Integrand size = 11, antiderivative size = 88 \[ \int \cosh ^3(a+b x) \log (x) \, dx=-\frac {3 \text {Chi}(b x) \sinh (a)}{4 b}-\frac {\text {Chi}(3 b x) \sinh (3 a)}{12 b}+\frac {\log (x) \sinh (a+b x)}{b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}-\frac {3 \cosh (a) \text {Shi}(b x)}{4 b}-\frac {\cosh (3 a) \text {Shi}(3 b x)}{12 b} \]
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Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2713, 2634, 12, 6874, 3384, 3379, 3382, 3393} \[ \int \cosh ^3(a+b x) \log (x) \, dx=-\frac {3 \sinh (a) \text {Chi}(b x)}{4 b}-\frac {\sinh (3 a) \text {Chi}(3 b x)}{12 b}-\frac {3 \cosh (a) \text {Shi}(b x)}{4 b}-\frac {\cosh (3 a) \text {Shi}(3 b x)}{12 b}+\frac {\log (x) \sinh ^3(a+b x)}{3 b}+\frac {\log (x) \sinh (a+b x)}{b} \]
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Rule 12
Rule 2634
Rule 2713
Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.75 \[ \int \cosh ^3(a+b x) \log (x) \, dx=-\frac {9 \text {Chi}(b x) \sinh (a)+\text {Chi}(3 b x) \sinh (3 a)-9 \log (x) \sinh (a+b x)-\log (x) \sinh (3 (a+b x))+9 \cosh (a) \text {Shi}(b x)+\cosh (3 a) \text {Shi}(3 b x)}{12 b} \]
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Time = 1.88 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {\ln \left (x \right ) {\mathrm e}^{3 b x +3 a}}{24 b}-\frac {3 \ln \left (x \right ) {\mathrm e}^{-b x -a}}{8 b}-\frac {\ln \left (x \right ) {\mathrm e}^{-3 b x -3 a}}{24 b}+\frac {{\mathrm e}^{3 a} \operatorname {Ei}_{1}\left (-3 b x \right )}{24 b}-\frac {{\mathrm e}^{-3 a} \operatorname {Ei}_{1}\left (3 b x \right )}{24 b}-\frac {3 \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{8 b}+\frac {3 \,{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{8 b}+\frac {3 \,{\mathrm e}^{b x +a} \ln \left (x \right )}{8 b}\) | \(120\) |
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Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (78) = 156\).
Time = 0.34 (sec) , antiderivative size = 587, normalized size of antiderivative = 6.67 \[ \int \cosh ^3(a+b x) \log (x) \, dx=\text {Too large to display} \]
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\[ \int \cosh ^3(a+b x) \log (x) \, dx=\int \log {\left (x \right )} \cosh ^{3}{\left (a + b x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.26 \[ \int \cosh ^3(a+b x) \log (x) \, dx=\frac {1}{24} \, {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \left (x\right ) - \frac {{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )}}{24 \, b} + \frac {3 \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )}}{8 \, b} + \frac {{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )}}{24 \, b} - \frac {3 \, {\rm Ei}\left (b x\right ) e^{a}}{8 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int \cosh ^3(a+b x) \log (x) \, dx=\frac {1}{24} \, {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \left (x\right ) - \frac {{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - 9 \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + 9 \, {\rm Ei}\left (b x\right ) e^{a}}{24 \, b} \]
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Timed out. \[ \int \cosh ^3(a+b x) \log (x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^3\,\ln \left (x\right ) \,d x \]
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