Optimal. Leaf size=33 \[ -b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right ) \]
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Rubi [A]
time = 0.05, 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 = {5410, 3378,
3384, 3379, 3382} \begin {gather*} -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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5410
Rubi steps
\begin {align*} \int \sinh \left (a+\frac {b}{x}\right ) \, dx &=-\text {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 \text {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)) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,\frac {1}{x}\right )-(b \sinh (a)) \text {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 \begin {gather*} -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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 56, normalized size = 1.70
method | result | size |
risch | \(\frac {b \,{\mathrm e}^{-a} \expIntegral \left (1, \frac {b}{x}\right )}{2}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} x}{2}+\frac {b \,{\mathrm e}^{a} \expIntegral \left (1, -\frac {b}{x}\right )}{2}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x}{2}\) | \(56\) |
meijerg | \(-\frac {\sqrt {\pi }\, \cosh \left (a \right ) b \left (\frac {4}{\sqrt {\pi }}-\frac {4 x \sinh \left (\frac {b}{x}\right )}{\sqrt {\pi }\, b}+\frac {4 \hyperbolicCosineIntegral \left (\frac {b}{x}\right )-4 \ln \left (\frac {b}{x}\right )-4 \gamma }{\sqrt {\pi }}+\frac {4 \gamma -4-4 \ln \left (x \right )+4 \ln \left (i b \right )}{\sqrt {\pi }}\right )}{4}-\frac {i \sqrt {\pi }\, \sinh \left (a \right ) b \left (\frac {4 i x \cosh \left (\frac {b}{x}\right )}{b \sqrt {\pi }}-\frac {4 i \hyperbolicSineIntegral \left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{4}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 36, normalized size = 1.09 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 58, normalized size = 1.76 \begin {gather*} -\frac {1}{2} \, {\left (b {\rm Ei}\left (\frac {b}{x}\right ) + b {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \left (a\right ) - \frac {1}{2} \, {\left (b {\rm Ei}\left (\frac {b}{x}\right ) - b {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \left (a\right ) + x \sinh \left (\frac {a x + b}{x}\right ) \end {gather*}
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 \begin {gather*} \int \sinh {\left (a + \frac {b}{x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (33) = 66\).
time = 0.46, size = 173, normalized size = 5.24 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int \mathrm {sinh}\left (a+\frac {b}{x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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