Optimal. Leaf size=84 \[ -\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{2} (m+n+1),\frac{1}{2} (m+n+3),\cos ^2(e+f x)\right )}{b f (m+n+1)} \]
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Rubi [A] time = 0.101382, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2602, 2576} \[ -\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(e+f x)\right )}{b f (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 2602
Rule 2576
Rubi steps
\begin{align*} \int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx &=-\frac{\left (a (a \cos (e+f x))^{-1-n} (b \cot (e+f x))^{1+n} (-\sin (e+f x))^{1+n}\right ) \int (a \cos (e+f x))^{m+n} (-\sin (e+f x))^{-n} \, dx}{b}\\ &=-\frac{(a \cos (e+f x))^m (b \cot (e+f x))^{1+n} \, _2F_1\left (\frac{1+n}{2},\frac{1}{2} (1+m+n);\frac{1}{2} (3+m+n);\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac{1+n}{2}}}{b f (1+m+n)}\\ \end{align*}
Mathematica [A] time = 0.510996, size = 83, normalized size = 0.99 \[ -\frac{b \sec ^2(e+f x)^{m/2} (a \cos (e+f x))^m (b \cot (e+f x))^{n-1} \text{Hypergeometric2F1}\left (\frac{m+2}{2},\frac{1-n}{2},\frac{3-n}{2},-\tan ^2(e+f x)\right )}{f (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.02, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( b\cot \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cot \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (a \cos \left (f x + e\right )\right )^{m} \left (b \cot \left (f x + e\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cot \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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