3.1.12 \(\int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx\) [12]

Optimal. Leaf size=65 \[ \frac {\sqrt [6]{1+\cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{\sqrt [6]{2} d (a+a \cos (c+d x))^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {2731, 2730} \begin {gather*} \frac {\sin (c+d x) \sqrt [6]{\cos (c+d x)+1} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right )}{\sqrt [6]{2} d (a \cos (c+d x)+a)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(-2/3),x]

[Out]

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

Rule 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 2731

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[a^IntPart[n]*((a + b*Sin[c + d*x])^FracPart
[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n]), Int[(1 + (b/a)*Sin[c + d*x])^n, 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 \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx &=\frac {(1+\cos (c+d x))^{2/3} \int \frac {1}{(1+\cos (c+d x))^{2/3}} \, dx}{(a+a \cos (c+d x))^{2/3}}\\ &=\frac {\sqrt [6]{1+\cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{\sqrt [6]{2} d (a+a \cos (c+d x))^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 67, normalized size = 1.03 \begin {gather*} \frac {6 \cot \left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{d (a (1+\cos (c+d x)))^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Cot[(c + d*x)/2]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(d*(a*(1 +
Cos[c + d*x]))^(2/3))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +a \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*cos(d*x+c))^(2/3),x)

[Out]

int(1/(a+a*cos(d*x+c))^(2/3),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \cos {\left (c + d x \right )} + a\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))^(2/3),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*cos(c + d*x))^(2/3),x)

[Out]

int(1/(a + a*cos(c + d*x))^(2/3), x)

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