Optimal. Leaf size=86 \[ \frac {1}{3} \sqrt {\pi } e^{-a} b^{3/2} \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} \sqrt {\pi } e^a b^{3/2} \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5346, 5326, 5327, 5298, 2204, 2205} \[ \frac {1}{3} \sqrt {\pi } e^{-a} b^{3/2} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} \sqrt {\pi } e^a b^{3/2} \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )+\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5326
Rule 5327
Rule 5346
Rubi steps
\begin {align*} \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx &=-\operatorname {Subst}\left (\int \frac {\sinh \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{3} \left (4 b^2\right ) \operatorname {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} \left (2 b^2\right ) \operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{3} \left (2 b^2\right ) \operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} b^{3/2} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} b^{3/2} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 0.98 \[ \frac {1}{3} \left (\sqrt {\pi } b^{3/2} (\cosh (a)-\sinh (a)) \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\sqrt {\pi } b^{3/2} (\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+2 b x \cosh \left (a+\frac {b}{x^2}\right )+x^3 \sinh \left (a+\frac {b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 267, normalized size = 3.10 \[ -\frac {x^{3} - {\left (x^{3} + 2 \, b x\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, \sqrt {\pi } {\left (b \cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (b \cosh \relax (a) + b \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - 2 \, \sqrt {\pi } {\left (b \cosh \relax (a) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \relax (a) + {\left (b \cosh \relax (a) - b \sinh \relax (a)\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) - 2 \, {\left (x^{3} + 2 \, b x\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - {\left (x^{3} + 2 \, b x\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, b x}{6 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]
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 \[ \int x^{2} \sinh \left (a + \frac {b}{x^{2}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 103, normalized size = 1.20 \[ -\frac {{\mathrm e}^{-a} x^{3} {\mathrm e}^{-\frac {b}{x^{2}}}}{6}+\frac {{\mathrm e}^{-a} \erf \left (\frac {\sqrt {b}}{x}\right ) b^{\frac {3}{2}} \sqrt {\pi }}{3}+\frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}} b x}{3}+\frac {{\mathrm e}^{a} x^{3} {\mathrm e}^{\frac {b}{x^{2}}}}{6}+\frac {{\mathrm e}^{a} b \,{\mathrm e}^{\frac {b}{x^{2}}} x}{3}-\frac {{\mathrm e}^{a} b^{2} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{3 \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 58, normalized size = 0.67 \[ \frac {1}{3} \, x^{3} \sinh \left (a + \frac {b}{x^{2}}\right ) + \frac {1}{6} \, {\left (x \sqrt {\frac {b}{x^{2}}} e^{\left (-a\right )} \Gamma \left (-\frac {1}{2}, \frac {b}{x^{2}}\right ) + x \sqrt {-\frac {b}{x^{2}}} e^{a} \Gamma \left (-\frac {1}{2}, -\frac {b}{x^{2}}\right )\right )} b \]
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 \[ \int x^2\,\mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]
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 \[ \int x^{2} \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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