\(\int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx\) [316]

Optimal result

Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\frac {3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(e+f x)\right ) (b \sin (e+f x))^{2/3}}{2 b f \sqrt {\cos ^2(e+f x)}} \]

[Out]

3/2*cos(f*x+e)*hypergeom([1/3, 1/2],[4/3],sin(f*x+e)^2)*(b*sin(f*x+e))^(2/3)/b/f/(cos(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\frac {3 \cos (e+f x) (b \sin (e+f x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(e+f x)\right )}{2 b f \sqrt {\cos ^2(e+f x)}} \]

[In]

Int[(b*Sin[e + f*x])^(-1/3),x]

[Out]

(3*Cos[e + f*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))/(2*b*f*Sqrt[Cos[e + f
*x]^2])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(e+f x)\right ) (b \sin (e+f x))^{2/3}}{2 b f \sqrt {\cos ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\frac {3 \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt [3]{b \sin (e+f x)}} \]

[In]

Integrate[(b*Sin[e + f*x])^(-1/3),x]

[Out]

(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(
1/3))

Maple [F]

\[\int \frac {1}{\left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\int { \frac {1}{\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt [3]{b \sin {\left (e + f x \right )}}}\, dx \]

[In]

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

[Out]

Integral((b*sin(e + f*x))**(-1/3), x)

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\int { \frac {1}{\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate((b*sin(f*x + e))^(-1/3), x)

Giac [F]

\[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\int { \frac {1}{\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate((b*sin(f*x + e))^(-1/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{b \sin (e+f x)}} \, dx=\int \frac {1}{{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \]

[In]

int(1/(b*sin(e + f*x))^(1/3),x)

[Out]

int(1/(b*sin(e + f*x))^(1/3), x)