Optimal. Leaf size=83 \[ -\frac {(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2617} \[ -\frac {(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2617
Rubi steps
\begin {align*} \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx &=-\frac {(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1+m+n);\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (1+m+n)}}{b f (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.89, size = 306, normalized size = 3.69 \[ -\frac {a (m+n-3) (a \csc (e+f x))^{m-1} (b \cot (e+f x))^n F_1\left (\frac {1}{2} (-m-n+1);-n,1-m;\frac {1}{2} (-m-n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (m+n-1) \left (2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (n F_1\left (\frac {1}{2} (-m-n+3);1-n,1-m;\frac {1}{2} (-m-n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-(m-1) F_1\left (\frac {1}{2} (-m-n+3);-n,2-m;\frac {1}{2} (-m-n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )+(m+n-3) F_1\left (\frac {1}{2} (-m-n+1);-n,1-m;\frac {1}{2} (-m-n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.95, size = 0, normalized size = 0.00 \[ \int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \csc \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________