Optimal. Leaf size=86 \[ \frac{\cos ^2(e+f x)^{\frac{m+1}{2}} \csc ^{n-1}(e+f x) (a \sec (e+f x))^{m+1} \text{Hypergeometric2F1}\left (\frac{m+1}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{a f (1-n)} \]
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Rubi [A] time = 0.0864896, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2631, 2577} \[ \frac{\cos ^2(e+f x)^{\frac{m+1}{2}} \csc ^{n-1}(e+f x) (a \sec (e+f x))^{m+1} \, _2F_1\left (\frac{m+1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2631
Rule 2577
Rubi steps
\begin{align*} \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx &=\frac{\left ((a \cos (e+f x))^{1+m} \csc ^{-1+n}(e+f x) (a \sec (e+f x))^{1+m} \sin ^{-1+n}(e+f x)\right ) \int (a \cos (e+f x))^{-m} \sin ^{-n}(e+f x) \, dx}{a^2}\\ &=\frac{\cos ^2(e+f x)^{\frac{1+m}{2}} \csc ^{-1+n}(e+f x) \, _2F_1\left (\frac{1+m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right ) (a \sec (e+f x))^{1+m}}{a f (1-n)}\\ \end{align*}
Mathematica [C] time = 0.549284, size = 280, normalized size = 3.26 \[ -\frac{(n-3) \csc ^{n-1}(e+f x) (a \sec (e+f x))^m F_1\left (\frac{1}{2}-\frac{n}{2};m,-m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (n-1) \left ((n-3) F_1\left (\frac{1}{2}-\frac{n}{2};m,-m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left ((m+n-1) F_1\left (\frac{3}{2}-\frac{n}{2};m,-m-n+2;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+m F_1\left (\frac{3}{2}-\frac{n}{2};m+1,-m-n+1;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.509, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \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 (a \sec \left (f x + e\right )\right )^{m} \csc \left (f x + e\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 \sec \left (f x + e\right )\right )^{m} \csc \left (f x + e\right )^{n}, 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} \csc ^{n}{\left (e + f x \right )}\, 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 (a \sec \left (f x + e\right )\right )^{m} \csc \left (f x + e\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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