Optimal. Leaf size=126 \[ \frac{i x \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac{x \cot (a+b x)}{b^2}+\frac{\log (\sin (a+b x))}{b^3}-\frac{x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2}{2 b}+\frac{i x^3}{3} \]
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Rubi [A] time = 0.186121, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3720, 3475, 30, 3717, 2190, 2531, 2282, 6589} \[ \frac{i x \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac{x \cot (a+b x)}{b^2}+\frac{\log (\sin (a+b x))}{b^3}-\frac{x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2}{2 b}+\frac{i x^3}{3} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3475
Rule 30
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \cot ^3(a+b x) \, dx &=-\frac{x^2 \cot ^2(a+b x)}{2 b}+\frac{\int x \cot ^2(a+b x) \, dx}{b}-\int x^2 \cot (a+b x) \, dx\\ &=\frac{i x^3}{3}-\frac{x \cot (a+b x)}{b^2}-\frac{x^2 \cot ^2(a+b x)}{2 b}+2 i \int \frac{e^{2 i (a+b x)} x^2}{1-e^{2 i (a+b x)}} \, dx+\frac{\int \cot (a+b x) \, dx}{b^2}-\frac{\int x \, dx}{b}\\ &=-\frac{x^2}{2 b}+\frac{i x^3}{3}-\frac{x \cot (a+b x)}{b^2}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac{\log (\sin (a+b x))}{b^3}+\frac{2 \int x \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac{x^2}{2 b}+\frac{i x^3}{3}-\frac{x \cot (a+b x)}{b^2}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac{\log (\sin (a+b x))}{b^3}+\frac{i x \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{i \int \text{Li}_2\left (e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{x^2}{2 b}+\frac{i x^3}{3}-\frac{x \cot (a+b x)}{b^2}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac{\log (\sin (a+b x))}{b^3}+\frac{i x \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^3}\\ &=-\frac{x^2}{2 b}+\frac{i x^3}{3}-\frac{x \cot (a+b x)}{b^2}-\frac{x^2 \cot ^2(a+b x)}{2 b}-\frac{x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac{\log (\sin (a+b x))}{b^3}+\frac{i x \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 5.14264, size = 221, normalized size = 1.75 \[ -\frac{2 e^{-i a} \sin (a) (\cot (a)+i) \left (6 i b x \text{PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b x \text{PolyLog}\left (2,e^{-i (a+b x)}\right )+6 \text{PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 \text{PolyLog}\left (3,e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )-b^3 x^3 \cot (a)+i b^3 x^3\right )+2 b^3 x^3 \cot (a)+3 b^2 x^2 \csc ^2(a+b x)+6 b x \cot (a)-6 \log (\sin (a+b x))-6 b x \csc (a) \sin (b x) \csc (a+b x)}{6 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.267, size = 293, normalized size = 2.3 \begin{align*}{\frac{2\,i{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}+2\,{\frac{x \left ( bx{{\rm e}^{2\,i \left ( bx+a \right ) }}-i{{\rm e}^{2\,i \left ( bx+a \right ) }}+i \right ) }{{b}^{2} \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}}}+{\frac{2\,i{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}-{\frac{\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ){x}^{2}}{b}}-{\frac{{\frac{4\,i}{3}}{a}^{3}}{{b}^{3}}}-{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ){x}^{2}}{b}}+{\frac{i}{3}}{x}^{3}-{\frac{2\,i{a}^{2}x}{{b}^{2}}}-2\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-2\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-2\,{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-1 \right ) }{{b}^{3}}}+{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) }{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ){a}^{2}}{{b}^{3}}}+2\,{\frac{{a}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-{\frac{{a}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-1 \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.69758, size = 1642, normalized size = 13.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.87122, size = 1077, normalized size = 8.55 \begin{align*} \frac{4 \, b^{2} x^{2} + 4 \, b x \sin \left (2 \, b x + 2 \, a\right ) +{\left (2 i \, b x \cos \left (2 \, b x + 2 \, a\right ) - 2 i \, b x\right )}{\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) +{\left (-2 i \, b x \cos \left (2 \, b x + 2 \, a\right ) + 2 i \, b x\right )}{\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \,{\left (a^{2} -{\left (a^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac{1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac{1}{2}\right ) + 2 \,{\left (a^{2} -{\left (a^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac{1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac{1}{2}\right ) + 2 \,{\left (b^{2} x^{2} - a^{2} -{\left (b^{2} x^{2} - a^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \,{\left (b^{2} x^{2} - a^{2} -{\left (b^{2} x^{2} - a^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) -{\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )}{\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) -{\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )}{\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{4 \,{\left (b^{3} \cos \left (2 \, b x + 2 \, a\right ) - b^{3}\right )}} \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} \cot ^{3}{\left (a + 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} \cot \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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