Optimal. Leaf size=171 \[ -\frac {2^{m+\frac {1}{2}} \left (A \left (m^2+3 m+2\right )+C \left (m^2+m+1\right )\right ) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1) (m+2)}+\frac {C \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left (m^2+3 m+2\right )}-\frac {C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3024, 2751, 2652, 2651} \[ -\frac {2^{m+\frac {1}{2}} \left (A \left (m^2+3 m+2\right )+C \left (m^2+m+1\right )\right ) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1) (m+2)}+\frac {C \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left (m^2+3 m+2\right )}-\frac {C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2651
Rule 2652
Rule 2751
Rule 3024
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right ) \, dx &=-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}+\frac {\int (a+a \sin (e+f x))^m (a (C (1+m)+A (2+m))-a C \sin (e+f x)) \, dx}{a (2+m)}\\ &=\frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}+\frac {\left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) \int (a+a \sin (e+f x))^m \, dx}{(1+m) (2+m)}\\ &=\frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}+\frac {\left (\left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^m \, dx}{(1+m) (2+m)}\\ &=\frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {2^{\frac {1}{2}+m} \left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) \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) (2+m)}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.54, size = 385, normalized size = 2.25 \[ -\frac {\sin ^{-2 m}\left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\frac {4 \sqrt {2} A \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \cos ^{2 m+1}\left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )}{(2 m+1) \sqrt {1-\sin (e+f x)}}+\frac {C 2^{-2 m-1} e^{-3 i (e+f x)} \left (1-i e^{i (e+f x)}\right ) \left (-(-1)^{3/4} e^{-\frac {1}{2} i (e+f x)} \left (e^{i (e+f x)}+i\right )\right )^{2 m} \left ((m-2) e^{4 i (e+f x)} \, _2F_1\left (1,m-1;-m-1;-i e^{-i (e+f x)}\right )+(m+2) \, _2F_1\left (1,m+3;3-m;-i e^{-i (e+f x)}\right )\right )}{m^2-4}+\frac {2 \sqrt {2} C \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \cos ^{2 m+1}\left (\frac {1}{4} (2 e+2 f x-\pi )\right ) \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )}{(2 m+1) \sqrt {1-\sin (e+f x)}}\right )}{2 f} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (C \cos \left (f x + e\right )^{2} - A - C\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 4.53, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (C\,{\sin \left (e+f\,x\right )}^2+A\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + C \sin ^{2}{\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________