Optimal. Leaf size=82 \[ -\frac {\text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {\text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.12, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5457, 4182, 2279, 2391, 4185} \[ -\frac {\text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {\text {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 4185
Rule 5457
Rubi steps
\begin {align*} \int x \coth ^2(a+b x) \text {csch}(a+b x) \, dx &=\int x \text {csch}(a+b x) \, dx+\int x \text {csch}^3(a+b x) \, dx\\ &=-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {x \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int x \text {csch}(a+b x) \, dx-\frac {\int \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {\int \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {x \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {\int \log \left (1-e^{a+b x}\right ) \, dx}{2 b}-\frac {\int \log \left (1+e^{a+b x}\right ) \, dx}{2 b}\\ &=-\frac {x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {x \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {\text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {\text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}-\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=-\frac {x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {x \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {\text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {\text {Li}_2\left (e^{a+b x}\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 2.03, size = 144, normalized size = 1.76 \[ -\frac {-4 \text {Li}_2\left (-e^{-a-b x}\right )+4 \text {Li}_2\left (e^{-a-b x}\right )-4 (a+b x) \left (\log \left (1-e^{-a-b x}\right )-\log \left (e^{-a-b x}+1\right )\right )-2 \tanh \left (\frac {1}{2} (a+b x)\right )+2 \coth \left (\frac {1}{2} (a+b x)\right )+b x \text {csch}^2\left (\frac {1}{2} (a+b x)\right )+b x \text {sech}^2\left (\frac {1}{2} (a+b x)\right )+4 a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 842, normalized size = 10.27 \[ \text {result too large to display} \]
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 \[ \int x \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 156, normalized size = 1.90 \[ -\frac {{\mathrm e}^{b x +a} \left (b x \,{\mathrm e}^{2 b x +2 a}+b x +{\mathrm e}^{2 b x +2 a}-1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 124, normalized size = 1.51 \[ -\frac {{\left (b x e^{\left (3 \, a\right )} + e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{2 \, b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{2 \, b^{2}} \]
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 \[ \int \frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int x \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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