3.428 \(\int (2-3 \sin ^2(x))^{3/5} \sin (4 x) \, dx\)

Optimal. Leaf size=33 \[ \frac{5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac{20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]

[Out]

(5*(2 - 3*Sin[x]^2)^(8/5))/36 - (20*(2 - 3*Sin[x]^2)^(13/5))/117

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Rubi [A]  time = 0.0608089, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {12, 444, 43} \[ \frac{5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac{20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(2 - 3*Sin[x]^2)^(8/5))/36 - (20*(2 - 3*Sin[x]^2)^(13/5))/117

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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 \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx &=\operatorname{Subst}\left (\int 4 x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=4 \operatorname{Subst}\left (\int x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int (2-3 x)^{3/5} (1-2 x) \, dx,x,\sin ^2(x)\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{1}{3} (2-3 x)^{3/5}+\frac{2}{3} (2-3 x)^{8/5}\right ) \, dx,x,\sin ^2(x)\right )\\ &=\frac{5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac{20}{117} \left (2-3 \sin ^2(x)\right )^{13/5}\\ \end{align*}

Mathematica [A]  time = 0.0413974, size = 29, normalized size = 0.88 \[ -\frac{5 (3 \cos (2 x)+1)^{8/5} (24 \cos (2 x)-5)}{936\ 2^{3/5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(1 + 3*Cos[2*x])^(8/5)*(-5 + 24*Cos[2*x]))/(936*2^(3/5))

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Maple [A]  time = 0.101, size = 38, normalized size = 1.2 \begin{align*}{\frac{5}{12} \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{{\frac{8}{5}}}}-{\frac{20}{117} \left ({\frac{1}{2}}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) ^{{\frac{13}{5}}}}-{\frac{5}{18} \left ({\frac{1}{2}}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) ^{{\frac{8}{5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/12*(3*cos(x)^2-1)^(8/5)-20/117*(1/2+3/2*cos(2*x))^(13/5)-5/18*(1/2+3/2*cos(2*x))^(8/5)

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Maxima [A]  time = 0.969511, size = 34, normalized size = 1.03 \begin{align*} -\frac{20}{117} \,{\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac{13}{5}} + \frac{5}{36} \,{\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac{8}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20/117*(-3*sin(x)^2 + 2)^(13/5) + 5/36*(-3*sin(x)^2 + 2)^(8/5)

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Fricas [A]  time = 3.46549, size = 89, normalized size = 2.7 \begin{align*} -\frac{5}{468} \,{\left (144 \, \cos \left (x\right )^{4} - 135 \, \cos \left (x\right )^{2} + 29\right )}{\left (3 \, \cos \left (x\right )^{2} - 1\right )}^{\frac{3}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/468*(144*cos(x)^4 - 135*cos(x)^2 + 29)*(3*cos(x)^2 - 1)^(3/5)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac{3}{5}} \sin \left (4 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-3*sin(x)^2 + 2)^(3/5)*sin(4*x), x)