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 = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5320
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*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 0.90 \[ \frac {1}{2} \left (b^2 \sinh (a) \left (-\text {Chi}\left (\frac {b}{x}\right )\right )-b^2 \cosh (a) \text {Shi}\left (\frac {b}{x}\right )+x \left (x \sinh \left (a+\frac {b}{x}\right )+b \cosh \left (a+\frac {b}{x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 83, normalized size = 1.38 \[ \frac {1}{2} \, b x \cosh \left (\frac {a x + b}{x}\right ) + \frac {1}{2} \, x^{2} \sinh \left (\frac {a x + b}{x}\right ) - \frac {1}{4} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x}\right ) - b^{2} {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \relax (a) - \frac {1}{4} \, {\left (b^{2} {\rm Ei}\left (\frac {b}{x}\right ) + b^{2} {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 313, normalized size = 5.22 \[ \frac {a^{2} b^{3} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} - a^{2} b^{3} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a} - \frac {2 \, {\left (a x + b\right )} a b^{3} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x} + \frac {2 \, {\left (a x + b\right )} a b^{3} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x} + \frac {{\left (a x + b\right )}^{2} b^{3} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x^{2}} - \frac {{\left (a x + b\right )}^{2} b^{3} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x^{2}} - a b^{3} e^{\left (\frac {a x + b}{x}\right )} - a b^{3} e^{\left (-\frac {a x + b}{x}\right )} + b^{3} e^{\left (\frac {a x + b}{x}\right )} + \frac {{\left (a x + b\right )} b^{3} e^{\left (\frac {a x + b}{x}\right )}}{x} - b^{3} e^{\left (-\frac {a x + b}{x}\right )} + \frac {{\left (a x + b\right )} b^{3} e^{\left (-\frac {a x + b}{x}\right )}}{x}}{4 \, {\left (a^{2} - \frac {2 \, {\left (a x + b\right )} a}{x} + \frac {{\left (a x + b\right )}^{2}}{x^{2}}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 93, normalized size = 1.55 \[ \frac {b \,{\mathrm e}^{-\frac {a x +b}{x}} x}{4}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} x^{2}}{4}-\frac {b^{2} {\mathrm e}^{-a} \Ei \left (1, \frac {b}{x}\right )}{4}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x^{2}}{4}+\frac {b \,{\mathrm e}^{\frac {a x +b}{x}} x}{4}+\frac {b^{2} {\mathrm e}^{a} \Ei \left (1, -\frac {b}{x}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 44, normalized size = 0.73 \[ \frac {1}{2} \, x^{2} \sinh \left (a + \frac {b}{x}\right ) + \frac {1}{4} \, {\left (b e^{\left (-a\right )} \Gamma \left (-1, \frac {b}{x}\right ) - b e^{a} \Gamma \left (-1, -\frac {b}{x}\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {sinh}\left (a+\frac {b}{x}\right ) \,d x \]
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 \[ \int x \sinh {\left (a + \frac {b}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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