3.335 \(\int \frac{\sin ^{\frac{2}{3}}(x)}{\cos ^{\frac{8}{3}}(x)} \, dx\)

Optimal. Leaf size=16 \[ \frac{3 \sin ^{\frac{5}{3}}(x)}{5 \cos ^{\frac{5}{3}}(x)} \]

[Out]

(3*Sin[x]^(5/3))/(5*Cos[x]^(5/3))

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Rubi [A]  time = 0.0231872, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2563} \[ \frac{3 \sin ^{\frac{5}{3}}(x)}{5 \cos ^{\frac{5}{3}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^(2/3)/Cos[x]^(8/3),x]

[Out]

(3*Sin[x]^(5/3))/(5*Cos[x]^(5/3))

Rule 2563

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

Rubi steps

\begin{align*} \int \frac{\sin ^{\frac{2}{3}}(x)}{\cos ^{\frac{8}{3}}(x)} \, dx &=\frac{3 \sin ^{\frac{5}{3}}(x)}{5 \cos ^{\frac{5}{3}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0152747, size = 16, normalized size = 1. \[ \frac{3 \sin ^{\frac{5}{3}}(x)}{5 \cos ^{\frac{5}{3}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^(2/3)/Cos[x]^(8/3),x]

[Out]

(3*Sin[x]^(5/3))/(5*Cos[x]^(5/3))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( x \right ) \right ) ^{{\frac{2}{3}}} \left ( \cos \left ( x \right ) \right ) ^{-{\frac{8}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^(2/3)/cos(x)^(8/3),x)

[Out]

int(sin(x)^(2/3)/cos(x)^(8/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )^{\frac{2}{3}}}{\cos \left (x\right )^{\frac{8}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^(2/3)/cos(x)^(8/3),x, algorithm="maxima")

[Out]

integrate(sin(x)^(2/3)/cos(x)^(8/3), x)

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Fricas [A]  time = 2.38153, size = 42, normalized size = 2.62 \begin{align*} \frac{3 \, \sin \left (x\right )^{\frac{5}{3}}}{5 \, \cos \left (x\right )^{\frac{5}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^(2/3)/cos(x)^(8/3),x, algorithm="fricas")

[Out]

3/5*sin(x)^(5/3)/cos(x)^(5/3)

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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(sin(x)**(2/3)/cos(x)**(8/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )^{\frac{2}{3}}}{\cos \left (x\right )^{\frac{8}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^(2/3)/cos(x)^(8/3),x, algorithm="giac")

[Out]

integrate(sin(x)^(2/3)/cos(x)^(8/3), x)