3.4.15 \(\int \frac {\cos ^2(e+f x)}{\sqrt [3]{b \sin (e+f x)}} \, dx\) [315]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2657} \begin {gather*} \frac {3 \cos (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt {\cos ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2657

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)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {\cos ^2(e+f x)}{\sqrt [3]{b \sin (e+f x)}} \, dx &=\frac {3 \cos (e+f x) \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\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*}

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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.95 \begin {gather*} \frac {3 \sqrt {\cos ^2(e+f x)} \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};\sin ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt [3]{b \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{2}\left (f x +e \right )}{\left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(cos(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{2}{\left (e + f x \right )}}{\sqrt [3]{b \sin {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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