3.38 \(\int \frac{1}{(c \sin (a+b x))^{4/3}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \cos (a+b x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b c \sqrt{\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0167059, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ -\frac{3 \cos (a+b x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b c \sqrt{\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(-4/3),x]

[Out]

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

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

Mathematica [A]  time = 0.0415464, size = 53, normalized size = 0.95 \[ -\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right )}{b (c \sin (a+b x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^(-4/3),x]

[Out]

(-3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sin[a + b*x]^2]*Tan[a + b*x])/(b*(c*Sin[a + b*x])^(
4/3))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ( c\sin \left ( bx+a \right ) \right ) ^{-{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(b*x+a))^(4/3),x)

[Out]

int(1/(c*sin(b*x+a))^(4/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \sin \left (b x + a\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(4/3),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(-4/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(4/3),x, algorithm="fricas")

[Out]

integral(-(c*sin(b*x + a))^(2/3)/(c^2*cos(b*x + a)^2 - c^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \sin{\left (a + b x \right )}\right )^{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))**(4/3),x)

[Out]

Integral((c*sin(a + b*x))**(-4/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \sin \left (b x + a\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(4/3),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(-4/3), x)