Optimal. Leaf size=151 \[ -\frac{i x \text{PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac{i x \text{PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac{i \text{PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac{i \text{PolyLog}\left (4,i e^{-a-b x}\right )}{b^3}-\frac{i x^2 \text{PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{PolyLog}\left (2,i e^{-a-b x}\right )}{2 b} \]
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Rubi [A] time = 0.0977789, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5144, 2531, 6609, 2282, 6589} \[ -\frac{i x \text{PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac{i x \text{PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac{i \text{PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac{i \text{PolyLog}\left (4,i e^{-a-b x}\right )}{b^3}-\frac{i x^2 \text{PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{PolyLog}\left (2,i e^{-a-b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5144
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \cot ^{-1}\left (e^{a+b x}\right ) \, dx &=\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 \text{Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{Li}_2\left (i e^{-a-b x}\right )}{2 b}+\frac{i \int x \text{Li}_2\left (-i e^{-a-b x}\right ) \, dx}{b}-\frac{i \int x \text{Li}_2\left (i e^{-a-b x}\right ) \, dx}{b}\\ &=-\frac{i x^2 \text{Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac{i x \text{Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac{i x \text{Li}_3\left (i e^{-a-b x}\right )}{b^2}+\frac{i \int \text{Li}_3\left (-i e^{-a-b x}\right ) \, dx}{b^2}-\frac{i \int \text{Li}_3\left (i e^{-a-b x}\right ) \, dx}{b^2}\\ &=-\frac{i x^2 \text{Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac{i x \text{Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac{i x \text{Li}_3\left (i e^{-a-b x}\right )}{b^2}-\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3}+\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{-a-b x}\right )}{b^3}\\ &=-\frac{i x^2 \text{Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac{i x \text{Li}_3\left (-i e^{-a-b x}\right )}{b^2}+\frac{i x \text{Li}_3\left (i e^{-a-b x}\right )}{b^2}-\frac{i \text{Li}_4\left (-i e^{-a-b x}\right )}{b^3}+\frac{i \text{Li}_4\left (i e^{-a-b x}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.008189, size = 151, normalized size = 1. \[ -\frac{i x \text{PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac{i x \text{PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}-\frac{i \text{PolyLog}\left (4,-i e^{-a-b x}\right )}{b^3}+\frac{i \text{PolyLog}\left (4,i e^{-a-b x}\right )}{b^3}-\frac{i x^2 \text{PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac{i x^2 \text{PolyLog}\left (2,i e^{-a-b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.253, size = 413, normalized size = 2.7 \begin{align*}{\frac{\pi \,{x}^{3}}{6}}-{\frac{{\frac{i}{2}}{a}^{3}\ln \left ( 1-i{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{{\frac{i}{2}}{\it polylog} \left ( 2,i{{\rm e}^{bx+a}} \right ){x}^{2}}{b}}-{\frac{{\frac{i}{2}}{x}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{bx+a}} \right ) }{b}}+{\frac{{\frac{i}{2}}\ln \left ( 1+i{{\rm e}^{bx+a}} \right ) x{a}^{2}}{{b}^{2}}}+{\frac{i{\it polylog} \left ( 4,i{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{{\frac{i}{2}}{\it polylog} \left ( 2,-i{{\rm e}^{bx+a}} \right ){a}^{2}}{{b}^{3}}}-{\frac{{\frac{i}{2}}{\it polylog} \left ( 2,i{{\rm e}^{bx+a}} \right ){a}^{2}}{{b}^{3}}}-{\frac{{\frac{i}{2}}\ln \left ( -i \left ( -{{\rm e}^{bx+a}}+i \right ) \right ) x{a}^{2}}{{b}^{2}}}-{\frac{i{\it polylog} \left ( 4,-i{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-{\frac{{\frac{i}{2}}\ln \left ( -i \left ( -{{\rm e}^{bx+a}}+i \right ) \right ){a}^{3}}{{b}^{3}}}+{\frac{{\frac{i}{2}}\ln \left ( -i \left ({{\rm e}^{bx+a}}+i \right ) \right ){a}^{3}}{{b}^{3}}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( -i \left ({{\rm e}^{bx+a}}+i \right ) \right ){a}^{2}}{{b}^{3}}}+{\frac{{\frac{i}{2}}{a}^{3}\ln \left ( 1+i{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{{\frac{i}{2}}\ln \left ( -i{{\rm e}^{bx+a}} \right ) \ln \left ( -i \left ( -{{\rm e}^{bx+a}}+i \right ) \right ){a}^{2}}{{b}^{3}}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( -i{{\rm e}^{bx+a}} \right ){a}^{2}}{{b}^{3}}}+{\frac{i{\it polylog} \left ( 3,-i{{\rm e}^{bx+a}} \right ) x}{{b}^{2}}}-{\frac{ix{\it polylog} \left ( 3,i{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{{\frac{i}{2}}\ln \left ( -i \left ({{\rm e}^{bx+a}}+i \right ) \right ) x{a}^{2}}{{b}^{2}}}-{\frac{{\frac{i}{2}}\ln \left ( 1-i{{\rm e}^{bx+a}} \right ) x{a}^{2}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (e^{\left (-b x - a\right )}\right ) + b \int \frac{x^{3} e^{\left (b x + a\right )}}{3 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.6963, size = 540, normalized size = 3.58 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acot}{\left (e^{a} e^{b x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arccot}\left (e^{\left (b x + a\right )}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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