3.5.28 \(\int (2-3 \sin ^2(x))^{3/5} \sin (4 x) \, dx\) [428]

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]

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

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Rubi [A]
time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.00, 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, 455, 45} \begin {gather*} \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 {gather*}

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 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 455

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]

Rubi steps

\begin {align*} \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx &=\text {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 \text {Subst}\left (\int x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int (2-3 x)^{3/5} (1-2 x) \, dx,x,\sin ^2(x)\right )\\ &=2 \text {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*}

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Mathematica [A]
time = 0.03, size = 29, normalized size = 0.88 \begin {gather*} -\frac {5 (1+3 \cos (2 x))^{8/5} (-5+24 \cos (2 x))}{936\ 2^{3/5}} \end {gather*}

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 = 0.14, size = 38, normalized size = 1.15

method	result	size
default	\(\frac {5 \left (3 \left (\cos ^{2}\left (x \right )\right )-1\right )^{\frac {8}{5}}}{12}-\frac {20 \left (\frac {1}{2}+\frac {3 \cos \left (2 x \right )}{2}\right )^{\frac {13}{5}}}{117}-\frac {5 \left (\frac {1}{2}+\frac {3 \cos \left (2 x \right )}{2}\right )^{\frac {8}{5}}}{18}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 4.52, size = 25, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.94, size = 26, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8008 deep

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Giac [A]
time = 1.72, size = 35, normalized size = 1.06 \begin {gather*} -\frac {20}{117} \, {\left (3 \, \sin \left (x\right )^{2} - 2\right )}^{2} {\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac {3}{5}} + \frac {5}{36} \, {\left (-3 \, \sin \left (x\right )^{2} + 2\right )}^{\frac {8}{5}} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sin \left (4\,x\right )\,{\left (2-3\,{\sin \left (x\right )}^2\right )}^{3/5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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