Optimal. Leaf size=81 \[ \frac{b \cos ^2(e+f x)^{5/4} (c \sec (e+f x))^{5/2} (b \csc (e+f x))^{n-1} \text{Hypergeometric2F1}\left (\frac{5}{4},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{c f (1-n)} \]
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Rubi [A] time = 0.114924, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2631, 2577} \[ \frac{b \cos ^2(e+f x)^{5/4} (c \sec (e+f x))^{5/2} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{5}{4},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{c f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2631
Rule 2577
Rubi steps
\begin{align*} \int (b \csc (e+f x))^n (c \sec (e+f x))^{3/2} \, dx &=\frac{\left (b^2 (c \cos (e+f x))^{5/2} (b \csc (e+f x))^{-1+n} (c \sec (e+f x))^{5/2} (b \sin (e+f x))^{-1+n}\right ) \int \frac{(b \sin (e+f x))^{-n}}{(c \cos (e+f x))^{3/2}} \, dx}{c^2}\\ &=\frac{b \cos ^2(e+f x)^{5/4} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac{5}{4},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right ) (c \sec (e+f x))^{5/2}}{c f (1-n)}\\ \end{align*}
Mathematica [A] time = 1.82734, size = 92, normalized size = 1.14 \[ \frac{2 \cot (e+f x) (c \sec (e+f x))^{3/2} \left (-\tan ^2(e+f x)\right )^{\frac{n+1}{2}} (b \csc (e+f x))^n \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{4} (2 n+3),\frac{1}{4} (2 n+7),\sec ^2(e+f x)\right )}{f (2 n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.159, size = 0, normalized size = 0. \begin{align*} \int \left ( b\csc \left ( fx+e \right ) \right ) ^{n} \left ( c\sec \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, 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 (c \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \csc \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 (\sqrt{c \sec \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{n} c \sec \left (f x + e\right ), 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 (c \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \csc \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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