Integrand size = 12, antiderivative size = 119 \[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b}+\frac {x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (4,-e^{-a-b x}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (4,e^{-a-b x}\right )}{b^3} \]
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Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6349, 2611, 6744, 2320, 6724} \[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (4,-e^{-a-b x}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (4,e^{-a-b x}\right )}{b^3}+\frac {x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )}{b^2}+\frac {x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]
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Rule 2320
Rule 2611
Rule 6349
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int x^2 \log \left (1-e^{-a-b x}\right ) \, dx\right )+\frac {1}{2} \int x^2 \log \left (1+e^{-a-b x}\right ) \, dx \\ & = \frac {x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b}-\frac {\int x \operatorname {PolyLog}\left (2,-e^{-a-b x}\right ) \, dx}{b}+\frac {\int x \operatorname {PolyLog}\left (2,e^{-a-b x}\right ) \, dx}{b} \\ & = \frac {x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b}+\frac {x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )}{b^2}-\frac {\int \operatorname {PolyLog}\left (3,-e^{-a-b x}\right ) \, dx}{b^2}+\frac {\int \operatorname {PolyLog}\left (3,e^{-a-b x}\right ) \, dx}{b^2} \\ & = \frac {x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b}+\frac {x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )}{b^2}+\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3} \\ & = \frac {x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b}+\frac {x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (4,-e^{-a-b x}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (4,e^{-a-b x}\right )}{b^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.25 \[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {2 b^3 x^3 \coth ^{-1}\left (e^{a+b x}\right )+b^3 x^3 \log \left (1-e^{a+b x}\right )-b^3 x^3 \log \left (1+e^{a+b x}\right )-3 b^2 x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+3 b^2 x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 b x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 b x \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )+6 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{6 b^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(109)=218\).
Time = 0.32 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.05
method | result | size |
risch | \(-\frac {x^{3} \ln \left (-1+{\mathrm e}^{b x +a}\right )}{6}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{6}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x \,a^{2}}{2 b^{2}}+\frac {x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b}-\frac {a^{3} \ln \left (-1+{\mathrm e}^{b x +a}\right )}{6 b^{3}}-\frac {a^{3} \ln \left (1-{\mathrm e}^{b x +a}\right )}{3 b^{3}}-\frac {x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {a^{2} \operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2 b^{3}}+\frac {\operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2 b}-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{2 b^{3}}+\frac {x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {\operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(244\) |
default | \(\frac {x^{3} \operatorname {arccoth}\left ({\mathrm e}^{b x +a}\right )}{3}+\frac {\frac {\left (b x +a \right )^{3} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}-3 \left (b x +a \right ) \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )+3 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{3} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}+3 \left (b x +a \right ) \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )-3 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )+a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\left (b x +a \right ) \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\left (b x +a \right ) \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}-\frac {\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}\right )}{3 b^{3}}\) | \(314\) |
parts | \(\frac {x^{3} \operatorname {arccoth}\left ({\mathrm e}^{b x +a}\right )}{3}+\frac {\frac {\left (b x +a \right )^{3} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}-3 \left (b x +a \right ) \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )+3 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{3} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}+3 \left (b x +a \right ) \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )-3 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )+a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\left (b x +a \right ) \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\left (b x +a \right ) \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}-\frac {\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}\right )}{3 b^{3}}\) | \(314\) |
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (107) = 214\).
Time = 0.27 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.08 \[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {b^{3} x^{3} \log \left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b^{3} x^{3} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 3 \, b^{2} x^{2} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - a^{3} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 6 \, b x {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, b x {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\rm polylog}\left (4, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, {\rm polylog}\left (4, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{6 \, b^{3}} \]
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\[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\int x^{2} \operatorname {acoth}{\left (e^{a} e^{b x} \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.19 \[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (e^{\left (b x + a\right )}\right ) - \frac {1}{6} \, b {\left (\frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} - \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}}\right )} \]
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\[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\int { x^{2} \operatorname {arcoth}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
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Timed out. \[ \int x^2 \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\int x^2\,\mathrm {acoth}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]
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