Optimal. Leaf size=47 \[ \frac {1}{4} \text {Chi}(b x) \sinh (a)+\frac {1}{4} \text {Chi}(3 b x) \sinh (3 a)+\frac {1}{4} \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (3 a) \text {Shi}(3 b x) \]
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Rubi [A]
time = 0.11, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5556, 3384,
3379, 3382} \begin {gather*} \frac {1}{4} \sinh (a) \text {Chi}(b x)+\frac {1}{4} \sinh (3 a) \text {Chi}(3 b x)+\frac {1}{4} \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (3 a) \text {Shi}(3 b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x} \, dx &=\int \left (\frac {\sinh (a+b x)}{4 x}+\frac {\sinh (3 a+3 b x)}{4 x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh (a+b x)}{x} \, dx+\frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{x} \, dx\\ &=\frac {1}{4} \cosh (a) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{4} \cosh (3 a) \int \frac {\sinh (3 b x)}{x} \, dx+\frac {1}{4} \sinh (a) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{4} \sinh (3 a) \int \frac {\cosh (3 b x)}{x} \, dx\\ &=\frac {1}{4} \text {Chi}(b x) \sinh (a)+\frac {1}{4} \text {Chi}(3 b x) \sinh (3 a)+\frac {1}{4} \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (3 a) \text {Shi}(3 b x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 39, normalized size = 0.83 \begin {gather*} \frac {1}{4} (\text {Chi}(b x) \sinh (a)+\text {Chi}(3 b x) \sinh (3 a)+\cosh (a) \text {Shi}(b x)+\cosh (3 a) \text {Shi}(3 b x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.12, size = 47, normalized size = 1.00
method | result | size |
risch | \(\frac {{\mathrm e}^{-3 a} \expIntegral \left (1, 3 b x \right )}{8}+\frac {{\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{8}-\frac {{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{8}-\frac {{\mathrm e}^{3 a} \expIntegral \left (1, -3 b x \right )}{8}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 42, normalized size = 0.89 \begin {gather*} \frac {1}{8} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - \frac {1}{8} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (b x\right ) e^{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 67, normalized size = 1.43 \begin {gather*} \frac {1}{8} \, {\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \sinh \left (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 ^{2}{\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 = 0.39, size = 42, normalized size = 0.89 \begin {gather*} \frac {1}{8} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - \frac {1}{8} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (b x\right ) e^{a} \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 = -0.02 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\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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