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

Optimal. Leaf size=51 \[ -\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [A]  time = 0.07, size = 54, normalized size = 1.06 \[ \frac {\cos (a+b x) \sqrt [3]{\csc (a+b x)} \left (2 \sqrt [6]{\sin ^2(a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\cos ^2(a+b x)\right )-3\right )}{b} \]

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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fricas [F]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{\frac {4}{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {4}{3}}\,{d x} \]

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)

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maple [F]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \csc ^{\frac {4}{3}}\left (b x +a \right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {4}{3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{4/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(a + b*x))^(4/3),x)

[Out]

int((1/sin(a + b*x))^(4/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{\frac {4}{3}}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(a + b*x)**(4/3), x)

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