Optimal. Leaf size=12 \[ \frac {1}{b (a+b \cot (x))} \]
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Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 32
Rule 3506
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*}
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Mathematica [A] time = 0.03, size = 17, normalized size = 1.42 \[ \frac {\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 39, normalized size = 3.25 \[ -\frac {a \cos \relax (x) - b \sin \relax (x)}{{\left (a^{2} b + b^{3}\right )} \cos \relax (x) + {\left (a^{3} + a b^{2}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 13, normalized size = 1.08 \[ -\frac {1}{{\left (a \tan \relax (x) + b\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 13, normalized size = 1.08 \[ \frac {1}{b \left (a +b \cot \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 12, normalized size = 1.00 \[ \frac {1}{{\left (b \cot \relax (x) + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 14, normalized size = 1.17 \[ -\frac {1}{\mathrm {tan}\relax (x)\,a^2+b\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\relax (x )}}{\left (a + b \cot {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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