Optimal. Leaf size=45 \[ \frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^2}{2} \]
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Rubi [A] time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3716, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rubi steps
\begin {align*} \int x \coth (a+b x) \, dx &=-\frac {x^2}{2}-2 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx\\ &=-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=-\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}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 1.04 \[ \frac {\text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}+\frac {x \log \left (1-e^{2 a+2 b x}\right )}{b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 112, normalized size = 2.49 \[ -\frac {b^{2} x^{2} - 2 \, b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b^{2}} \]
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 \[ \int x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 122, normalized size = 2.71 \[ -\frac {x^{2}}{2}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 58, normalized size = 1.29 \[ -\frac {1}{2} \, x^{2} + \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x \cosh {\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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