Integrand size = 16, antiderivative size = 93 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4} \]
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Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5527, 4267, 2611, 2320, 6724} \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}+\frac {6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {x^3 \text {csch}(a+b x)}{b} \]
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Rule 2320
Rule 2611
Rule 4267
Rule 5527
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 \int x^2 \text {csch}(a+b x) \, dx}{b} \\ & = -\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {6 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b^2} \\ & = -\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \int \operatorname {PolyLog}\left (2,-e^{a+b x}\right ) \, dx}{b^3}-\frac {6 \int \operatorname {PolyLog}\left (2,e^{a+b x}\right ) \, dx}{b^3} \\ & = -\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{a+b x}\right )}{b^4} \\ & = -\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.17 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\frac {-b^3 x^3 \text {csch}(a+b x)+3 b^2 x^2 \log \left (1-e^{a+b x}\right )-3 b^2 x^2 \log \left (1+e^{a+b x}\right )-6 b x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+6 b x \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4} \]
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Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.87
method | result | size |
risch | \(-\frac {2 x^{3} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {6 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}+\frac {6 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{4}}-\frac {6 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}\) | \(174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (86) = 172\).
Time = 0.27 (sec) , antiderivative size = 551, normalized size of antiderivative = 5.92 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, b^{3} x^{3} \cosh \left (b x + a\right ) + 2 \, b^{3} x^{3} \sinh \left (b x + a\right ) - 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 3 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2} - b^{2} x^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 3 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 3 \, {\left (b^{2} x^{2} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{4} \cosh \left (b x + a\right )^{2} + 2 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )^{2} - b^{4}} \]
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\[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\int x^{3} \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, x^{3} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}} \]
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\[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\int { x^{3} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\int \frac {x^3\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
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