Optimal. Leaf size=112 \[ \frac{c 2^{\frac{1}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left (\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.157636, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2745, 2689, 70, 69} \[ \frac{c 2^{\frac{1}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left (\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2745
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx &=\left (\cos ^{-2 m}(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \cos ^{2 m}(e+f x) (c-c \sin (e+f x))^{-2 m} \, dx\\ &=\frac{\left (c^2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac{1}{2} (-1-2 m)+m} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m)}\right ) \operatorname{Subst}\left (\int (c-c x)^{-2 m+\frac{1}{2} (-1+2 m)} (c+c x)^{\frac{1}{2} (-1+2 m)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (2^{-\frac{1}{2}-m} c^2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-\frac{1}{2}+\frac{1}{2} (-1-2 m)} \left (\frac{c-c \sin (e+f x)}{c}\right )^{\frac{1}{2}+m} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{-2 m+\frac{1}{2} (-1+2 m)} (c+c x)^{\frac{1}{2} (-1+2 m)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{2^{\frac{1}{2}-m} c \cos (e+f x) \, _2F_1\left (\frac{1}{2} (1+2 m),\frac{1}{2} (1+2 m);\frac{1}{2} (3+2 m);\frac{1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac{1}{2}+m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{f (1+2 m)}\\ \end{align*}
Mathematica [C] time = 2.93898, size = 388, normalized size = 3.46 \[ \frac{2^{1-m} (2 m-3) \sin ^2\left (\frac{1}{8} (2 e+2 f x+3 \pi )\right ) \cos ^{1-2 m}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m F_1\left (\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left (\frac{1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right )\right ) (c-c \sin (e+f x))^{-m} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{2 m}}{f (2 m-1) \left (2 \sin ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right ) \left (2 m F_1\left (\frac{3}{2}-m;1-2 m,1;\frac{5}{2}-m;\tan ^2\left (\frac{1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right )\right )+F_1\left (\frac{3}{2}-m;-2 m,2;\frac{5}{2}-m;\tan ^2\left (\frac{1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right )\right )\right )+(2 m-3) \cos ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right ) F_1\left (\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left (\frac{1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.463, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}}{ \left ( c-c\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 \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\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 (\frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\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 \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]