Integrand size = 12, antiderivative size = 151 \[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=-\frac {i x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i x \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac {i \operatorname {PolyLog}\left (4,i e^{-a-b x}\right )}{b^3} \]
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Time = 0.07 (sec) , antiderivative size = 151, 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 = {5252, 2611, 6744, 2320, 6724} \[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=-\frac {i \operatorname {PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac {i \operatorname {PolyLog}\left (4,i e^{-a-b x}\right )}{b^3}-\frac {i x \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac {i x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b} \]
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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^{-a-b x}\right ) \, dx-\frac {1}{2} i \int x^2 \log \left (1+i e^{-a-b x}\right ) \, dx \\ & = -\frac {i x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}+\frac {i \int x \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right ) \, dx}{b}-\frac {i \int x \operatorname {PolyLog}\left (2,i e^{-a-b x}\right ) \, dx}{b} \\ & = -\frac {i x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i x \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}+\frac {i \int \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right ) \, dx}{b^2}-\frac {i \int \operatorname {PolyLog}\left (3,i e^{-a-b x}\right ) \, dx}{b^2} \\ & = -\frac {i x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i x \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3}+\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3} \\ & = -\frac {i x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i x \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {i x \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac {i \operatorname {PolyLog}\left (4,i e^{-a-b x}\right )}{b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=-\frac {i \left (b^2 x^2 \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )-b^2 x^2 \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )+2 \left (b x \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )-b x \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )+\operatorname {PolyLog}\left (4,-i e^{-a-b x}\right )-\operatorname {PolyLog}\left (4,i e^{-a-b x}\right )\right )\right )}{2 b^3} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (129 ) = 258\).
Time = 0.80 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.74
method | result | size |
risch | \(\frac {i \ln \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a^{3}}{2 b^{3}}+\frac {\pi \,x^{3}}{6}+\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a^{2} x}{2 b^{2}}-\frac {i \operatorname {polylog}\left (3, i {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{3}}+\frac {i \operatorname {polylog}\left (3, -i {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {i \operatorname {polylog}\left (4, i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {i \ln \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) x \,a^{2}}{2 b^{2}}-\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x \,a^{2}}{2 b^{2}}-\frac {i \operatorname {polylog}\left (2, -i {\mathrm e}^{b x +a}\right ) x^{2}}{2 b}+\frac {i a^{3} \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{3}}-\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a^{2} x}{2 b^{2}}-\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a^{3}}{2 b^{3}}+\frac {i \operatorname {polylog}\left (2, -i {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {i \operatorname {polylog}\left (2, i {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {i \operatorname {dilog}\left (-i {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {i \operatorname {polylog}\left (4, -i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {i \operatorname {dilog}\left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a^{2}}{2 b^{3}}+\frac {i \operatorname {polylog}\left (2, i {\mathrm e}^{b x +a}\right ) x^{2}}{2 b}+\frac {i \ln \left (-i {\mathrm e}^{b x +a}\right ) \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a^{2}}{2 b^{3}}\) | \(413\) |
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Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.24 \[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {2 \, b^{3} x^{3} \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) + 3 i \, b^{2} x^{2} {\rm Li}_2\left (i \, e^{\left (b x + a\right )}\right ) - 3 i \, b^{2} x^{2} {\rm Li}_2\left (-i \, e^{\left (b x + a\right )}\right ) - i \, a^{3} \log \left (e^{\left (b x + a\right )} + i\right ) + i \, a^{3} \log \left (e^{\left (b x + a\right )} - i\right ) - 6 i \, b x {\rm polylog}\left (3, i \, e^{\left (b x + a\right )}\right ) + 6 i \, b x {\rm polylog}\left (3, -i \, e^{\left (b x + a\right )}\right ) + {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \log \left (i \, e^{\left (b x + a\right )} + 1\right ) + {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \log \left (-i \, e^{\left (b x + a\right )} + 1\right ) + 6 i \, {\rm polylog}\left (4, i \, e^{\left (b x + a\right )}\right ) - 6 i \, {\rm polylog}\left (4, -i \, e^{\left (b x + a\right )}\right )}{6 \, b^{3}} \]
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\[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int x^{2} \operatorname {acot}{\left (e^{a} e^{b x} \right )}\, dx \]
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\[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int { x^{2} \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
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\[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int { x^{2} \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
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Timed out. \[ \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int x^2\,\mathrm {acot}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]
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