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

Optimal. Leaf size=67 \[ \frac {2\ 2^{5/6} a \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{5/6}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 67, 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 = {2652, 2651} \[ \frac {2\ 2^{5/6} a \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(4/3),x]

[Out]

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

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]

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]

Rubi steps

\begin {align*} \int (a+a \cos (c+d x))^{4/3} \, dx &=\frac {\left (a \sqrt [3]{a+a \cos (c+d x)}\right ) \int (1+\cos (c+d x))^{4/3} \, dx}{\sqrt [3]{1+\cos (c+d x)}}\\ &=\frac {2\ 2^{5/6} a \sqrt [3]{a+a \cos (c+d x)} \, _2F_1\left (-\frac {5}{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))^{5/6}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 69, normalized size = 1.03 \[ -\frac {6 \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )} \cot \left (\frac {1}{2} (c+d x)\right ) (a (\cos (c+d x)+1))^{4/3} \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(a*(1 + Cos[c + d*x]))^(4/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[(c + d*x)/2]^2]*Sqrt[
Sin[(c + d*x)/2]^2])/(11*d)

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {4}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {4}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (a +a \cos \left (d x +c \right )\right )^{\frac {4}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {4}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{4/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cos(c + d*x))^(4/3),x)

[Out]

int((a + a*cos(c + d*x))^(4/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos {\left (c + d x \right )} + a\right )^{\frac {4}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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