Optimal. Leaf size=84 \[ \frac {b \cos ^2(e+f x)^{\frac {m+1}{2}} \sec ^{m+1}(e+f x) (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {m+1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2631, 2577} \[ \frac {b \cos ^2(e+f x)^{\frac {m+1}{2}} \sec ^{m+1}(e+f x) (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {m+1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2577
Rule 2631
Rubi steps
\begin {align*} \int (b \csc (e+f x))^n \sec ^m(e+f x) \, dx &=\left (b^2 \cos ^{1+m}(e+f x) (b \csc (e+f x))^{-1+n} \sec ^{1+m}(e+f x) (b \sin (e+f x))^{-1+n}\right ) \int \cos ^{-m}(e+f x) (b \sin (e+f x))^{-n} \, dx\\ &=\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 ) \sec ^{1+m}(e+f x)}{f (1-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.32, size = 281, normalized size = 3.35 \[ -\frac {b (n-3) \sec ^m(e+f x) (b \csc (e+f x))^{n-1} F_1\left (\frac {1-n}{2};m,-m-n+1;\frac {3-n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (n-1) \left ((n-3) F_1\left (\frac {1-n}{2};m,-m-n+1;\frac {3-n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left ((m+n-1) F_1\left (\frac {3-n}{2};m,-m-n+2;\frac {5-n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+m F_1\left (\frac {3-n}{2};m+1,-m-n+1;\frac {5-n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 3.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.88, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x +e \right )\right )^{n} \left (\sec ^{m}\left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc {\left (e + f x \right )}\right )^{n} \sec ^{m}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________