Optimal. Leaf size=117 \[ -\frac {2^{m+\frac {1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1)}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2751, 2652, 2651} \[ -\frac {2^{m+\frac {1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1)}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2651
Rule 2652
Rule 2751
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}+\frac {(A+A m+B m) \int (a+a \sin (e+f x))^m \, dx}{1+m}\\ &=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}+\frac {\left ((A+A m+B m) (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^m \, dx}{1+m}\\ &=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} (A+A m+B m) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\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}{f (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.82, size = 275, normalized size = 2.35 \[ -\frac {\sin ^{-2 m}\left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\frac {2 \sqrt {2} A \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \cos ^{2 m+1}\left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )}{(2 m+1) \sqrt {1-\sin (e+f x)}}+\frac {\sqrt [4]{-1} B 2^{-2 m-1} e^{-\frac {3}{2} i (e+f x)} \left (-(-1)^{3/4} e^{-\frac {1}{2} i (e+f x)} \left (e^{i (e+f x)}+i\right )\right )^{2 m+1} \left ((m-1) e^{2 i (e+f x)} \, _2F_1\left (1,m;-m;-i e^{-i (e+f x)}\right )-(m+1) \, _2F_1\left (1,m+2;2-m;-i e^{-i (e+f x)}\right )\right )}{m^2-1}\right )}{f} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________