Optimal. Leaf size=89 \[ \frac{b \cos ^2(e+f x)^{\frac{m+1}{2}} (a \sec (e+f x))^{m+1} (b \csc (e+f x))^{n-1} \text{Hypergeometric2F1}\left (\frac{m+1}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{a f (1-n)} \]
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Rubi [A] time = 0.0942339, antiderivative size = 89, 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 = {2631, 2577} \[ \frac{b \cos ^2(e+f x)^{\frac{m+1}{2}} (a \sec (e+f x))^{m+1} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{m+1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2631
Rule 2577
Rubi steps
\begin{align*} \int (b \csc (e+f x))^n (a \sec (e+f x))^m \, dx &=\frac{\left (b^2 (a \cos (e+f x))^{1+m} (b \csc (e+f x))^{-1+n} (a \sec (e+f x))^{1+m} (b \sin (e+f x))^{-1+n}\right ) \int (a \cos (e+f x))^{-m} (b \sin (e+f x))^{-n} \, dx}{a^2}\\ &=\frac{b \cos ^2(e+f x)^{\frac{1+m}{2}} (b \csc (e+f x))^{-1+n} \, _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 = 0.187567, size = 283, normalized size = 3.18 \[ -\frac{b (n-3) (a \sec (e+f x))^m (b \csc (e+f x))^{n-1} F_1\left (\frac{1-n}{2};m,-m-n+1;\frac{3-n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (n-1) \left ((n-3) F_1\left (\frac{1-n}{2};m,-m-n+1;\frac{3-n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left ((m+n-1) F_1\left (\frac{3-n}{2};m,-m-n+2;\frac{5-n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+m F_1\left (\frac{3-n}{2};m+1,-m-n+1;\frac{5-n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.558, size = 0, normalized size = 0. \begin{align*} \int \left ( b\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 (b \csc \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 \csc \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 \csc{\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 \csc \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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