Optimal. Leaf size=90 \[ -\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \sec (e+f x))^m (b \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{2} (-m+n+1),\frac{1}{2} (-m+n+3),\cos ^2(e+f x)\right )}{b f (-m+n+1)} \]
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Rubi [A] time = 0.153449, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2618, 2602, 2576} \[ -\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \sec (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);\cos ^2(e+f x)\right )}{b f (-m+n+1)} \]
Antiderivative was successfully verified.
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Rule 2618
Rule 2602
Rule 2576
Rubi steps
\begin{align*} \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx &=\left (\left (\frac{\cos (e+f x)}{a}\right )^m (a \sec (e+f x))^m\right ) \int \left (\frac{\cos (e+f x)}{a}\right )^{-m} (b \cot (e+f x))^n \, dx\\ &=-\frac{\left (\left (\frac{\cos (e+f x)}{a}\right )^{-1+m-n} (b \cot (e+f x))^{1+n} (a \sec (e+f x))^m (-\sin (e+f x))^{1+n}\right ) \int \left (\frac{\cos (e+f x)}{a}\right )^{-m+n} (-\sin (e+f x))^{-n} \, dx}{a b}\\ &=-\frac{(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 ) (a \sec (e+f x))^m \sin ^2(e+f x)^{\frac{1+n}{2}}}{b f (1-m+n)}\\ \end{align*}
Mathematica [A] time = 0.44545, size = 83, normalized size = 0.92 \[ -\frac{b \sec ^2(e+f x)^{-m/2} (a \sec (e+f x))^m (b \cot (e+f x))^{n-1} \text{Hypergeometric2F1}\left (1-\frac{m}{2},\frac{1-n}{2},\frac{3-n}{2},-\tan ^2(e+f x)\right )}{f (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.068, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cot \left ( fx+e \right ) \right ) ^{n} \left ( a\sec \left ( fx+e \right ) \right ) ^{m}\, 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 (b \cot \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m}\,{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 (b \cot \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m}, 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 \left (a \sec{\left (e + f x \right )}\right )^{m} \left (b \cot{\left (e + f x \right )}\right )^{n}\, dx \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 (b \cot \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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