Optimal. Leaf size=58 \[ -\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5569, 4267,
2317, 2438} \begin {gather*} -\frac {\text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rubi steps
\begin {align*} \int x \text {csch}(a+b x) \text {sech}(a+b x) \, dx &=2 \int x \text {csch}(2 a+2 b x) \, dx\\ &=-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\int \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b}+\frac {\int \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^2}\\ &=-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 110, normalized size = 1.90 \begin {gather*} \frac {2 a \log \left (1-e^{-2 (a+b x)}\right )+2 b x \log \left (1-e^{-2 (a+b x)}\right )-2 a \log \left (1+e^{-2 (a+b x)}\right )-2 b x \log \left (1+e^{-2 (a+b x)}\right )-2 a \log (\tanh (a+b x))+\text {PolyLog}\left (2,-e^{-2 (a+b x)}\right )-\text {PolyLog}\left (2,e^{-2 (a+b x)}\right )}{2 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. \(124\) vs.
\(2(51)=102\).
time = 1.70, size = 125, normalized size = 2.16
method | result | size |
risch | \(\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {x \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}-\frac {\polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}+\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 87, normalized size = 1.50 \begin {gather*} -\frac {2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, 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 )}{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 [C] Result contains complex when optimal does not.
time = 0.46, size = 224, normalized size = 3.86 \begin {gather*} \frac {b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (b x + a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - {\left (b x + a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{2}} \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 \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}{\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 )\,\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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