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

Optimal. Leaf size=58 \[ -\frac{3 \sin (a+b x) (c \cos (a+b x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(a+b x)\right )}{5 b c \sqrt{\sin ^2(a+b x)}} \]

[Out]

(-3*(c*Cos[a + b*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*c*Sqrt[Sin[a +
b*x]^2])

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Rubi [A]  time = 0.01522, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ -\frac{3 \sin (a+b x) (c \cos (a+b x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(a+b x)\right )}{5 b c \sqrt{\sin ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(2/3),x]

[Out]

(-3*(c*Cos[a + b*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*c*Sqrt[Sin[a +
b*x]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (c \cos (a+b x))^{2/3} \, dx &=-\frac{3 (c \cos (a+b x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{5 b c \sqrt{\sin ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0381831, size = 55, normalized size = 0.95 \[ -\frac{3 \sqrt{\sin ^2(a+b x)} \cot (a+b x) (c \cos (a+b x))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(a+b x)\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(2/3),x]

[Out]

(-3*(c*Cos[a + b*x])^(2/3)*Cot[a + b*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sqrt[Sin[a + b*x]^2]
)/(5*b)

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int \left ( c\cos \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*cos(b*x + a))^(2/3), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(2/3), x)