Optimal. Leaf size=88 \[ \frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \sinh ^2(a+b x)}{2 b}+\frac {x}{4 b}-\frac {x^2}{2} \]
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Rubi [A] time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5450, 5372, 2635, 8, 3716, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \sinh ^2(a+b x)}{2 b}+\frac {x}{4 b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 3716
Rule 5372
Rule 5450
Rubi steps
\begin {align*} \int x \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int x \coth (a+b x) \, dx+\int x \cosh (a+b x) \sinh (a+b x) \, dx\\ &=-\frac {x^2}{2}+\frac {x \sinh ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx-\frac {\int \sinh ^2(a+b x) \, dx}{2 b}\\ &=-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {x \sinh ^2(a+b x)}{2 b}+\frac {\int 1 \, dx}{4 b}-\frac {\int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x}{4 b}-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {x \sinh ^2(a+b x)}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=\frac {x}{4 b}-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac {\cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {x \sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 82, normalized size = 0.93 \[ -\frac {4 \left (\text {Li}_2\left (e^{-2 (a+b x)}\right )-(a+b x)^2\right )-8 (a+b x) \log \left (1-e^{-2 (a+b x)}\right )+\sinh (2 (a+b x))-2 b x \cosh (2 (a+b x))+8 a \log (\sinh (a+b x))}{8 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.11, size = 488, normalized size = 5.55 \[ \frac {{\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (2 \, b x - 1\right )} \sinh \left (b x + a\right )^{4} - 8 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (4 \, b^{2} x^{2} - 3 \, {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} - 8 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 16 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 16 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 16 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 16 \, {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 16 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 4 \, {\left ({\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{3} - 4 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{16 \, {\left (b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.61, size = 162, normalized size = 1.84 \[ -\frac {x^{2}}{2}+\frac {\left (2 b x -1\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{2}}+\frac {\left (2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{2}}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 113, normalized size = 1.28 \[ -x^{2} + \frac {{\left (8 \, b^{2} x^{2} e^{\left (2 \, a\right )} + {\left (2 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} + {\left (2 \, b x + 1\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{16 \, b^{2}} + \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
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 \[ \int \frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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