3.198 \(\int (a+a \cos (e+f x))^m (A+C \cos ^2(e+f x)) \, dx\)

Optimal. Leaf size=170 \[ \frac{2^{m+\frac{1}{2}} \left (A \left (m^2+3 m+2\right )+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac{C \sin (e+f x) (a \cos (e+f x)+a)^m}{f \left (m^2+3 m+2\right )}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]

[Out]

-((C*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(2 + 3*m + m^2))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])
/(a*f*(2 + m)) + (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2))*(1 + Cos[e + f*x])^(-1/2 - m)*(a + a*Cos[e
+ f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Cos[e + f*x])/2]*Sin[e + f*x])/(f*(1 + m)*(2 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.208652, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3024, 2751, 2652, 2651} \[ \frac{2^{m+\frac{1}{2}} \left (A \left (m^2+3 m+2\right )+C \left (m^2+m+1\right )\right ) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right )}{f (m+1) (m+2)}-\frac{C \sin (e+f x) (a \cos (e+f x)+a)^m}{f \left (m^2+3 m+2\right )}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[e + f*x])^m*(A + C*Cos[e + f*x]^2),x]

[Out]

-((C*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(2 + 3*m + m^2))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])
/(a*f*(2 + m)) + (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2))*(1 + Cos[e + f*x])^(-1/2 - m)*(a + a*Cos[e
+ f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Cos[e + f*x])/2]*Sin[e + f*x])/(f*(1 + m)*(2 + m))

Rule 3024

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp
[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x]
)^m*Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] &&  !LtQ[
m, -1]

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2652

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart
[n])/(1 + (b*Sin[c + d*x])/a)^FracPart[n], Int[(1 + (b*Sin[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 2651

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(2^(n + 1/2)*a^(n - 1/2)*b*Cos[c + d*x]*Hy
pergeometric2F1[1/2, 1/2 - n, 3/2, (1*(1 - (b*Sin[c + d*x])/a))/2])/(d*Sqrt[a + b*Sin[c + d*x]]), x] /; FreeQ[
{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rubi steps

\begin{align*} \int (a+a \cos (e+f x))^m \left (A+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 (2+m))-a C \cos (e+f x)) \, dx}{a (2+m)}\\ &=-\frac{C (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\frac{C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{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 \cos (e+f x))^m \, dx}{(1+m) (2+m)}\\ &=-\frac{C (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\frac{C (a+a \cos (e+f x))^{1+m} \sin (e+f x)}{a f (2+m)}+\frac{\left (\left (C \left (1+m+m^2\right )+A \left (2+3 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 (a+a \cos (e+f x))^m \sin (e+f x)}{f \left (2+3 m+m^2\right )}+\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 (C \left (1+m+m^2\right )+A \left (2+3 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 = 1.4133, size = 242, normalized size = 1.42 \[ \frac{i 4^{-m-1} \left (1+e^{i (e+f x)}\right ) e^{-i (m+2) (e+f x)} \left (e^{-\frac{1}{2} i (e+f x)} \left (1+e^{i (e+f x)}\right )\right )^{2 m} \cos ^{-2 m}\left (\frac{1}{2} (e+f x)\right ) (a (\cos (e+f x)+1))^m \left ((m+2) e^{i (m+2) (e+f x)} \left (2 (m-2) (2 A+C) \, _2F_1\left (1,m+1;1-m;-e^{i (e+f x)}\right )+C m e^{2 i (e+f x)} \, _2F_1\left (1,m+3;3-m;-e^{i (e+f x)}\right )\right )+C (m-2) m e^{i m (e+f x)} \, _2F_1\left (1,m-1;-m-1;-e^{i (e+f x)}\right )\right )}{f (m-2) m (m+2)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Cos[e + f*x])^m*(A + C*Cos[e + f*x]^2),x]

[Out]

(I*4^(-1 - m)*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))/E^((I/2)*(e + f*x)))^(2*m)*(a*(1 + Cos[e + f*x]))^m
*(C*E^(I*m*(e + f*x))*(-2 + m)*m*Hypergeometric2F1[1, -1 + m, -1 - m, -E^(I*(e + f*x))] + E^(I*(2 + m)*(e + f*
x))*(2 + m)*(2*(2*A + C)*(-2 + m)*Hypergeometric2F1[1, 1 + m, 1 - m, -E^(I*(e + f*x))] + C*E^((2*I)*(e + f*x))
*m*Hypergeometric2F1[1, 3 + m, 3 - m, -E^(I*(e + f*x))])))/(E^(I*(2 + m)*(e + f*x))*f*(-2 + m)*m*(2 + m)*Cos[(
e + f*x)/2]^(2*m))

________________________________________________________________________________________

Maple [F]  time = 1.687, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\cos \left ( fx+e \right ) \right ) ^{m} \left ( A+C \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(f*x+e))^m*(A+C*cos(f*x+e)^2),x)

[Out]

int((a+a*cos(f*x+e))^m*(A+C*cos(f*x+e)^2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (f x + e\right )^{2} + A\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]

integrate((a+a*cos(f*x+e))^m*(A+C*cos(f*x+e)^2),x, algorithm="maxima")

[Out]

integrate((C*cos(f*x + e)^2 + A)*(a*cos(f*x + e) + a)^m, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (f x + e\right )^{2} + A\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]

integrate((a+a*cos(f*x+e))^m*(A+C*cos(f*x+e)^2),x, algorithm="fricas")

[Out]

integral((C*cos(f*x + e)^2 + A)*(a*cos(f*x + e) + a)^m, x)

________________________________________________________________________________________

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 (A + C \cos ^{2}{\left (e + f x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(f*x+e))**m*(A+C*cos(f*x+e)**2),x)

[Out]

Integral((a*(cos(e + f*x) + 1))**m*(A + C*cos(e + f*x)**2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (f x + e\right )^{2} + A\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]

integrate((a+a*cos(f*x+e))^m*(A+C*cos(f*x+e)^2),x, algorithm="giac")

[Out]

integrate((C*cos(f*x + e)^2 + A)*(a*cos(f*x + e) + a)^m, x)