3.307 \(\int \sec ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx\)

Optimal. Leaf size=58 \[ \frac {3 \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right )}{4 b f} \]

[Out]

3/4*hypergeom([2/3, 3/2],[5/3],sin(f*x+e)^2)*sec(f*x+e)*(b*sin(f*x+e))^(4/3)*(cos(f*x+e)^2)^(1/2)/b/f

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Rubi [A]  time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2577} \[ \frac {3 \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right )}{4 b f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^2*(b*Sin[e + f*x])^(1/3),x]

[Out]

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

Rule 2577

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b^(2*IntPart
[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2
, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \sec ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx &=\frac {3 \sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right ) \sec (e+f x) (b \sin (e+f x))^{4/3}}{4 b f}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 55, normalized size = 0.95 \[ \frac {3 \sqrt {\cos ^2(e+f x)} \tan (e+f x) \sqrt [3]{b \sin (e+f x)} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right )}{4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^2*(b*Sin[e + f*x])^(1/3),x]

[Out]

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \sec \left (f x + e\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(b*sin(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(b*sin(f*x+e))^(1/3),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e))^(1/3)*sec(f*x + e)^2, x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{2}\left (f x +e \right )\right ) \left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^2*(b*sin(f*x+e))^(1/3),x)

[Out]

int(sec(f*x+e)^2*(b*sin(f*x+e))^(1/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(b*sin(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e))^(1/3)*sec(f*x + e)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3}}{{\cos \left (e+f\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{b \sin {\left (e + f x \right )}} \sec ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**2*(b*sin(f*x+e))**(1/3),x)

[Out]

Integral((b*sin(e + f*x))**(1/3)*sec(e + f*x)**2, x)

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