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]

1/2*3^(3/4)*((c^(2/3)-(c*sin(b*x+a))^(2/3)*(1-3^(1/2)))^2/(c^(2/3)-(c*sin(b*x+a))^(2/3)*(1+3^(1/2)))^2)^(1/2)/
(c^(2/3)-(c*sin(b*x+a))^(2/3)*(1-3^(1/2)))*(c^(2/3)-(c*sin(b*x+a))^(2/3)*(1+3^(1/2)))*EllipticF((1-(c^(2/3)-(c
*sin(b*x+a))^(2/3)*(1-3^(1/2)))^2/(c^(2/3)-(c*sin(b*x+a))^(2/3)*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))
*sec(b*x+a)*(c*sin(b*x+a))^(1/3)*(c^(2/3)-(c*sin(b*x+a))^(2/3))*(c^(4/3)*(1+(c*sin(b*x+a))^(2/3)/c^(2/3)+(c*si
n(b*x+a))^(4/3)/c^(4/3))/(c^(2/3)-(c*sin(b*x+a))^(2/3)*(1+3^(1/2)))^2)^(1/2)/b/c^(5/3)/(-(c*sin(b*x+a))^(2/3)*
(c^(2/3)-(c*sin(b*x+a))^(2/3))/(c^(2/3)-(c*sin(b*x+a))^(2/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [C]  time = 0.02, 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*}

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Mathematica [C]  time = 0.04, size = 53, normalized size = 0.20 \[ \frac {3 \sqrt {\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b (c \sin (a+b x))^{2/3}} \]

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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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{c \sin \left (b x + a\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]

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)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x +a \right )\right )^{\frac {2}{3}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(a + b*x))^(2/3),x)

[Out]

int(1/(c*sin(a + b*x))^(2/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {2}{3}}}\, dx \]

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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