Optimal. Leaf size=88 \[ -\frac {3 \text {Chi}(b x) \sinh (a)}{4 b}-\frac {\text {Chi}(3 b x) \sinh (3 a)}{12 b}+\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {3 \cosh (a) \text {Shi}(b x)}{4 b}-\frac {\cosh (3 a) \text {Shi}(3 b x)}{12 b} \]
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Rubi [A]
time = 0.32, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2713, 2634,
12, 6874, 3384, 3379, 3382, 3393} \begin {gather*} -\frac {3 \sinh (a) \text {Chi}(b x)}{4 b}-\frac {\sinh (3 a) \text {Chi}(3 b x)}{12 b}-\frac {3 \cosh (a) \text {Shi}(b x)}{4 b}-\frac {\cosh (3 a) \text {Shi}(3 b x)}{12 b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}+\frac {\log (x) \sinh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2634
Rule 2713
Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 6874
Rubi steps
\begin {align*} \int \cosh ^3(a+b x) \log (x) \, dx &=\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\int \frac {\sinh (a+b x) \left (3+\sinh ^2(a+b x)\right )}{3 b x} \, dx\\ &=\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {\int \frac {\sinh (a+b x) \left (3+\sinh ^2(a+b x)\right )}{x} \, dx}{3 b}\\ &=\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {\int \left (\frac {3 \sinh (a+b x)}{x}+\frac {\sinh ^3(a+b x)}{x}\right ) \, dx}{3 b}\\ &=\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {\int \frac {\sinh ^3(a+b x)}{x} \, dx}{3 b}-\frac {\int \frac {\sinh (a+b x)}{x} \, dx}{b}\\ &=\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {i \int \left (\frac {3 i \sinh (a+b x)}{4 x}-\frac {i \sinh (3 a+3 b x)}{4 x}\right ) \, dx}{3 b}-\frac {\cosh (a) \int \frac {\sinh (b x)}{x} \, dx}{b}-\frac {\sinh (a) \int \frac {\cosh (b x)}{x} \, dx}{b}\\ &=-\frac {\text {Chi}(b x) \sinh (a)}{b}+\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {\cosh (a) \text {Shi}(b x)}{b}-\frac {\int \frac {\sinh (3 a+3 b x)}{x} \, dx}{12 b}+\frac {\int \frac {\sinh (a+b x)}{x} \, dx}{4 b}\\ &=-\frac {\text {Chi}(b x) \sinh (a)}{b}+\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {\cosh (a) \text {Shi}(b x)}{b}+\frac {\cosh (a) \int \frac {\sinh (b x)}{x} \, dx}{4 b}-\frac {\cosh (3 a) \int \frac {\sinh (3 b x)}{x} \, dx}{12 b}+\frac {\sinh (a) \int \frac {\cosh (b x)}{x} \, dx}{4 b}-\frac {\sinh (3 a) \int \frac {\cosh (3 b x)}{x} \, dx}{12 b}\\ &=-\frac {3 \text {Chi}(b x) \sinh (a)}{4 b}-\frac {\text {Chi}(3 b x) \sinh (3 a)}{12 b}+\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {3 \cosh (a) \text {Shi}(b x)}{4 b}-\frac {\cosh (3 a) \text {Shi}(3 b x)}{12 b}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 66, normalized size = 0.75 \begin {gather*} -\frac {9 \text {Chi}(b x) \sinh (a)+\text {Chi}(3 b x) \sinh (3 a)-9 \log (x) \sinh (a+b x)-\log (x) \sinh (3 (a+b x))+9 \cosh (a) \text {Shi}(b x)+\cosh (3 a) \text {Shi}(3 b x)}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 116, normalized size = 1.32
method | result | size |
risch | \(\left (\frac {{\mathrm e}^{3 b x +3 a}}{24 b}+\frac {3 \,{\mathrm e}^{b x +a}}{8 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{8 b}-\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}\right ) \ln \left (x \right )+\frac {{\mathrm e}^{3 a} \expIntegral \left (1, -3 b x \right )}{24 b}-\frac {{\mathrm e}^{-3 a} \expIntegral \left (1, 3 b x \right )}{24 b}-\frac {3 \,{\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{8 b}+\frac {3 \,{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{8 b}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 111, normalized size = 1.26 \begin {gather*} \frac {1}{24} \, {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \left (x\right ) - \frac {{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )}}{24 \, b} + \frac {3 \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )}}{8 \, b} + \frac {{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )}}{24 \, b} - \frac {3 \, {\rm Ei}\left (b x\right ) e^{a}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 587 vs.
\(2 (78) = 156\).
time = 0.38, size = 587, normalized size = 6.67 \begin {gather*} \frac {6 \, \cosh \left (b x + a\right ) \log \left (x\right ) \sinh \left (b x + a\right )^{5} + \log \left (x\right ) \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} + 3\right )} \log \left (x\right ) \sinh \left (b x + a\right )^{4} - {\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (b x + a\right )^{3} \sinh \left (3 \, a\right ) - 9 \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right )^{3} \sinh \left (a\right ) - {\left ({\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right )\right )} \cosh \left (b x + a\right )^{3} - {\left ({\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) - 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right )\right )} \log \left (x\right ) + {\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left ({\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (b x + a\right ) \sinh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \sinh \left (a\right ) + {\left ({\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right )\right )} \cosh \left (b x + a\right ) - {\left (5 \, \cosh \left (b x + a\right )^{4} + 18 \, \cosh \left (b x + a\right )^{2} - 3\right )} \log \left (x\right )\right )} \sinh \left (b x + a\right )^{2} + {\left (\cosh \left (b x + a\right )^{6} + 9 \, \cosh \left (b x + a\right )^{4} - 9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \log \left (x\right ) - 3 \, {\left ({\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (b x + a\right )^{2} \sinh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right )^{2} \sinh \left (a\right ) + {\left ({\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 9 \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right )\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (\cosh \left (b x + a\right )^{5} + 6 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \log \left (x\right )\right )} \sinh \left (b x + a\right )}{24 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\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 \log {\left (x \right )} \cosh ^{3}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.43, size = 104, normalized size = 1.18 \begin {gather*} \frac {1}{24} \, {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \left (x\right ) - \frac {{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - 9 \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + 9 \, {\rm Ei}\left (b x\right ) e^{a}}{24 \, 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 = -0.01 \begin {gather*} \int {\mathrm {cosh}\left (a+b\,x\right )}^3\,\ln \left (x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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