Optimal. Leaf size=20 \[ -\frac {(a+b \cot (x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 20, 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} \[ -\frac {(a+b \cot (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3506
Rubi steps
\begin {align*} \int (a+b \cot (x))^n \csc ^2(x) \, dx &=-\frac {\operatorname {Subst}\left (\int (a+x)^n \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac {(a+b \cot (x))^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 19, normalized size = 0.95 \[ -\frac {(a+b \cot (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 38, normalized size = 1.90 \[ -\frac {{\left (b \cos \relax (x) + a \sin \relax (x)\right )} \left (\frac {b \cos \relax (x) + a \sin \relax (x)}{\sin \relax (x)}\right )^{n}}{{\left (b n + b\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cot \relax (x) + a\right )}^{n} \csc \relax (x)^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 21, normalized size = 1.05 \[ -\frac {\left (a +b \cot \relax (x )\right )^{1+n}}{b \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 20, normalized size = 1.00 \[ -\frac {{\left (b \cot \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 43, normalized size = 2.15 \[ \left \{\begin {array}{cl} -\frac {\ln \left (a+\frac {b}{\mathrm {tan}\relax (x)}\right )}{b} & \text {\ if\ \ }n=-1\\ -\frac {{\left (a+\frac {b}{\mathrm {tan}\relax (x)}\right )}^{n+1}}{b\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
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 \[ \int \left (a + b \cot {\relax (x )}\right )^{n} \csc ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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