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \sinh \left (b x^{2} + a\right ) - \cosh \left (b x^{2} + a\right ) \sinh \relax (a)\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} x\right ) - 2 \, b x}{8 \, {\left (b^{2} \cosh \left (b x^{2} + a\right ) + b^{2} \sinh \left (b x^{2} + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 75, normalized size = 1.09 \[ \frac {x e^{\left (b x^{2} + a\right )}}{4 \, b} - \frac {x e^{\left (-b x^{2} - a\right )}}{4 \, b} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{8 \, b^{\frac {3}{2}}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{8 \, \sqrt {-b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 74, normalized size = 1.07 \[ -\frac {{\mathrm e}^{-a} x \,{\mathrm e}^{-b \,x^{2}}}{4 b}+\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \erf \left (x \sqrt {b}\right )}{8 b^{\frac {3}{2}}}+\frac {{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}} x}{4 b}-\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{8 b \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 110, normalized size = 1.59 \[ \frac {1}{3} \, x^{3} \cosh \left (b x^{2} + a\right ) - \frac {1}{24} \, b {\left (\frac {2 \, {\left (2 \, b x^{3} e^{a} - 3 \, x e^{a}\right )} e^{\left (b x^{2}\right )}}{b^{2}} + \frac {2 \, {\left (2 \, b x^{3} + 3 \, x\right )} e^{\left (-b x^{2} - a\right )}}{b^{2}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{b^{\frac {5}{2}}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b} b^{2}}\right )} \]
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 \[ \int x^2\,\mathrm {cosh}\left (b\,x^2+a\right ) \,d x \]
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 \[ \int x^{2} \cosh {\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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