Optimal. Leaf size=28 \[ -\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b} \]
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Rubi [A] time = 0.0129672, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 3475} \[ -\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \cot ^3(a+b x) \, dx &=-\frac{\cot ^2(a+b x)}{2 b}-\int \cot (a+b x) \, dx\\ &=-\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0948977, size = 34, normalized size = 1.21 \[ -\frac{\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 27, normalized size = 1. \begin{align*} -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{2}}{2\,b}}-{\frac{\ln \left ( \sin \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977599, size = 31, normalized size = 1.11 \begin{align*} -\frac{\frac{1}{\sin \left (b x + a\right )^{2}} + \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39515, size = 108, normalized size = 3.86 \begin{align*} -\frac{2 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (b x + a\right )\right ) - 1}{2 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0051, size = 42, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{\log{\left (\sin{\left (a + b x \right )} \right )}}{b} - \frac{\cos ^{2}{\left (a + b x \right )}}{2 b \sin ^{2}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{3}{\left (a \right )}}{\sin ^{3}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13655, size = 49, normalized size = 1.75 \begin{align*} \frac{\frac{\sin \left (b x + a\right )^{2} - 1}{\sin \left (b x + a\right )^{2}} - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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