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

Optimal. Leaf size=56 \[ \frac{3 c \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\sin ^2(a+b x)\right )}{4 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.0262917, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3772, 2643} \[ \frac{3 c \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right )}{4 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(-1/3),x]

[Out]

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

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

Mathematica [A]  time = 0.0661538, size = 59, normalized size = 1.05 \[ -\frac{\sin (a+b x) \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sin ^2(a+b x)^{2/3} \sqrt [3]{c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(-1/3),x]

[Out]

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

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Maple [F]  time = 0.142, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{c\csc \left ( bx+a \right ) }}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*csc(b*x + a))^(2/3)/(c*csc(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*csc(a + b*x))**(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(-1/3), x)