3.4.35 \(\int \frac {\sin ^{\frac {2}{3}}(x)}{\cos ^{\frac {8}{3}}(x)} \, dx\) [335]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {2643} \begin {gather*} \frac {3 \sin ^{\frac {5}{3}}(x)}{5 \cos ^{\frac {5}{3}}(x)} \end {gather*}

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 2643

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*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 \sin ^{\frac {5}{3}}(x)}{5 \cos ^{\frac {5}{3}}(x)} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin ^{\frac {2}{3}}\left (x \right )}{\cos \left (x \right )^{\frac {8}{3}}}\, dx\]

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 = 0.00, 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(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 = 0.39, size = 10, normalized size = 0.62 \begin {gather*} \frac {3 \, \sin \left (x\right )^{\frac {5}{3}}}{5 \, \cos \left (x\right )^{\frac {5}{3}}} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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

[Out]

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

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Mupad [B]
time = 0.82, size = 94, normalized size = 5.88 \begin {gather*} \frac {6\,2^{2/3}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{5/3}\,{\left (1-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\right )}^{1/3}+6\,2^{2/3}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{11/3}\,{\left (1-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\right )}^{1/3}}{5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+10\right )+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*2^(2/3)*tan(x/2)^(5/3)*(1 - tan(x/2)^2)^(1/3) + 6*2^(2/3)*tan(x/2)^(11/3)*(1 - tan(x/2)^2)^(1/3))/(5*tan(x/
2)^2 - tan(x/2)^2*(10*tan(x/2)^2 - 5*tan(x/2)^2*(tan(x/2)^2 + 1) + 10) + 5)

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