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

Optimal. Leaf size=66 \[ \frac{2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]

[Out]

(2*2^(1/6)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(d
*(1 + Cos[c + d*x])^(7/6))

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Rubi [A]  time = 0.0344978, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2652, 2651} \[ \frac{2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(2/3),x]

[Out]

(2*2^(1/6)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Cos[c + d*x])/2]*Sin[c + d*x])/(d
*(1 + Cos[c + d*x])^(7/6))

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 (c+d x))^{2/3} \, dx &=\frac{(a+a \cos (c+d x))^{2/3} \int (1+\cos (c+d x))^{2/3} \, dx}{(1+\cos (c+d x))^{2/3}}\\ &=\frac{2 \sqrt [6]{2} (a+a \cos (c+d x))^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}}\\ \end{align*}

Mathematica [A]  time = 0.0649784, size = 69, normalized size = 1.05 \[ -\frac{6 \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) (a (\cos (c+d x)+1))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(a*(1 + Cos[c + d*x]))^(2/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[(c + d*x)/2]^2]*Sqrt[S
in[(c + d*x)/2]^2])/(7*d)

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Maple [F]  time = 0.21, size = 0, normalized size = 0. \begin{align*} \int \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((a+cos(d*x+c)*a)^(2/3),x)

[Out]

int((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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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