3.450 \(\int \frac{\cos (x) \sin ^8(x)}{(2-5 \sin ^3(x))^{4/3}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{625} \left (2-5 \sin ^3(x)\right )^{5/3}+\frac{2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}+\frac{4}{125 \sqrt [3]{2-5 \sin ^3(x)}} \]

[Out]

4/(125*(2 - 5*Sin[x]^3)^(1/3)) + (2*(2 - 5*Sin[x]^3)^(2/3))/125 - (2 - 5*Sin[x]^3)^(5/3)/625

________________________________________________________________________________________

Rubi [A]  time = 0.108198, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4334, 266, 43} \[ -\frac{1}{625} \left (2-5 \sin ^3(x)\right )^{5/3}+\frac{2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}+\frac{4}{125 \sqrt [3]{2-5 \sin ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]*Sin[x]^8)/(2 - 5*Sin[x]^3)^(4/3),x]

[Out]

4/(125*(2 - 5*Sin[x]^3)^(1/3)) + (2*(2 - 5*Sin[x]^3)^(2/3))/125 - (2 - 5*Sin[x]^3)^(5/3)/625

Rule 4334

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cos (x) \sin ^8(x)}{\left (2-5 \sin ^3(x)\right )^{4/3}} \, dx &=\operatorname{Subst}\left (\int \frac{x^8}{\left (2-5 x^3\right )^{4/3}} \, dx,x,\sin (x)\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(2-5 x)^{4/3}} \, dx,x,\sin ^3(x)\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{4}{25 (2-5 x)^{4/3}}-\frac{4}{25 \sqrt [3]{2-5 x}}+\frac{1}{25} (2-5 x)^{2/3}\right ) \, dx,x,\sin ^3(x)\right )\\ &=\frac{4}{125 \sqrt [3]{2-5 \sin ^3(x)}}+\frac{2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}-\frac{1}{625} \left (2-5 \sin ^3(x)\right )^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.461911, size = 30, normalized size = 0.61 \[ \frac{-25 \sin ^6(x)-30 \sin ^3(x)+36}{625 \sqrt [3]{2-5 \sin ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]*Sin[x]^8)/(2 - 5*Sin[x]^3)^(4/3),x]

[Out]

(36 - 30*Sin[x]^3 - 25*Sin[x]^6)/(625*(2 - 5*Sin[x]^3)^(1/3))

________________________________________________________________________________________

Maple [F]  time = 0.61, size = 0, normalized size = 0. \begin{align*} \int{\cot \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{9} \left ( 2-5\, \left ( \sin \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)*sin(x)^9/(2-5*sin(x)^3)^(4/3),x)

[Out]

int(cot(x)*sin(x)^9/(2-5*sin(x)^3)^(4/3),x)

________________________________________________________________________________________

Maxima [A]  time = 0.929972, size = 50, normalized size = 1.02 \begin{align*} -\frac{1}{625} \,{\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac{5}{3}} + \frac{2}{125} \,{\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac{2}{3}} + \frac{4}{125 \,{\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*sin(x)^9/(2-5*sin(x)^3)^(4/3),x, algorithm="maxima")

[Out]

-1/625*(-5*sin(x)^3 + 2)^(5/3) + 2/125*(-5*sin(x)^3 + 2)^(2/3) + 4/125/(-5*sin(x)^3 + 2)^(1/3)

________________________________________________________________________________________

Fricas [A]  time = 3.06118, size = 158, normalized size = 3.22 \begin{align*} \frac{25 \, \cos \left (x\right )^{6} - 75 \, \cos \left (x\right )^{4} + 75 \, \cos \left (x\right )^{2} + 30 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + 11}{625 \,{\left (5 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + 2\right )}^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*sin(x)^9/(2-5*sin(x)^3)^(4/3),x, algorithm="fricas")

[Out]

1/625*(25*cos(x)^6 - 75*cos(x)^4 + 75*cos(x)^2 + 30*(cos(x)^2 - 1)*sin(x) + 11)/(5*(cos(x)^2 - 1)*sin(x) + 2)^
(1/3)

________________________________________________________________________________________

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(cot(x)*sin(x)**9/(2-5*sin(x)**3)**(4/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.0856, size = 50, normalized size = 1.02 \begin{align*} -\frac{1}{625} \,{\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac{5}{3}} + \frac{2}{125} \,{\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac{2}{3}} + \frac{4}{125 \,{\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*sin(x)^9/(2-5*sin(x)^3)^(4/3),x, algorithm="giac")

[Out]

-1/625*(-5*sin(x)^3 + 2)^(5/3) + 2/125*(-5*sin(x)^3 + 2)^(2/3) + 4/125/(-5*sin(x)^3 + 2)^(1/3)