\(\int (a+b \cos (e+f x))^m (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)) \, dx\) [392]

Optimal result

Integrand size = 35, antiderivative size = 215 \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\frac {4 \sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}} \]

[Out]

4*C*AppellF1(1/2,-m,-3/2,3/2,b*(1-cos(f*x+e))/(a+b),1/2-1/2*cos(f*x+e))*(a+b*cos(f*x+e))^m*sin(f*x+e)*2^(1/2)/
f/(((a+b*cos(f*x+e))/(a+b))^m)/(1+cos(f*x+e))^(1/2)+2*(A-C)*AppellF1(1/2,-m,-1/2,3/2,b*(1-cos(f*x+e))/(a+b),1/
2-1/2*cos(f*x+e))*(a+b*cos(f*x+e))^m*sin(f*x+e)*2^(1/2)/f/(((a+b*cos(f*x+e))/(a+b))^m)/(1+cos(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3096, 2834, 144, 143, 2863} \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}+\frac {4 \sqrt {2} C \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}} \]

[In]

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

[Out]

(4*Sqrt[2]*C*AppellF1[1/2, -3/2, -m, 3/2, (1 - Cos[e + f*x])/2, (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e +
f*x])^m*Sin[e + f*x])/(f*Sqrt[1 + Cos[e + f*x]]*((a + b*Cos[e + f*x])/(a + b))^m) + (2*Sqrt[2]*(A - C)*Appell
F1[1/2, -1/2, -m, 3/2, (1 - Cos[e + f*x])/2, (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*
x])/(f*Sqrt[1 + Cos[e + f*x]]*((a + b*Cos[e + f*x])/(a + b))^m)

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 2834

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

Rule 2863

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

Rule 3096

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Dist[A - C, Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x], x] + Dist[C, Int[(a
+ b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A - B + C, 0] &&
!IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = (A-C) \int (1+\cos (e+f x)) (a+b \cos (e+f x))^m \, dx+C \int (1+\cos (e+f x))^2 (a+b \cos (e+f x))^m \, dx \\ & = -\frac {(C \sin (e+f x)) \text {Subst}\left (\int \frac {(1+x)^{3/2} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}+\frac {((-A+C) \sin (e+f x)) \text {Subst}\left (\int \frac {\sqrt {1+x} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}} \\ & = -\frac {\left (C (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \text {Subst}\left (\int \frac {(1+x)^{3/2} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}+\frac {\left ((-A+C) (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}} \\ & = \frac {4 \sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}} \\ \end{align*}

Mathematica [F]

\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (A +\left (A +C \right ) \cos \left (f x +e \right )+C \left (\cos ^{2}\left (f x +e \right )\right )\right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + {\left (A + C\right )} \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + {\left (A + C\right )} \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int { {\left (C \cos \left (f x + e\right )^{2} + {\left (A + C\right )} \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int {\left (a+b\,\cos \left (e+f\,x\right )\right )}^m\,\left (C\,{\cos \left (e+f\,x\right )}^2+\left (A+C\right )\,\cos \left (e+f\,x\right )+A\right ) \,d x \]

[In]

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

[Out]

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