Optimal. Leaf size=104 \[ -\frac{2}{15} \sqrt{\pi } e^{-a} b^{5/2} \text{Erf}\left (\frac{\sqrt{b}}{x}\right )-\frac{2}{15} \sqrt{\pi } e^a b^{5/2} \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )+\frac{4}{15} b^2 x \sinh \left (a+\frac{b}{x^2}\right )+\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )+\frac{2}{15} b x^3 \cosh \left (a+\frac{b}{x^2}\right ) \]
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Rubi [A] time = 0.0903547, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5346, 5326, 5327, 5299, 2204, 2205} \[ -\frac{2}{15} \sqrt{\pi } e^{-a} b^{5/2} \text{Erf}\left (\frac{\sqrt{b}}{x}\right )-\frac{2}{15} \sqrt{\pi } e^a b^{5/2} \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )+\frac{4}{15} b^2 x \sinh \left (a+\frac{b}{x^2}\right )+\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )+\frac{2}{15} b x^3 \cosh \left (a+\frac{b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 5346
Rule 5326
Rule 5327
Rule 5299
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^4 \sinh \left (a+\frac{b}{x^2}\right ) \, dx &=-\operatorname{Subst}\left (\int \frac{\sinh \left (a+b x^2\right )}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )-\frac{1}{5} (2 b) \operatorname{Subst}\left (\int \frac{\cosh \left (a+b x^2\right )}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2}{15} b x^3 \cosh \left (a+\frac{b}{x^2}\right )+\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )-\frac{1}{15} \left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (a+b x^2\right )}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2}{15} b x^3 \cosh \left (a+\frac{b}{x^2}\right )+\frac{4}{15} b^2 x \sinh \left (a+\frac{b}{x^2}\right )+\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )-\frac{1}{15} \left (8 b^3\right ) \operatorname{Subst}\left (\int \cosh \left (a+b x^2\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2}{15} b x^3 \cosh \left (a+\frac{b}{x^2}\right )+\frac{4}{15} b^2 x \sinh \left (a+\frac{b}{x^2}\right )+\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )-\frac{1}{15} \left (4 b^3\right ) \operatorname{Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{15} \left (4 b^3\right ) \operatorname{Subst}\left (\int e^{a+b x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2}{15} b x^3 \cosh \left (a+\frac{b}{x^2}\right )-\frac{2}{15} b^{5/2} e^{-a} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b}}{x}\right )-\frac{2}{15} b^{5/2} e^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b}}{x}\right )+\frac{4}{15} b^2 x \sinh \left (a+\frac{b}{x^2}\right )+\frac{1}{5} x^5 \sinh \left (a+\frac{b}{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.110222, size = 102, normalized size = 0.98 \[ \frac{1}{15} \left (2 \sqrt{\pi } b^{5/2} (\sinh (a)-\cosh (a)) \text{Erf}\left (\frac{\sqrt{b}}{x}\right )-2 \sqrt{\pi } b^{5/2} (\sinh (a)+\cosh (a)) \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )+4 b^2 x \sinh \left (a+\frac{b}{x^2}\right )+3 x^5 \sinh \left (a+\frac{b}{x^2}\right )+2 b x^3 \cosh \left (a+\frac{b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 138, normalized size = 1.3 \begin{align*} -{\frac{{{\rm e}^{-a}}{x}^{5}}{10}{{\rm e}^{-{\frac{b}{{x}^{2}}}}}}+{\frac{{{\rm e}^{-a}}b{x}^{3}}{15}{{\rm e}^{-{\frac{b}{{x}^{2}}}}}}-{\frac{2\,{{\rm e}^{-a}}\sqrt{\pi }}{15}{b}^{{\frac{5}{2}}}{\it Erf} \left ({\frac{1}{x}\sqrt{b}} \right ) }-{\frac{2\,{{\rm e}^{-a}}{b}^{2}x}{15}{{\rm e}^{-{\frac{b}{{x}^{2}}}}}}+{\frac{{{\rm e}^{a}}{x}^{5}}{10}{{\rm e}^{{\frac{b}{{x}^{2}}}}}}+{\frac{{{\rm e}^{a}}b{x}^{3}}{15}{{\rm e}^{{\frac{b}{{x}^{2}}}}}}+{\frac{2\,{{\rm e}^{a}}{b}^{2}x}{15}{{\rm e}^{{\frac{b}{{x}^{2}}}}}}-{\frac{2\,{{\rm e}^{a}}{b}^{3}\sqrt{\pi }}{15}{\it Erf} \left ({\frac{1}{x}\sqrt{-b}} \right ){\frac{1}{\sqrt{-b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24816, size = 84, normalized size = 0.81 \begin{align*} \frac{1}{5} \, x^{5} \sinh \left (a + \frac{b}{x^{2}}\right ) + \frac{1}{10} \,{\left (x^{3} \left (\frac{b}{x^{2}}\right )^{\frac{3}{2}} e^{\left (-a\right )} \Gamma \left (-\frac{3}{2}, \frac{b}{x^{2}}\right ) + x^{3} \left (-\frac{b}{x^{2}}\right )^{\frac{3}{2}} e^{a} \Gamma \left (-\frac{3}{2}, -\frac{b}{x^{2}}\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80541, size = 795, normalized size = 7.64 \begin{align*} -\frac{3 \, x^{5} - 2 \, b x^{3} + 4 \, b^{2} x -{\left (3 \, x^{5} + 2 \, b x^{3} + 4 \, b^{2} x\right )} \cosh \left (\frac{a x^{2} + b}{x^{2}}\right )^{2} - 4 \, \sqrt{\pi }{\left (b^{2} \cosh \left (a\right ) \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) + b^{2} \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) +{\left (b^{2} \cosh \left (a\right ) + b^{2} \sinh \left (a\right )\right )} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )\right )} \sqrt{-b} \operatorname{erf}\left (\frac{\sqrt{-b}}{x}\right ) + 4 \, \sqrt{\pi }{\left (b^{2} \cosh \left (a\right ) \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) - b^{2} \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) +{\left (b^{2} \cosh \left (a\right ) - b^{2} \sinh \left (a\right )\right )} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )\right )} \sqrt{b} \operatorname{erf}\left (\frac{\sqrt{b}}{x}\right ) - 2 \,{\left (3 \, x^{5} + 2 \, b x^{3} + 4 \, b^{2} x\right )} \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac{a x^{2} + b}{x^{2}}\right ) -{\left (3 \, x^{5} + 2 \, b x^{3} + 4 \, b^{2} x\right )} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )^{2}}{30 \,{\left (\cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \sinh{\left (a + \frac{b}{x^{2}} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \sinh \left (a + \frac{b}{x^{2}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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