Optimal. Leaf size=12 \[ \frac{1}{b (a+b \cot (x))} \]
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Rubi [A] time = 0.0400216, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3506, 32} \[ \frac{1}{b (a+b \cot (x))} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 32
Rubi steps
\begin{align*} \int \frac{\csc ^2(x)}{(a+b \cot (x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,b \cot (x)\right )}{b}\\ &=\frac{1}{b (a+b \cot (x))}\\ \end{align*}
Mathematica [A] time = 0.0275174, size = 17, normalized size = 1.42 \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 13, normalized size = 1.1 \begin{align*}{\frac{1}{b \left ( a+b\cot \left ( x \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18473, size = 16, normalized size = 1.33 \begin{align*} \frac{1}{{\left (b \cot \left (x\right ) + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59851, size = 95, normalized size = 7.92 \begin{align*} -\frac{a \cos \left (x\right ) - b \sin \left (x\right )}{{\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) +{\left (a^{3} + a b^{2}\right )} \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 \frac{\csc ^{2}{\left (x \right )}}{\left (a + b \cot{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27932, size = 18, normalized size = 1.5 \begin{align*} -\frac{1}{{\left (a \tan \left (x\right ) + b\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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