Integrand size = 8, antiderivative size = 103 \[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=-\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )-i x \operatorname {PolyLog}\left (3,-i e^{-x}\right )+i x \operatorname {PolyLog}\left (3,i e^{-x}\right )-i \operatorname {PolyLog}\left (4,-i e^{-x}\right )+i \operatorname {PolyLog}\left (4,i e^{-x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5252, 2611, 6744, 2320, 6724} \[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=-\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )-i x \operatorname {PolyLog}\left (3,-i e^{-x}\right )+i x \operatorname {PolyLog}\left (3,i e^{-x}\right )-i \operatorname {PolyLog}\left (4,-i e^{-x}\right )+i \operatorname {PolyLog}\left (4,i e^{-x}\right ) \]
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Rule 2320
Rule 2611
Rule 5252
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} i \int x^2 \log \left (1-i e^{-x}\right ) \, dx-\frac {1}{2} i \int x^2 \log \left (1+i e^{-x}\right ) \, dx \\ & = -\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )+i \int x \operatorname {PolyLog}\left (2,-i e^{-x}\right ) \, dx-i \int x \operatorname {PolyLog}\left (2,i e^{-x}\right ) \, dx \\ & = -\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )-i x \operatorname {PolyLog}\left (3,-i e^{-x}\right )+i x \operatorname {PolyLog}\left (3,i e^{-x}\right )+i \int \operatorname {PolyLog}\left (3,-i e^{-x}\right ) \, dx-i \int \operatorname {PolyLog}\left (3,i e^{-x}\right ) \, dx \\ & = -\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )-i x \operatorname {PolyLog}\left (3,-i e^{-x}\right )+i x \operatorname {PolyLog}\left (3,i e^{-x}\right )-i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{-x}\right )+i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{-x}\right ) \\ & = -\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )-i x \operatorname {PolyLog}\left (3,-i e^{-x}\right )+i x \operatorname {PolyLog}\left (3,i e^{-x}\right )-i \operatorname {PolyLog}\left (4,-i e^{-x}\right )+i \operatorname {PolyLog}\left (4,i e^{-x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=-\frac {1}{2} i \left (x^2 \operatorname {PolyLog}\left (2,-i e^{-x}\right )-x^2 \operatorname {PolyLog}\left (2,i e^{-x}\right )+2 \left (x \operatorname {PolyLog}\left (3,-i e^{-x}\right )-x \operatorname {PolyLog}\left (3,i e^{-x}\right )+\operatorname {PolyLog}\left (4,-i e^{-x}\right )-\operatorname {PolyLog}\left (4,i e^{-x}\right )\right )\right ) \]
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Time = 1.69 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {\pi \,x^{3}}{6}+\frac {i x^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{x}\right )}{2}-i x \operatorname {polylog}\left (3, i {\mathrm e}^{x}\right )+i \operatorname {polylog}\left (4, i {\mathrm e}^{x}\right )-\frac {i x^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{x}\right )}{2}+i x \operatorname {polylog}\left (3, -i {\mathrm e}^{x}\right )-i \operatorname {polylog}\left (4, -i {\mathrm e}^{x}\right )\) | \(76\) |
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Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arccot}\left (e^{x}\right ) - \frac {1}{6} i \, x^{3} \log \left (i \, e^{x} + 1\right ) + \frac {1}{6} i \, x^{3} \log \left (-i \, e^{x} + 1\right ) + \frac {1}{2} i \, x^{2} {\rm Li}_2\left (i \, e^{x}\right ) - \frac {1}{2} i \, x^{2} {\rm Li}_2\left (-i \, e^{x}\right ) - i \, x {\rm polylog}\left (3, i \, e^{x}\right ) + i \, x {\rm polylog}\left (3, -i \, e^{x}\right ) + i \, {\rm polylog}\left (4, i \, e^{x}\right ) - i \, {\rm polylog}\left (4, -i \, e^{x}\right ) \]
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\[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=\int x^{2} \operatorname {acot}{\left (e^{x} \right )}\, dx \]
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\[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=\int { x^{2} \operatorname {arccot}\left (e^{x}\right ) \,d x } \]
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\[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=\int { x^{2} \operatorname {arccot}\left (e^{x}\right ) \,d x } \]
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Timed out. \[ \int x^2 \cot ^{-1}\left (e^x\right ) \, dx=\int x^2\,\mathrm {acot}\left ({\mathrm {e}}^x\right ) \,d x \]
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