Optimal. Leaf size=83 \[ -\frac {3 x^2}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {3 \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4} \]
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Rubi [A]
time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5527, 4269,
3797, 2221, 2317, 2438} \begin {gather*} \frac {3 \text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 x^2}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 4269
Rule 5527
Rubi steps
\begin {align*} \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx &=-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 \int x^2 \text {csch}^2(a+b x) \, dx}{2 b}\\ &=-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 \int x \coth (a+b x) \, dx}{b^2}\\ &=-\frac {3 x^2}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {6 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx}{b^2}\\ &=-\frac {3 x^2}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac {3 \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {3 x^2}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=-\frac {3 x^2}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {3 \text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.03, size = 175, normalized size = 2.11 \begin {gather*} \frac {3 i b \pi x-b^3 x^3 \text {csch}^2(a+b x)-3 i \pi \log \left (1+e^{2 b x}\right )+6 b x \log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )+3 i \pi \log (\cosh (b x))+6 \tanh ^{-1}(\tanh (a)) \left (b x+\log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )-\log \left (i \sinh \left (b x+\tanh ^{-1}(\tanh (a))\right )\right )\right )-3 \text {PolyLog}\left (2,e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )-3 b^2 e^{-\tanh ^{-1}(\tanh (a))} x^2 \coth (a) \sqrt {\text {sech}^2(a)}+3 b^2 x^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{2 b^4} \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. \(176\) vs.
\(2(75)=150\).
time = 1.35, size = 177, normalized size = 2.13
method | result | size |
risch | \(-\frac {x^{2} \left (2 b x \,{\mathrm e}^{2 b x +2 a}+3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 x^{2}}{b^{2}}-\frac {6 a x}{b^{3}}-\frac {3 a^{2}}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}+\frac {3 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {3 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 130, normalized size = 1.57 \begin {gather*} \frac {3 \, x^{2} - {\left (2 \, b x^{3} e^{\left (2 \, a\right )} + 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, 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 {3 \, x^{2}}{b^{2}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \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. 979 vs.
\(2 (74) = 148\).
time = 0.39, size = 979, normalized size = 11.80 \begin {gather*} -\frac {3 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{4} + 12 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{4} + {\left (2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} + {\left (2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} + 18 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} + 6 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} - 3 \, a^{2} - 3 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\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 ) - 3 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\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 ) - 3 \, {\left (b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, b x \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (a \cosh \left (b x + a\right )^{4} + 4 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + a \sinh \left (b x + a\right )^{4} - 2 \, a \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, a \cosh \left (b x + a\right )^{2} - a\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (a \cosh \left (b x + a\right )^{3} - a \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 3 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} - b x - a\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} - {\left (b x + a\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (6 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{3} + {\left (2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b^{4} \cosh \left (b x + a\right )^{4} + 4 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{4} \sinh \left (b x + a\right )^{4} - 2 \, b^{4} \cosh \left (b x + a\right )^{2} + b^{4} + 2 \, {\left (3 \, b^{4} \cosh \left (b x + a\right )^{2} - b^{4}\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b^{4} \cosh \left (b x + a\right )^{3} - b^{4} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int \frac {x^3\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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