3.1.26 \(\int \cos ^{\frac {2}{3}}(a+b x) \, dx\) [26]

Optimal. Leaf size=53 \[ -\frac {3 \cos ^{\frac {5}{3}}(a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(a+b x)\right ) \sin (a+b x)}{5 b \sqrt {\sin ^2(a+b x)}} \]

[Out]

-3/5*cos(b*x+a)^(5/3)*hypergeom([1/2, 5/6],[11/6],cos(b*x+a)^2)*sin(b*x+a)/b/(sin(b*x+a)^2)^(1/2)

________________________________________________________________________________________

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 {5}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(a+b x)\right )}{5 b \sqrt {\sin ^2(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(2/3),x]

[Out]

(-3*Cos[a + b*x]^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*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 \cos ^{\frac {2}{3}}(a+b x) \, dx &=-\frac {3 \cos ^{\frac {5}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{5 b \sqrt {\sin ^2(a+b x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 53, normalized size = 1.00 \begin {gather*} -\frac {3 \cos ^{\frac {5}{3}}(a+b x) \csc (a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(2/3),x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \cos ^{\frac {2}{3}}\left (b x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^(2/3),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^(2/3), x)

________________________________________________________________________________________

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)^(2/3),x, algorithm="fricas")

[Out]

integral(cos(b*x + a)^(2/3), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^{\frac {2}{3}}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x)**(2/3), x)

________________________________________________________________________________________

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)^(2/3),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^(2/3), x)

________________________________________________________________________________________

Mupad [B]
time = 0.19, size = 42, normalized size = 0.79 \begin {gather*} -\frac {3\,{\cos \left (a+b\,x\right )}^{5/3}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{6};\ \frac {11}{6};\ {\cos \left (a+b\,x\right )}^2\right )}{5\,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)^(2/3),x)

[Out]

-(3*cos(a + b*x)^(5/3)*sin(a + b*x)*hypergeom([1/2, 5/6], 11/6, cos(a + b*x)^2))/(5*b*(sin(a + b*x)^2)^(1/2))

________________________________________________________________________________________