3.1.27 \(\int \sqrt [3]{\cos (a+b x)} \, dx\) [27]

Optimal. Leaf size=53 \[ -\frac {3 \cos ^{\frac {4}{3}}(a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(a+b x)\right ) \sin (a+b x)}{4 b \sqrt {\sin ^2(a+b x)}} \]

[Out]

-3/4*cos(b*x+a)^(4/3)*hypergeom([1/2, 2/3],[5/3],cos(b*x+a)^2)*sin(b*x+a)/b/(sin(b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \begin {gather*} -\frac {3 \sin (a+b x) \cos ^{\frac {4}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(a+b x)\right )}{4 b \sqrt {\sin ^2(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(1/3),x]

[Out]

(-3*Cos[a + b*x]^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[a + b*x]^2]*Sin[a + b*x])/(4*b*Sqrt[Sin[a + b*x]^2
])

Rule 2722

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 53, normalized size = 1.00 \begin {gather*} -\frac {3 \cos ^{\frac {4}{3}}(a+b x) \csc (a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(1/3),x]

[Out]

(-3*Cos[a + b*x]^(4/3)*Csc[a + b*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[a + b*x]^2]*Sqrt[Sin[a + b*x]^2])/(4*
b)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \cos ^{\frac {1}{3}}\left (b x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^(1/3),x)

[Out]

int(cos(b*x+a)^(1/3),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(cos(b*x+a)^(1/3),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^(1/3), 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(cos(b*x+a)^(1/3),x, algorithm="fricas")

[Out]

integral(cos(b*x + a)^(1/3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{\cos {\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**(1/3),x)

[Out]

Integral(cos(a + b*x)**(1/3), 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(cos(b*x+a)^(1/3),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^(1/3), x)

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Mupad [B]
time = 0.18, size = 42, normalized size = 0.79 \begin {gather*} -\frac {3\,{\cos \left (a+b\,x\right )}^{4/3}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {2}{3};\ \frac {5}{3};\ {\cos \left (a+b\,x\right )}^2\right )}{4\,b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^(1/3),x)

[Out]

-(3*cos(a + b*x)^(4/3)*sin(a + b*x)*hypergeom([1/2, 2/3], 5/3, cos(a + b*x)^2))/(4*b*(sin(a + b*x)^2)^(1/2))

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