Optimal. Leaf size=15 \[ -\frac{\cot ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.029278, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 30} \[ -\frac{\cot ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^2(a+b x) \csc ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (a+b x)\right )}{b}\\ &=-\frac{\cot ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0095005, size = 15, normalized size = 1. \[ -\frac{\cot ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 22, normalized size = 1.5 \begin{align*} -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{3\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992593, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{3 \, b \tan \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1128, size = 78, normalized size = 5.2 \begin{align*} \frac{\cos \left (b x + a\right )^{3}}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.84079, size = 71, normalized size = 4.73 \begin{align*} \begin{cases} \frac{\tan ^{3}{\left (\frac{a}{2} + \frac{b x}{2} \right )}}{24 b} - \frac{\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )}}{8 b} + \frac{1}{8 b \tan{\left (\frac{a}{2} + \frac{b x}{2} \right )}} - \frac{1}{24 b \tan ^{3}{\left (\frac{a}{2} + \frac{b x}{2} \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{2}{\left (a \right )}}{\sin ^{4}{\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 = 1.17765, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{3 \, b \tan \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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