Optimal. Leaf size=87 \[ -\frac{(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac{1}{2} (-m+n+1)} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{2} (-m+n+1),\frac{n+3}{2},\cos ^2(e+f x)\right )}{b f (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0966597, antiderivative size = 87, normalized size of antiderivative = 1., 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 = {2603, 2617} \[ -\frac{(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac{1}{2} (-m+n+1)} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2603
Rule 2617
Rubi steps
\begin{align*} \int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx &=\left (\left (\frac{\csc (e+f x)}{a}\right )^m (a \sin (e+f x))^m\right ) \int (b \cot (e+f x))^n \left (\frac{\csc (e+f x)}{a}\right )^{-m} \, dx\\ &=-\frac{(b \cot (e+f x))^{1+n} \, _2F_1\left (\frac{1+n}{2},\frac{1}{2} (1-m+n);\frac{3+n}{2};\cos ^2(e+f x)\right ) (a \sin (e+f x))^m \sin ^2(e+f x)^{\frac{1}{2} (1-m+n)}}{b f (1+n)}\\ \end{align*}
Mathematica [C] time = 1.72175, size = 289, normalized size = 3.32 \[ \frac{(m-n+3) \sin (e+f x) (a \sin (e+f x))^m (b \cot (e+f x))^n F_1\left (\frac{1}{2} (m-n+1);-n,m+1;\frac{1}{2} (m-n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (m-n+1) \left ((m-n+3) F_1\left (\frac{1}{2} (m-n+1);-n,m+1;\frac{1}{2} (m-n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left (n F_1\left (\frac{1}{2} (m-n+3);1-n,m+1;\frac{1}{2} (m-n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(m+1) F_1\left (\frac{1}{2} (m-n+3);-n,m+2;\frac{1}{2} (m-n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.175, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cot \left ( fx+e \right ) \right ) ^{n} \left ( a\sin \left ( fx+e \right ) \right ) ^{m}\, 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 \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{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 (\left (b \cot \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]