3.29 \(\int \frac{1}{\csc ^{\frac{2}{3}}(a+b x)} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0212965, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ \frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right )}{5 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{5}{3}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^(-2/3),x]

[Out]

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

Rule 3772

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

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

Mathematica [A]  time = 0.0934955, size = 51, normalized size = 0.96 \[ -\frac{\cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sin ^2(a+b x)^{5/6} \csc ^{\frac{5}{3}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^(-2/3),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.142, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( bx+a \right ) \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/csc(b*x+a)^(2/3),x)

[Out]

int(1/csc(b*x+a)^(2/3),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\csc \left (b x + a\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csc(b*x+a)^(2/3),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^(-2/3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\csc \left (b x + a\right )^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csc(b*x+a)^(2/3),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^(-2/3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\csc ^{\frac{2}{3}}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csc(b*x+a)**(2/3),x)

[Out]

Integral(csc(a + b*x)**(-2/3), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\csc \left (b x + a\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csc(b*x+a)^(2/3),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^(-2/3), x)