3.37 \(\int \frac{1}{(c \sin (a+b x))^{2/3}} \, dx\)

Optimal. Leaf size=271 \[ \frac{3^{3/4} \sec (a+b x) \sqrt [3]{c \sin (a+b x)} \left (c^{2/3}-(c \sin (a+b x))^{2/3}\right ) \sqrt{\frac{c^{4/3} \left (\frac{(c \sin (a+b x))^{4/3}}{c^{4/3}}+\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}+1\right )}{\left (c^{2/3}-\left (1+\sqrt{3}\right ) (c \sin (a+b x))^{2/3}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\left (1-\sqrt{3}\right ) (c \sin (a+b x))^{2/3}}{c^{2/3}-\left (1+\sqrt{3}\right ) (c \sin (a+b x))^{2/3}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2 b c^{5/3} \sqrt{-\frac{(c \sin (a+b x))^{2/3} \left (c^{2/3}-(c \sin (a+b x))^{2/3}\right )}{\left (c^{2/3}-\left (1+\sqrt{3}\right ) (c \sin (a+b x))^{2/3}\right )^2}}} \]

[Out]

(3^(3/4)*EllipticF[ArcCos[(c^(2/3) - (1 - Sqrt[3])*(c*Sin[a + b*x])^(2/3))/(c^(2/3) - (1 + Sqrt[3])*(c*Sin[a +
b*x])^(2/3))], (2 + Sqrt[3])/4]*Sec[a + b*x]*(c*Sin[a + b*x])^(1/3)*(c^(2/3) - (c*Sin[a + b*x])^(2/3))*Sqrt[(
c^(4/3)*(1 + (c*Sin[a + b*x])^(2/3)/c^(2/3) + (c*Sin[a + b*x])^(4/3)/c^(4/3)))/(c^(2/3) - (1 + Sqrt[3])*(c*Sin
[a + b*x])^(2/3))^2])/(2*b*c^(5/3)*Sqrt[-(((c*Sin[a + b*x])^(2/3)*(c^(2/3) - (c*Sin[a + b*x])^(2/3)))/(c^(2/3)
- (1 + Sqrt[3])*(c*Sin[a + b*x])^(2/3))^2)])

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Rubi [C]  time = 0.0155328, antiderivative size = 56, normalized size of antiderivative = 0.21, 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} \[ \frac{3 \cos (a+b x) \sqrt [3]{c \sin (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right )}{b c \sqrt{\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(-2/3),x]

[Out]

(3*Cos[a + b*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1/3))/(b*c*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]

Rubi steps

\begin{align*} \int \frac{1}{(c \sin (a+b x))^{2/3}} \, dx &=\frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right ) \sqrt [3]{c \sin (a+b x)}}{b c \sqrt{\cos ^2(a+b x)}}\\ \end{align*}

Mathematica [C]  time = 0.0407623, size = 53, normalized size = 0.2 \[ \frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right )}{b (c \sin (a+b x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^(-2/3),x]

[Out]

(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[a + b*x]^2]*Tan[a + b*x])/(b*(c*Sin[a + b*x])^(2/
3))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \left ( c\sin \left ( bx+a \right ) \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(b*x+a))^(2/3),x)

[Out]

int(1/(c*sin(b*x+a))^(2/3),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{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(2/3),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(-2/3), 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{1}{3}}}{c \sin \left (b x + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(2/3),x, algorithm="fricas")

[Out]

integral((c*sin(b*x + a))^(1/3)/(c*sin(b*x + a)), 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{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))**(2/3),x)

[Out]

Integral((c*sin(a + b*x))**(-2/3), 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{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(2/3),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(-2/3), x)