Optimal. Leaf size=68 \[ \frac {\cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left (1,m+\frac {1}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt {a-a \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2745, 2667, 68} \[ \frac {\cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left (1,m+\frac {1}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt {a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 2667
Rule 2745
Rubi steps
\begin {align*} \int \frac {(c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx &=\frac {\cos (e+f x) \int \sec (e+f x) (c+c \sin (e+f x))^{\frac {1}{2}+m} \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {(c \cos (e+f x)) \operatorname {Subst}\left (\int \frac {(c+x)^{-\frac {1}{2}+m}}{c-x} \, dx,x,c \sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) \, _2F_1\left (1,\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt {a-a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.44, size = 157, normalized size = 2.31 \[ \frac {2^{-2 m-\frac {3}{2}} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c (\sin (e+f x)+1))^m \left (4^m \, _2F_1\left (1,2 m;2 m+1;\sin \left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )-\sec ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right )^{2 m} \, _2F_1\left (2 m,2 m;2 m+1;\frac {1}{2} \left (1-\tan ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right )\right )\right )\right )}{f m \sqrt {a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{a \sin \left (f x + e\right ) - a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt {-a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (c +c \sin \left (f x +e \right )\right )^{m}}{\sqrt {a -a \sin \left (f x +e \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt {-a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+c\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {a-a\,\sin \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________