3.321 \(\int \frac{1}{(b \sin (e+f x))^{5/3}} \, dx\)

Optimal. Leaf size=58 \[ -\frac{3 \cos (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt{\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}} \]

[Out]

(-3*Cos[e + f*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sin[e + f*x]^2])/(2*b*f*Sqrt[Cos[e + f*x]^2]*(b*Sin[e + f*x
])^(2/3))

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Rubi [A]  time = 0.0144688, antiderivative size = 58, 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} \[ -\frac{3 \cos (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt{\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sin[e + f*x])^(-5/3),x]

[Out]

(-3*Cos[e + f*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sin[e + f*x]^2])/(2*b*f*Sqrt[Cos[e + f*x]^2]*(b*Sin[e + f*x
])^(2/3))

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}{(b \sin (e+f x))^{5/3}} \, dx &=-\frac{3 \cos (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt{\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0337511, size = 55, normalized size = 0.95 \[ -\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 f (b \sin (e+f x))^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sin[e + f*x])^(-5/3),x]

[Out]

(-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])
^(5/3))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sin \left ( fx+e \right ) \right ) ^{-{\frac{5}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*sin(f*x+e))^(5/3),x)

[Out]

int(1/(b*sin(f*x+e))^(5/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \sin \left (f x + e\right )\right )^{\frac{5}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sin(f*x+e))^(5/3),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e))^(-5/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (b \sin \left (f x + e\right )\right )^{\frac{1}{3}}}{b^{2} \cos \left (f x + e\right )^{2} - b^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sin(f*x+e))^(5/3),x, algorithm="fricas")

[Out]

integral(-(b*sin(f*x + e))^(1/3)/(b^2*cos(f*x + e)^2 - b^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \sin{\left (e + f x \right )}\right )^{\frac{5}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sin(f*x+e))**(5/3),x)

[Out]

Integral((b*sin(e + f*x))**(-5/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \sin \left (f x + e\right )\right )^{\frac{5}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sin(f*x+e))^(5/3),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e))^(-5/3), x)