3.289 \(\int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx\)

Optimal. Leaf size=79 \[ \frac{2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]

[Out]

(2*2^(1/6)*AppellF1[1/2, -4/3, -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))

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Rubi [A]  time = 0.121835, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2787, 2785, 133} \[ \frac{2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3),x]

[Out]

(2*2^(1/6)*AppellF1[1/2, -4/3, -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 2787

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a^In
tPart[m]*(a + b*Sin[e + f*x])^FracPart[m])/(1 + (b*Sin[e + f*x])/a)^FracPart[m], Int[(1 + (b*Sin[e + f*x])/a)^
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]

Rule 2785

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] &&  !In
tegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]

Rule 133

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

Rubi steps

\begin{align*} \int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx &=\frac{(a+a \cos (c+d x))^{2/3} \int \cos ^{\frac{4}{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 ) \operatorname{Subst}\left (\int \frac{(1-x)^{4/3} \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} F_1\left (\frac{1}{2};-\frac{4}{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]  time = 3.28333, size = 0, normalized size = 0. \[ \int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3),x]

[Out]

Integrate[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3), x]

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Maple [F]  time = 0.168, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}} \left ( a+\cos \left ( dx+c \right ) a \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(4/3)*(a+cos(d*x+c)*a)^(2/3),x)

[Out]

int(cos(d*x+c)^(4/3)*(a+cos(d*x+c)*a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}} \cos \left (d x + c\right )^{\frac{4}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(4/3)*(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)^(4/3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(4/3)*(a+a*cos(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(4/3)*(a+a*cos(d*x+c))**(2/3),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(4/3)*(a+a*cos(d*x+c))^(2/3),x, algorithm="giac")

[Out]

Timed out