\(\int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx\) [28]

Optimal result

Integrand size = 10, antiderivative size = 53 \[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(a+b x)\right )}{4 b \sqrt {\cos ^2(a+b x)} \csc ^{\frac {4}{3}}(a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3857, 2722} \[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(a+b x)\right )}{4 b \sqrt {\cos ^2(a+b x)} \csc ^{\frac {4}{3}}(a+b x)} \]

[In]

Int[Csc[a + b*x]^(-1/3),x]

[Out]

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

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]

Rule 3857

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]

Rubi steps \begin{align*} \text {integral}& = \csc ^{\frac {2}{3}}(a+b x) \sin ^{\frac {2}{3}}(a+b x) \int \sqrt [3]{\sin (a+b x)} \, dx \\ & = \frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(a+b x)\right )}{4 b \sqrt {\cos ^2(a+b x)} \csc ^{\frac {4}{3}}(a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=-\frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\cos ^2(a+b x)\right )}{b \csc ^{\frac {4}{3}}(a+b x) \sin ^2(a+b x)^{2/3}} \]

[In]

Integrate[Csc[a + b*x]^(-1/3),x]

[Out]

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

Maple [F]

\[\int \frac {1}{\csc \left (x b +a \right )^{\frac {1}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integral(csc(b*x + a)^(-1/3), x)

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\int \frac {1}{\sqrt [3]{\csc {\left (a + b x \right )}}}\, dx \]

[In]

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

[Out]

Integral(csc(a + b*x)**(-1/3), x)

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(csc(b*x + a)^(-1/3), x)

Giac [F]

\[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(csc(b*x + a)^(-1/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{\csc (a+b x)}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{1/3}} \,d x \]

[In]

int(1/(1/sin(a + b*x))^(1/3),x)

[Out]

int(1/(1/sin(a + b*x))^(1/3), x)