Optimal. Leaf size=33 \[ -b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5302, 3297, 3303, 3298, 3301} \[ -b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5302
Rubi steps
\begin {align*} \int \sinh \left (a+\frac {b}{x}\right ) \, dx &=-\operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=x \sinh \left (a+\frac {b}{x}\right )-b \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=x \sinh \left (a+\frac {b}{x}\right )-(b \cosh (a)) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,\frac {1}{x}\right )-(b \sinh (a)) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=-b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ -b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 58, normalized size = 1.76 \[ -\frac {1}{2} \, {\left (b {\rm Ei}\left (\frac {b}{x}\right ) + b {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \relax (a) - \frac {1}{2} \, {\left (b {\rm Ei}\left (\frac {b}{x}\right ) - b {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \relax (a) + x \sinh \left (\frac {a x + b}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 173, normalized size = 5.24 \[ -\frac {a b^{2} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} - \frac {{\left (a x + b\right )} b^{2} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x} - b^{2} e^{\left (-\frac {a x + b}{x}\right )}}{2 \, {\left (a - \frac {a x + b}{x}\right )} b} - \frac {a b^{2} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a} - \frac {{\left (a x + b\right )} b^{2} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x} + b^{2} e^{\left (\frac {a x + b}{x}\right )}}{2 \, {\left (a - \frac {a x + b}{x}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 1.70 \[ \frac {b \,{\mathrm e}^{-a} \Ei \left (1, \frac {b}{x}\right )}{2}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} x}{2}+\frac {b \,{\mathrm e}^{a} \Ei \left (1, -\frac {b}{x}\right )}{2}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 36, normalized size = 1.09 \[ -\frac {1}{2} \, {\left ({\rm Ei}\left (-\frac {b}{x}\right ) e^{\left (-a\right )} + {\rm Ei}\left (\frac {b}{x}\right ) e^{a}\right )} b + x \sinh \left (a + \frac {b}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {sinh}\left (a+\frac {b}{x}\right ) \,d x \]
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 \[ \int \sinh {\left (a + \frac {b}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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