Optimal. Leaf size=89 \[ \frac {b \cos ^2(e+f x)^{\frac {m+1}{2}} (a \sec (e+f x))^{m+1} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {m+1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 2577
Rule 2631
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 = 0.19, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.87, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x +e \right )\right )^{n} \left (a \sec \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sec {\left (e + f x \right )}\right )^{m} \left (b \csc {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________