Optimal. Leaf size=173 \[ \frac{2^{m+\frac{1}{2}} \left (B m (m+2)+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac{(C-B (m+2)) \sin (e+f x) (a \cos (e+f x)+a)^m}{f (m+1) (m+2)}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.209168, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3023, 2751, 2652, 2651} \[ \frac{2^{m+\frac{1}{2}} \left (B m (m+2)+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac{(C-B (m+2)) \sin (e+f x) (a \cos (e+f x)+a)^m}{f (m+1) (m+2)}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3023
Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a+a \cos (e+f x))^m \left (B \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx &=\frac{C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac{\int (a+a \cos (e+f x))^m (a C (1+m)-a (C-B (2+m)) \cos (e+f x)) \, dx}{a (2+m)}\\ &=-\frac{(C-B (2+m)) (a+a \cos (e+f x))^m \sin (e+f x)}{f (1+m) (2+m)}+\frac{C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac{\left (B m (2+m)+C \left (1+m+m^2\right )\right ) \int (a+a \cos (e+f x))^m \, dx}{(1+m) (2+m)}\\ &=-\frac{(C-B (2+m)) (a+a \cos (e+f x))^m \sin (e+f x)}{f (1+m) (2+m)}+\frac{C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac{\left (\left (B m (2+m)+C \left (1+m+m^2\right )\right ) (1+\cos (e+f x))^{-m} (a+a \cos (e+f x))^m\right ) \int (1+\cos (e+f x))^m \, dx}{(1+m) (2+m)}\\ &=-\frac{(C-B (2+m)) (a+a \cos (e+f x))^m \sin (e+f x)}{f (1+m) (2+m)}+\frac{C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac{2^{\frac{1}{2}+m} \left (B m (2+m)+C \left (1+m+m^2\right )\right ) (1+\cos (e+f x))^{-\frac{1}{2}-m} (a+a \cos (e+f x))^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right ) \sin (e+f x)}{f (1+m) (2+m)}\\ \end{align*}
Mathematica [C] time = 45.8027, size = 356, normalized size = 2.06 \[ \frac{i 4^{-m-1} e^{-2 i (e+f x)} \left (1+e^{i (e+f x)}\right )^{-2 m} \left (e^{-\frac{1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right )\right )^{2 m} \cos ^{-2 m}\left (\frac{1}{2} (e+f x)\right ) (a (\cos (e+f x)+1))^m \left ((m+2) e^{i (e+f x)} \left ((m+1) e^{i (e+f x)} \left (2 B (m-2) m e^{i (e+f x)} \, _2F_1\left (1-m,-2 m;2-m;-e^{i (e+f x)}\right )+C (m-1) \left (m e^{2 i (e+f x)} \, _2F_1\left (2-m,-2 m;3-m;-e^{i (e+f x)}\right )+2 (m-2) \, _2F_1\left (-2 m,-m;1-m;-e^{i (e+f x)}\right )\right )\right )+2 B m \left (m^2-3 m+2\right ) \, _2F_1\left (-m-1,-2 m;-m;-e^{i (e+f x)}\right )\right )+C m \left (m^3-2 m^2-m+2\right ) \, _2F_1\left (-m-2,-2 m;-m-1;-e^{i (e+f x)}\right )\right )}{f (m-2) (m-1) m (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.662, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\cos \left ( fx+e \right ) \right ) ^{m} \left ( B\cos \left ( fx+e \right ) +C \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \, 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 (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right )\right )}{\left (a \cos \left (f x + e\right ) + a\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 (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right )\right )}{\left (a \cos \left (f x + e\right ) + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\cos{\left (e + f x \right )} + 1\right )\right )^{m} \left (B + C \cos{\left (e + f x \right )}\right ) \cos{\left (e + f x \right )}\, dx \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 (C \cos \left (f x + e\right )^{2} + B \cos \left (f x + e\right )\right )}{\left (a \cos \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]