Optimal. Leaf size=165 \[ -\frac {6 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {6 \text {Li}_4\left (e^{a+b x}\right )}{b^4}-\frac {6 \sinh (a+b x)}{b^4}+\frac {6 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}+\frac {6 x \cosh (a+b x)}{b^3}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \sinh (a+b x)}{b^2}+\frac {x^3 \cosh (a+b x)}{b}-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.18, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5450, 3296, 2637, 4182, 2531, 6609, 2282, 6589} \[ -\frac {3 x^2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {6 x \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {6 x \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {6 \text {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {6 \text {PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2}-\frac {6 \sinh (a+b x)}{b^4}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 4182
Rule 5450
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \cosh (a+b x) \coth (a+b x) \, dx &=\int x^3 \text {csch}(a+b x) \, dx+\int x^3 \sinh (a+b x) \, dx\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x^3 \cosh (a+b x)}{b}-\frac {3 \int x^2 \cosh (a+b x) \, dx}{b}-\frac {3 \int x^2 \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {3 \int x^2 \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x^3 \cosh (a+b x)}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \sinh (a+b x)}{b^2}+\frac {6 \int x \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {6 \int x \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}+\frac {6 \int x \sinh (a+b x) \, dx}{b^2}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \sinh (a+b x)}{b^2}-\frac {6 \int \cosh (a+b x) \, dx}{b^3}-\frac {6 \int \text {Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}+\frac {6 \int \text {Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2}-\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {6 \text {Li}_4\left (e^{a+b x}\right )}{b^4}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 4.18, size = 202, normalized size = 1.22 \[ \frac {b^3 x^3 \cosh (a+b x)-2 b^3 x^3 \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))-3 b^2 x^2 \text {Li}_2(-\cosh (a+b x)-\sinh (a+b x))+3 b^2 x^2 \text {Li}_2(\cosh (a+b x)+\sinh (a+b x))-3 b^2 x^2 \sinh (a+b x)+6 b x \text {Li}_3(-\cosh (a+b x)-\sinh (a+b x))-6 b x \text {Li}_3(\cosh (a+b x)+\sinh (a+b x))-6 \text {Li}_4(-\cosh (a+b x)-\sinh (a+b x))+6 \text {Li}_4(\cosh (a+b x)+\sinh (a+b x))-6 \sinh (a+b x)+6 b x \cosh (a+b x)}{b^4} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.80, size = 511, normalized size = 3.10 \[ \frac {b^{3} x^{3} + 3 \, b^{2} x^{2} + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x + 6 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \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^{3} x^{3} \cosh \left (b x + a\right ) + b^{3} x^{3} \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^{3} \cosh \left (b x + a\right ) + a^{3} \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^{3} x^{3} + a^{3}\right )} \cosh \left (b x + a\right ) + {\left (b^{3} x^{3} + a^{3}\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 12 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm polylog}\left (4, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 12 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm polylog}\left (4, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 12 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 12 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 6}{2 \, {\left (b^{4} \cosh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )\right )}} \]
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 \[ \int x^{3} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 246, normalized size = 1.49 \[ \frac {\left (x^{3} b^{3}-3 x^{2} b^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{2 b^{4}}+\frac {\left (x^{3} b^{3}+3 x^{2} b^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{2 b^{4}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{3}}{b}-\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {6 x \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}+\frac {3 x^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {6 x \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {2 a^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 206, normalized size = 1.25 \[ \frac {{\left ({\left (b^{3} x^{3} e^{\left (2 \, a\right )} - 3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 6 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{4}} - \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} \]
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 \[ \int \frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{3} \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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