Optimal. Leaf size=60 \[ -\frac{1}{2} b^2 \sinh (a) \text{Chi}\left (\frac{b}{x}\right )-\frac{1}{2} b^2 \cosh (a) \text{Shi}\left (\frac{b}{x}\right )+\frac{1}{2} x^2 \sinh \left (a+\frac{b}{x}\right )+\frac{1}{2} b x \cosh \left (a+\frac{b}{x}\right ) \]
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Rubi [A] time = 0.107096, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5320, 3297, 3303, 3298, 3301} \[ -\frac{1}{2} b^2 \sinh (a) \text{Chi}\left (\frac{b}{x}\right )-\frac{1}{2} b^2 \cosh (a) \text{Shi}\left (\frac{b}{x}\right )+\frac{1}{2} x^2 \sinh \left (a+\frac{b}{x}\right )+\frac{1}{2} b x \cosh \left (a+\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5320
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int x \sinh \left (a+\frac{b}{x}\right ) \, dx &=-\operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} x^2 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{2} b \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} b x \cosh \left (a+\frac{b}{x}\right )+\frac{1}{2} x^2 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} b x \cosh \left (a+\frac{b}{x}\right )+\frac{1}{2} x^2 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{2} \left (b^2 \cosh (a)\right ) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (b^2 \sinh (a)\right ) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} b x \cosh \left (a+\frac{b}{x}\right )-\frac{1}{2} b^2 \text{Chi}\left (\frac{b}{x}\right ) \sinh (a)+\frac{1}{2} x^2 \sinh \left (a+\frac{b}{x}\right )-\frac{1}{2} b^2 \cosh (a) \text{Shi}\left (\frac{b}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0440879, size = 54, normalized size = 0.9 \[ \frac{1}{2} \left (b^2 \sinh (a) \left (-\text{Chi}\left (\frac{b}{x}\right )\right )-b^2 \cosh (a) \text{Shi}\left (\frac{b}{x}\right )+x \left (x \sinh \left (a+\frac{b}{x}\right )+b \cosh \left (a+\frac{b}{x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 93, normalized size = 1.6 \begin{align*}{\frac{bx}{4}{{\rm e}^{-{\frac{ax+b}{x}}}}}-{\frac{{x}^{2}}{4}{{\rm e}^{-{\frac{ax+b}{x}}}}}-{\frac{{b}^{2}{{\rm e}^{-a}}}{4}{\it Ei} \left ( 1,{\frac{b}{x}} \right ) }+{\frac{{x}^{2}}{4}{{\rm e}^{{\frac{ax+b}{x}}}}}+{\frac{bx}{4}{{\rm e}^{{\frac{ax+b}{x}}}}}+{\frac{{b}^{2}{{\rm e}^{a}}}{4}{\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 = 1.15938, size = 59, normalized size = 0.98 \begin{align*} \frac{1}{2} \, x^{2} \sinh \left (a + \frac{b}{x}\right ) + \frac{1}{4} \,{\left (b e^{\left (-a\right )} \Gamma \left (-1, \frac{b}{x}\right ) - b e^{a} \Gamma \left (-1, -\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 = 1.72197, size = 190, normalized size = 3.17 \begin{align*} \frac{1}{2} \, b x \cosh \left (\frac{a x + b}{x}\right ) + \frac{1}{2} \, x^{2} \sinh \left (\frac{a x + b}{x}\right ) - \frac{1}{4} \,{\left (b^{2}{\rm Ei}\left (\frac{b}{x}\right ) - b^{2}{\rm Ei}\left (-\frac{b}{x}\right )\right )} \cosh \left (a\right ) - \frac{1}{4} \,{\left (b^{2}{\rm Ei}\left (\frac{b}{x}\right ) + b^{2}{\rm Ei}\left (-\frac{b}{x}\right )\right )} \sinh \left (a\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 \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 \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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