3.25 \(\int \csc ^{\frac{4}{3}}(a+b x) \, dx\)

Optimal. Leaf size=51 \[ -\frac{3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0223791, antiderivative size = 51, 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} \[ -\frac{3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^(4/3),x]

[Out]

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

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 \csc ^{\frac{4}{3}}(a+b x) \, dx &=\sqrt [3]{\csc (a+b x)} \sqrt [3]{\sin (a+b x)} \int \frac{1}{\sin ^{\frac{4}{3}}(a+b x)} \, dx\\ &=-\frac{3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0775629, size = 54, normalized size = 1.06 \[ \frac{\cos (a+b x) \sqrt [3]{\csc (a+b x)} \left (2 \sqrt [6]{\sin ^2(a+b x)} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{3}{2},\cos ^2(a+b x)\right )-3\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^(4/3),x]

[Out]

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

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( bx+a \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^(4/3),x)

[Out]

int(csc(b*x+a)^(4/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (b x + a\right )^{\frac{4}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(4/3),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^(4/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\csc \left (b x + a\right )^{\frac{4}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(4/3),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^(4/3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**(4/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (b x + a\right )^{\frac{4}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(4/3),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^(4/3), x)