Optimal. Leaf size=33 \[ -\frac{285 x}{2}+(3-2 \cot (x))^3+5 (3-2 \cot (x))^2-42 \cot (x)+4 \log (\sin (x)) \]
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Rubi [A] time = 0.0624847, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3528, 3525, 3475} \[ -\frac{285 x}{2}+(3-2 \cot (x))^3+5 (3-2 \cot (x))^2-42 \cot (x)+4 \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \left (\frac{1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx &=(3-2 \cot (x))^3+\int \left (-\frac{9}{2}-10 \cot (x)\right ) (3-2 \cot (x))^2 \, dx\\ &=5 (3-2 \cot (x))^2+(3-2 \cot (x))^3+\int \left (-\frac{67}{2}-21 \cot (x)\right ) (3-2 \cot (x)) \, dx\\ &=-\frac{285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \int \cot (x) \, dx\\ &=-\frac{285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.023241, size = 29, normalized size = 0.88 \[ -\frac{285 x}{2}-148 \cot (x)+56 \csc ^2(x)+4 \log (\sin (x))-8 \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 1. \begin{align*} -8\, \left ( \cot \left ( x \right ) \right ) ^{3}+56\, \left ( \cot \left ( x \right ) \right ) ^{2}-156\,\cot \left ( x \right ) -2\,\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) +{\frac{285\,\pi }{4}}-{\frac{285\,x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41367, size = 49, normalized size = 1.48 \begin{align*} -\frac{285}{2} \, x - \frac{4 \,{\left (39 \, \tan \left (x\right )^{2} - 14 \, \tan \left (x\right ) + 2\right )}}{\tan \left (x\right )^{3}} - 2 \, \log \left (\tan \left (x\right )^{2} + 1\right ) + 4 \, \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1512, size = 220, normalized size = 6.67 \begin{align*} \frac{4 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \sin \left (2 \, x\right ) - 296 \, \cos \left (2 \, x\right )^{2} -{\left (285 \, x \cos \left (2 \, x\right ) - 285 \, x + 224\right )} \sin \left (2 \, x\right ) + 32 \, \cos \left (2 \, x\right ) + 328}{2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.612355, size = 39, normalized size = 1.18 \begin{align*} - \frac{285 x}{2} - 2 \log{\left (\tan ^{2}{\left (x \right )} + 1 \right )} + 4 \log{\left (\tan{\left (x \right )} \right )} - \frac{156}{\tan{\left (x \right )}} + \frac{56}{\tan ^{2}{\left (x \right )}} - \frac{8}{\tan ^{3}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14233, size = 101, normalized size = 3.06 \begin{align*} \tan \left (\frac{1}{2} \, x\right )^{3} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \frac{285}{2} \, x - \frac{22 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 225 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 42 \, \tan \left (\frac{1}{2} \, x\right ) + 3}{3 \, \tan \left (\frac{1}{2} \, x\right )^{3}} - 4 \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + 4 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) + 75 \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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