\(\int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx\) [290]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{3},-\frac {1}{6},\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}} \]

[Out]

2*2^(1/6)*AppellF1(1/2,-5/3,-1/6,3/2,1-cos(d*x+c),1/2-1/2*cos(d*x+c))*(a+a*cos(d*x+c))^(2/3)*sin(d*x+c)/d/(1+c
os(d*x+c))^(7/6)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2866, 2864, 138} \[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{3},-\frac {1}{6},\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]

[In]

Int[Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3),x]

[Out]

(2*2^(1/6)*AppellF1[1/2, -5/3, -1/6, 3/2, 1 - Cos[c + d*x], (1 - Cos[c + d*x])/2]*(a + a*Cos[c + d*x])^(2/3)*S
in[c + d*x])/(d*(1 + Cos[c + d*x])^(7/6))

Rule 138

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

Rule 2864

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

Rule 2866

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

Rubi steps \begin{align*} \text {integral}& = \frac {(a+a \cos (c+d x))^{2/3} \int \cos ^{\frac {5}{3}}(c+d x) (1+\cos (c+d x))^{2/3} \, dx}{(1+\cos (c+d x))^{2/3}} \\ & = \frac {\left ((a+a \cos (c+d x))^{2/3} \sin (c+d x)\right ) \text {Subst}\left (\int \frac {(1-x)^{5/3} \sqrt [6]{2-x}}{\sqrt {x}} \, dx,x,1-\cos (c+d x)\right )}{d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{7/6}} \\ & = \frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{3},-\frac {1}{6},\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}} \\ \end{align*}

Mathematica [F]

\[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx \]

[In]

Integrate[Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3),x]

[Out]

Integrate[Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3), x]

Maple [F]

\[\int \left (\cos ^{\frac {5}{3}}\left (d x +c \right )\right ) \left (a +\cos \left (d x +c \right ) a \right )^{\frac {2}{3}}d x\]

[In]

int(cos(d*x+c)^(5/3)*(a+cos(d*x+c)*a)^(2/3),x)

[Out]

int(cos(d*x+c)^(5/3)*(a+cos(d*x+c)*a)^(2/3),x)

Fricas [F]

\[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \cos \left (d x + c\right )^{\frac {5}{3}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/3)*(a+a*cos(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

integral((a*cos(d*x + c) + a)^(2/3)*cos(d*x + c)^(5/3), x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(5/3)*(a+a*cos(d*x+c))**(2/3),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \cos \left (d x + c\right )^{\frac {5}{3}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/3)*(a+a*cos(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((a*cos(d*x + c) + a)^(2/3)*cos(d*x + c)^(5/3), x)

Giac [F]

\[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \cos \left (d x + c\right )^{\frac {5}{3}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/3)*(a+a*cos(d*x+c))^(2/3),x, algorithm="giac")

[Out]

integrate((a*cos(d*x + c) + a)^(2/3)*cos(d*x + c)^(5/3), x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^{\frac {5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx=\int {\cos \left (c+d\,x\right )}^{5/3}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]

[In]

int(cos(c + d*x)^(5/3)*(a + a*cos(c + d*x))^(2/3),x)

[Out]

int(cos(c + d*x)^(5/3)*(a + a*cos(c + d*x))^(2/3), x)