Optimal. Leaf size=27 \[ \frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \]
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Rubi [A]
time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5556, 12, 3384,
3379, 3382} \begin {gather*} \frac {1}{2} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rubi steps
\begin {align*} \int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx &=\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx\\ &=\frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx\\ &=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{2} (\text {Chi}(2 b x) \sinh (2 a)+\cosh (2 a) \text {Shi}(2 b x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.57, size = 26, normalized size = 0.96
method | result | size |
risch | \(\frac {{\mathrm e}^{-2 a} \expIntegral \left (1, 2 b x \right )}{4}-\frac {{\mathrm e}^{2 a} \expIntegral \left (1, -2 b x \right )}{4}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 37, normalized size = 1.37 \begin {gather*} \frac {1}{4} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{4} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\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 \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \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 = -0.04 \begin {gather*} \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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