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(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)

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Rubi [A]  time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 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])

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 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]

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*}

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Mathematica [F]  time = 3.43, size = 0, normalized size = 0.00 \[ \int \cos ^{\frac {4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{\frac {4}{3}}\left (d x +c \right )\right ) \left (a +a \cos \left (d x +c \right )\right )^{\frac {2}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \cos \left (d x + c\right )^{\frac {4}{3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^{4/3}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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