3.31 \(\int (c \csc (a+b x))^{4/3} \, dx\)

Optimal. Leaf size=54 \[ -\frac {3 c \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]

[Out]

-3*c*cos(b*x+a)*(c*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 = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 c \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(4/3),x]

[Out]

(-3*c*Cos[a + b*x]*(c*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 (c \csc (a+b x))^{4/3} \, dx &=\sqrt [3]{c \csc (a+b x)} \sqrt [3]{\frac {\sin (a+b x)}{c}} \int \frac {1}{\left (\frac {\sin (a+b x)}{c}\right )^{4/3}} \, dx\\ &=-\frac {3 c \cos (a+b x) \sqrt [3]{c \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 = 0.05, size = 57, normalized size = 1.06 \[ \frac {c \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \left (2 \sqrt [6]{\sin ^2(a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\cos ^2(a+b x)\right )-3\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(4/3),x]

[Out]

(c*Cos[a + b*x]*(c*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 = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c \csc \left (b x + a\right )\right )^{\frac {1}{3}} c \csc \left (b x + a\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(4/3), x)

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maple [F]  time = 0.58, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x +a \right )\right )^{\frac {4}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(4/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/sin(a + b*x))^(4/3),x)

[Out]

int((c/sin(a + b*x))^(4/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*csc(a + b*x))**(4/3), x)

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