Optimal. Leaf size=81 \[ \frac {b \cos ^2(e+f x)^{5/4} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {5}{4},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right ) (c \sec (e+f x))^{5/2}}{c f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 = {2711, 2657}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2657
Rule 2711
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 = 11.93, size = 92, normalized size = 1.14 \begin {gather*} \frac {2 \cot (e+f x) (b \csc (e+f x))^n \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (3+2 n);\frac {1}{4} (7+2 n);\sec ^2(e+f x)\right ) (c \sec (e+f x))^{3/2} \left (-\tan ^2(e+f x)\right )^{\frac {1+n}{2}}}{f (3+2 n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.45, size = 0, normalized size = 0.00 \[\int \left (b \csc \left (f x +e \right )\right )^{n} \left (c \sec \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________