Optimal. Leaf size=78 \[ -\frac{1}{6} b^3 \cosh (a) \text{Chi}\left (\frac{b}{x}\right )-\frac{1}{6} b^3 \sinh (a) \text{Shi}\left (\frac{b}{x}\right )+\frac{1}{6} b^2 x \sinh \left (a+\frac{b}{x}\right )+\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )+\frac{1}{6} b x^2 \cosh \left (a+\frac{b}{x}\right ) \]
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Rubi [A] time = 0.143453, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5320, 3297, 3303, 3298, 3301} \[ -\frac{1}{6} b^3 \cosh (a) \text{Chi}\left (\frac{b}{x}\right )-\frac{1}{6} b^3 \sinh (a) \text{Shi}\left (\frac{b}{x}\right )+\frac{1}{6} b^2 x \sinh \left (a+\frac{b}{x}\right )+\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )+\frac{1}{6} b x^2 \cosh \left (a+\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5320
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int x^2 \sinh \left (a+\frac{b}{x}\right ) \, dx &=-\operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{3} b \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} b x^2 \cosh \left (a+\frac{b}{x}\right )+\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{6} b^2 \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} b x^2 \cosh \left (a+\frac{b}{x}\right )+\frac{1}{6} b^2 x \sinh \left (a+\frac{b}{x}\right )+\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{6} b^3 \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} b x^2 \cosh \left (a+\frac{b}{x}\right )+\frac{1}{6} b^2 x \sinh \left (a+\frac{b}{x}\right )+\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{6} \left (b^3 \cosh (a)\right ) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (b^3 \sinh (a)\right ) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} b x^2 \cosh \left (a+\frac{b}{x}\right )-\frac{1}{6} b^3 \cosh (a) \text{Chi}\left (\frac{b}{x}\right )+\frac{1}{6} b^2 x \sinh \left (a+\frac{b}{x}\right )+\frac{1}{3} x^3 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{6} b^3 \sinh (a) \text{Shi}\left (\frac{b}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0649958, size = 70, normalized size = 0.9 \[ \frac{1}{6} \left (b^3 (-\cosh (a)) \text{Chi}\left (\frac{b}{x}\right )-b^3 \sinh (a) \text{Shi}\left (\frac{b}{x}\right )+x \left (b^2 \sinh \left (a+\frac{b}{x}\right )+2 x^2 \sinh \left (a+\frac{b}{x}\right )+b x \cosh \left (a+\frac{b}{x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 130, normalized size = 1.7 \begin{align*} -{\frac{{b}^{2}x}{12}{{\rm e}^{-{\frac{ax+b}{x}}}}}+{\frac{b{x}^{2}}{12}{{\rm e}^{-{\frac{ax+b}{x}}}}}-{\frac{{x}^{3}}{6}{{\rm e}^{-{\frac{ax+b}{x}}}}}+{\frac{{b}^{3}{{\rm e}^{-a}}}{12}{\it Ei} \left ( 1,{\frac{b}{x}} \right ) }+{\frac{{x}^{3}}{6}{{\rm e}^{{\frac{ax+b}{x}}}}}+{\frac{b{x}^{2}}{12}{{\rm e}^{{\frac{ax+b}{x}}}}}+{\frac{{b}^{2}x}{12}{{\rm e}^{{\frac{ax+b}{x}}}}}+{\frac{{b}^{3}{{\rm e}^{a}}}{12}{\it Ei} \left ( 1,-{\frac{b}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16563, size = 63, normalized size = 0.81 \begin{align*} \frac{1}{3} \, x^{3} \sinh \left (a + \frac{b}{x}\right ) + \frac{1}{6} \,{\left (b^{2} e^{\left (-a\right )} \Gamma \left (-2, \frac{b}{x}\right ) + b^{2} e^{a} \Gamma \left (-2, -\frac{b}{x}\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7202, size = 212, normalized size = 2.72 \begin{align*} \frac{1}{6} \, b x^{2} \cosh \left (\frac{a x + b}{x}\right ) - \frac{1}{12} \,{\left (b^{3}{\rm Ei}\left (\frac{b}{x}\right ) + b^{3}{\rm Ei}\left (-\frac{b}{x}\right )\right )} \cosh \left (a\right ) - \frac{1}{12} \,{\left (b^{3}{\rm Ei}\left (\frac{b}{x}\right ) - b^{3}{\rm Ei}\left (-\frac{b}{x}\right )\right )} \sinh \left (a\right ) + \frac{1}{6} \,{\left (b^{2} x + 2 \, x^{3}\right )} \sinh \left (\frac{a x + b}{x}\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^{2} \sinh{\left (a + \frac{b}{x} \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^{2} \sinh \left (a + \frac{b}{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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