Optimal. Leaf size=66 \[ -\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \cosh (a+b x)}{b}-\frac {\text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {\text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {\sinh (a+b x)}{b^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5558, 3377,
2717, 4267, 2317, 2438} \begin {gather*} -\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 {\sinh (a+b x)}{b^2}+\frac {x \cosh (a+b x)}{b}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 2717
Rule 3377
Rule 4267
Rule 5558
Rubi steps
\begin {align*} \int x \cosh (a+b x) \coth (a+b x) \, dx &=\int x \text {csch}(a+b x) \, dx+\int x \sinh (a+b x) \, dx\\ &=-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \cosh (a+b x)}{b}-\frac {\int \cosh (a+b x) \, dx}{b}-\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 {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \cosh (a+b x)}{b}-\frac {\sinh (a+b x)}{b^2}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \cosh (a+b x)}{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 {\sinh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 131, normalized size = 1.98 \begin {gather*} -\frac {-b x \cosh (a+b x)-a \log \left (1-e^{-a-b x}\right )-b x \log \left (1-e^{-a-b x}\right )+a \log \left (1+e^{-a-b x}\right )+b x \log \left (1+e^{-a-b x}\right )+a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )-\text {PolyLog}\left (2,-e^{-a-b x}\right )+\text {PolyLog}\left (2,e^{-a-b x}\right )+\sinh (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(138\) vs.
\(2(63)=126\).
time = 2.29, size = 139, normalized size = 2.11
method | result | size |
risch | \(\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}+\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}-\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 94, normalized size = 1.42 \begin {gather*} \frac {{\left ({\left (b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b x + 1\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\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 )}{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}} \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. 255 vs.
\(2 (61) = 122\).
time = 0.41, size = 255, normalized size = 3.86 \begin {gather*} \frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{2} + b x + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (a \cosh \left (b x + a\right ) + a \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\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 x \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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