Optimal. Leaf size=112 \[ -3 i x \text{PolyLog}\left (2,e^{2 i x}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 i x}\right )-\frac{3 x^4}{8}-i x^3+\frac{3 x^2}{8}+3 x^2 \log \left (1-e^{2 i x}\right )-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)-\frac{1}{2} x^3 \sin (x) \cos (x)+\frac{3 \cos ^2(x)}{8}+\frac{3}{4} x \sin (x) \cos (x) \]
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Rubi [A] time = 0.185145, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {4408, 3311, 30, 3310, 3720, 3717, 2190, 2531, 2282, 6589} \[ -3 i x \text{PolyLog}\left (2,e^{2 i x}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 i x}\right )-\frac{3 x^4}{8}-i x^3+\frac{3 x^2}{8}+3 x^2 \log \left (1-e^{2 i x}\right )-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)-\frac{1}{2} x^3 \sin (x) \cos (x)+\frac{3 \cos ^2(x)}{8}+\frac{3}{4} x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 4408
Rule 3311
Rule 30
Rule 3310
Rule 3720
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \cos ^2(x) \cot ^2(x) \, dx &=-\int x^3 \cos ^2(x) \, dx+\int x^3 \cot ^2(x) \, dx\\ &=-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)-\frac{1}{2} x^3 \cos (x) \sin (x)-\frac{\int x^3 \, dx}{2}+\frac{3}{2} \int x \cos ^2(x) \, dx+3 \int x^2 \cot (x) \, dx-\int x^3 \, dx\\ &=-i x^3-\frac{3 x^4}{8}+\frac{3 \cos ^2(x)}{8}-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+\frac{3}{4} x \cos (x) \sin (x)-\frac{1}{2} x^3 \cos (x) \sin (x)-6 i \int \frac{e^{2 i x} x^2}{1-e^{2 i x}} \, dx+\frac{3 \int x \, dx}{4}\\ &=\frac{3 x^2}{8}-i x^3-\frac{3 x^4}{8}+\frac{3 \cos ^2(x)}{8}-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )+\frac{3}{4} x \cos (x) \sin (x)-\frac{1}{2} x^3 \cos (x) \sin (x)-6 \int x \log \left (1-e^{2 i x}\right ) \, dx\\ &=\frac{3 x^2}{8}-i x^3-\frac{3 x^4}{8}+\frac{3 \cos ^2(x)}{8}-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \text{Li}_2\left (e^{2 i x}\right )+\frac{3}{4} x \cos (x) \sin (x)-\frac{1}{2} x^3 \cos (x) \sin (x)+3 i \int \text{Li}_2\left (e^{2 i x}\right ) \, dx\\ &=\frac{3 x^2}{8}-i x^3-\frac{3 x^4}{8}+\frac{3 \cos ^2(x)}{8}-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \text{Li}_2\left (e^{2 i x}\right )+\frac{3}{4} x \cos (x) \sin (x)-\frac{1}{2} x^3 \cos (x) \sin (x)+\frac{3}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac{3 x^2}{8}-i x^3-\frac{3 x^4}{8}+\frac{3 \cos ^2(x)}{8}-\frac{3}{4} x^2 \cos ^2(x)-x^3 \cot (x)+3 x^2 \log \left (1-e^{2 i x}\right )-3 i x \text{Li}_2\left (e^{2 i x}\right )+\frac{3}{2} \text{Li}_3\left (e^{2 i x}\right )+\frac{3}{4} x \cos (x) \sin (x)-\frac{1}{2} x^3 \cos (x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.175528, size = 104, normalized size = 0.93 \[ \frac{1}{16} \left (48 i x \text{PolyLog}\left (2,e^{-2 i x}\right )+24 \text{PolyLog}\left (3,e^{-2 i x}\right )-6 x^4+16 i x^3+48 x^2 \log \left (1-e^{-2 i x}\right )-4 x^3 \sin (2 x)-6 x^2 \cos (2 x)-16 x^3 \cot (x)+6 x \sin (2 x)+3 \cos (2 x)-2 i \pi ^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 150, normalized size = 1.3 \begin{align*} -{\frac{3\,{x}^{4}}{8}}+{\frac{i}{32}} \left ( 6\,i{x}^{2}+4\,{x}^{3}-3\,i-6\,x \right ){{\rm e}^{2\,ix}}-{\frac{i}{32}} \left ( -6\,i{x}^{2}+4\,{x}^{3}+3\,i-6\,x \right ){{\rm e}^{-2\,ix}}-{\frac{2\,i{x}^{3}}{{{\rm e}^{2\,ix}}-1}}-2\,i{x}^{3}+3\,{x}^{2}\ln \left ( 1-{{\rm e}^{ix}} \right ) -6\,ix{\it polylog} \left ( 2,{{\rm e}^{ix}} \right ) +6\,{\it polylog} \left ( 3,{{\rm e}^{ix}} \right ) +3\,{x}^{2}\ln \left ( 1+{{\rm e}^{ix}} \right ) -6\,ix{\it polylog} \left ( 2,-{{\rm e}^{ix}} \right ) +6\,{\it polylog} \left ( 3,-{{\rm e}^{ix}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.594582, size = 851, normalized size = 7.6 \begin{align*} \frac{4 \,{\left (2 \, x^{3} - 3 \, x\right )} \cos \left (x\right )^{3} + 24 \, x^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 24 \, x^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 48 i \, x{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 i \, x{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 i \, x{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 48 i \, x{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) - 12 \,{\left (2 \, x^{3} - x\right )} \cos \left (x\right ) - 3 \,{\left (2 \, x^{4} + 2 \,{\left (2 \, x^{2} - 1\right )} \cos \left (x\right )^{2} - 2 \, x^{2} + 1\right )} \sin \left (x\right ) + 48 \,{\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \,{\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \,{\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) \sin \left (x\right ) + 48 \,{\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \sin \left (x\right )}{16 \, \sin \left (x\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^{3} \cos ^{2}{\left (x \right )} \cot ^{2}{\left (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^{3} \cos \left (x\right )^{2} \cot \left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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