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

Optimal. Leaf size=58 \[ \frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right )}{8 b f \sqrt{\cos ^2(e+f x)}} \]

[Out]

(3*Cos[e + f*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(8/3))/(8*b*f*Sqrt[Cos[e + f
*x]^2])

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Rubi [A]  time = 0.0141124, 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} \[ \frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right )}{8 b f \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sin[e + f*x])^(5/3),x]

[Out]

(3*Cos[e + f*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(8/3))/(8*b*f*Sqrt[Cos[e + f
*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 (b \sin (e+f x))^{5/3} \, dx &=\frac{3 \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{8/3}}{8 b f \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.041237, size = 55, normalized size = 0.95 \[ \frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right )}{8 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sin[e + f*x])^(5/3),x]

[Out]

(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(5/3)*Tan[e + f*x])/
(8*f)

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Maple [F]  time = 0.02, 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((b*sin(f*x+e))^(5/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 (\left (b \sin \left (f x + e\right )\right )^{\frac{2}{3}} b \sin \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((b*sin(f*x+e))^(5/3),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e))^(5/3), x)