\(\int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx\) [454]

Optimal result

Integrand size = 15, antiderivative size = 25 \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=-\frac {3}{40} \cos ^{\frac {5}{3}}(2 x)-\frac {3}{64} \cos ^{\frac {8}{3}}(2 x) \]

[Out]

-3/40*cos(2*x)^(5/3)-3/64*cos(2*x)^(8/3)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4442, 272, 45} \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=-\frac {3}{64} \cos ^{\frac {8}{3}}(2 x)-\frac {3}{40} \cos ^{\frac {5}{3}}(2 x) \]

[In]

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

[Out]

(-3*Cos[2*x]^(5/3))/40 - (3*Cos[2*x]^(8/3))/64

Rule 45

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])

Rule 272

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 4442

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^3 \left (-1+2 x^2\right )^{2/3} \, dx,x,\cos (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int x (-1+2 x)^{2/3} \, dx,x,\cos ^2(x)\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{2} (-1+2 x)^{2/3}+\frac {1}{2} (-1+2 x)^{5/3}\right ) \, dx,x,\cos ^2(x)\right )\right ) \\ & = -\frac {3}{40} \left (-1+2 \cos ^2(x)\right )^{5/3}-\frac {3}{64} \left (-1+2 \cos ^2(x)\right )^{8/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=-\frac {3}{320} \cos ^{\frac {5}{3}}(2 x) (8+5 \cos (2 x)) \]

[In]

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

[Out]

(-3*Cos[2*x]^(5/3)*(8 + 5*Cos[2*x]))/320

Maple [F]

\[\int \left (\cos ^{4}\left (x \right )\right ) \left (\cos ^{\frac {2}{3}}\left (2 x \right )\right ) \tan \left (x \right )d x\]

[In]

int(cos(x)^4*cos(2*x)^(2/3)*tan(x),x)

[Out]

int(cos(x)^4*cos(2*x)^(2/3)*tan(x),x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=-\frac {3}{320} \, {\left (20 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} - 3\right )} {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {2}{3}} \]

[In]

integrate(cos(x)^4*cos(2*x)^(2/3)*tan(x),x, algorithm="fricas")

[Out]

-3/320*(20*cos(x)^4 - 4*cos(x)^2 - 3)*(2*cos(x)^2 - 1)^(2/3)

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=\text {Timed out} \]

[In]

integrate(cos(x)**4*cos(2*x)**(2/3)*tan(x),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=\int { \cos \left (2 \, x\right )^{\frac {2}{3}} \cos \left (x\right )^{4} \tan \left (x\right ) \,d x } \]

[In]

integrate(cos(x)^4*cos(2*x)^(2/3)*tan(x),x, algorithm="maxima")

[Out]

integrate(cos(2*x)^(2/3)*cos(x)^4*tan(x), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=-\frac {3}{64} \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {8}{3}} - \frac {3}{40} \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {5}{3}} \]

[In]

integrate(cos(x)^4*cos(2*x)^(2/3)*tan(x),x, algorithm="giac")

[Out]

-3/64*(2*cos(x)^2 - 1)^(8/3) - 3/40*(2*cos(x)^2 - 1)^(5/3)

Mupad [F(-1)]

Timed out. \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx=\int {\cos \left (2\,x\right )}^{2/3}\,{\cos \left (x\right )}^4\,\mathrm {tan}\left (x\right ) \,d x \]

[In]

int(cos(2*x)^(2/3)*cos(x)^4*tan(x),x)

[Out]

int(cos(2*x)^(2/3)*cos(x)^4*tan(x), x)