Optimal. Leaf size=81 \[ \frac{b \cos ^2(e+f x)^{3/4} (c \sec (e+f x))^{3/2} (b \csc (e+f x))^{n-1} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{c f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0971722, 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} \[ \frac{b \cos ^2(e+f x)^{3/4} (c \sec (e+f x))^{3/2} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{3}{4},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{c f (1-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2631
Rule 2577
Rubi steps
\begin{align*} \int (b \csc (e+f x))^n \sqrt{c \sec (e+f x)} \, dx &=\frac{\left (b^2 (c \cos (e+f x))^{3/2} (b \csc (e+f x))^{-1+n} (c \sec (e+f x))^{3/2} (b \sin (e+f x))^{-1+n}\right ) \int \frac{(b \sin (e+f x))^{-n}}{\sqrt{c \cos (e+f x)}} \, dx}{c^2}\\ &=\frac{b \cos ^2(e+f x)^{3/4} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac{3}{4},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right ) (c \sec (e+f x))^{3/2}}{c f (1-n)}\\ \end{align*}
Mathematica [A] time = 1.61025, size = 90, normalized size = 1.11 \[ \frac{2 \cot (e+f x) \sqrt{c \sec (e+f x)} \left (-\tan ^2(e+f x)\right )^{\frac{n+1}{2}} (b \csc (e+f x))^n \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{4} (2 n+1),\frac{1}{4} (2 n+5),\sec ^2(e+f x)\right )}{2 f n+f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.152, size = 0, normalized size = 0. \begin{align*} \int \left ( b\csc \left ( fx+e \right ) \right ) ^{n}\sqrt{c\sec \left ( fx+e \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \sec \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc{\left (e + f x \right )}\right )^{n} \sqrt{c \sec{\left (e + f x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \sec \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]