Integrand size = 10, antiderivative size = 87 \[ \int x^3 \coth (a+b x) \, dx=-\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4} \]
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Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3797, 2221, 2611, 6744, 2320, 6724} \[ \int x^3 \coth (a+b x) \, dx=\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^4}{4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4}{4}-2 \int \frac {e^{2 (a+b x)} x^3}{1-e^{2 (a+b x)}} \, dx \\ & = -\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {3 \int x^2 \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b} \\ & = -\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 \int x \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \int \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right ) \, dx}{2 b^3} \\ & = -\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{4 b^4} \\ & = -\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05 \[ \int x^3 \coth (a+b x) \, dx=-\frac {x^4}{4}+\frac {x^3 \log \left (1-e^{2 a+2 b x}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 a+2 b x}\right )}{4 b^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs. \(2(79)=158\).
Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.30
method | result | size |
risch | \(-\frac {x^{4}}{4}-\frac {2 a^{3} x}{b^{3}}-\frac {3 a^{4}}{2 b^{4}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}+\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}-\frac {a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}\) | \(200\) |
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (78) = 156\).
Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.48 \[ \int x^3 \coth (a+b x) \, dx=-\frac {b^{4} x^{4} - 4 \, b^{3} x^{3} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 12 \, b^{2} x^{2} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 12 \, b^{2} x^{2} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 4 \, a^{3} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 24 \, b x {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 24 \, b x {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 4 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 24 \, {\rm polylog}\left (4, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 24 \, {\rm polylog}\left (4, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{4 \, b^{4}} \]
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\[ \int x^3 \coth (a+b x) \, dx=\int x^{3} \coth {\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (78) = 156\).
Time = 0.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.95 \[ \int x^3 \coth (a+b x) \, dx=\frac {1}{4} \, x^{4} \coth \left (b x + a\right ) - \frac {1}{2} \, {\left (\frac {x^{4}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} + \frac {x^{4}}{b} - \frac {2 \, {\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )}}{b^{5}} - \frac {2 \, {\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )}}{b^{5}}\right )} b \]
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\[ \int x^3 \coth (a+b x) \, dx=\int { x^{3} \coth \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^3 \coth (a+b x) \, dx=\int x^3\,\mathrm {coth}\left (a+b\,x\right ) \,d x \]
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