Optimal. Leaf size=201 \[ -\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}-\frac {3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.36, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5457, 4182, 2531, 6609, 2282, 6589, 4186, 2279, 2391} \[ -\frac {3 x^2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {3 x^2 \text {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {3 x \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^4}+\frac {3 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^4}-\frac {3 \text {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {3 \text {PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4182
Rule 4186
Rule 5457
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx &=\int x^3 \text {csch}(a+b x) \, dx+\int x^3 \text {csch}^3(a+b x) \, dx\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int x^3 \text {csch}(a+b x) \, dx+\frac {3 \int x \text {csch}(a+b x) \, dx}{b^2}-\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 {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 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 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^3}+\frac {3 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^3}+\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 {3 \int x^2 \log \left (1-e^{a+b x}\right ) \, dx}{2 b}-\frac {3 \int x^2 \log \left (1+e^{a+b x}\right ) \, dx}{2 b}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 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 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\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 {3 \int x \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac {3 \int x \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\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 {3 \int \text {Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac {3 \int \text {Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 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 {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}\\ \end {align*}
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Mathematica [A] time = 6.92, size = 280, normalized size = 1.39 \[ -\frac {-4 b^3 x^3 \log \left (1-e^{a+b x}\right )+4 b^3 x^3 \log \left (e^{a+b x}+1\right )+b^3 x^3 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )+b^3 x^3 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )+12 \left (b^2 x^2+2\right ) \text {Li}_2\left (-e^{a+b x}\right )-12 \left (b^2 x^2+2\right ) \text {Li}_2\left (e^{a+b x}\right )+12 b^2 x^2 \text {csch}(a)-6 b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right )-6 b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right )-24 b x \text {Li}_3\left (-e^{a+b x}\right )+24 b x \text {Li}_3\left (e^{a+b x}\right )+24 \text {Li}_4\left (-e^{a+b x}\right )-24 \text {Li}_4\left (e^{a+b x}\right )-24 b x \log \left (1-e^{a+b x}\right )+24 b x \log \left (e^{a+b x}+1\right )}{8 b^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.46, size = 1802, normalized size = 8.97 \[ \text {result too large to display} \]
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 \[ \int x^{3} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 340, normalized size = 1.69 \[ -\frac {x^{2} {\mathrm e}^{b x +a} \left (b x \,{\mathrm e}^{2 b x +2 a}+b x +3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 a \ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{3}}{2 b}-\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {3 x \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{2 b}+\frac {3 x^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {3 x \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{4}}+\frac {6 a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {a^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 262, normalized size = 1.30 \[ -\frac {{\left (b x^{3} e^{\left (3 \, a\right )} + 3 \, x^{2} e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x^{3} e^{a} - 3 \, x^{2} e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{2 \, b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{2 \, b^{4}} - \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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