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

Optimal. Leaf size=49 \[ \frac{1}{8} \sqrt{4 \sin ^2(x)-5}-\frac{5}{8 \sqrt{4 \sin ^2(x)-5}}-\frac{1}{4 \left (4 \sin ^2(x)-5\right )^{3/2}} \]

[Out]

-1/(4*(-5 + 4*Sin[x]^2)^(3/2)) - 5/(8*Sqrt[-5 + 4*Sin[x]^2]) + Sqrt[-5 + 4*Sin[x]^2]/8

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Rubi [A]  time = 0.115832, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4356, 1247, 698} \[ \frac{1}{8} \sqrt{4 \sin ^2(x)-5}-\frac{5}{8 \sqrt{4 \sin ^2(x)-5}}-\frac{1}{4 \left (4 \sin ^2(x)-5\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*(-5 + 4*Sin[x]^2)^(3/2)) - 5/(8*Sqrt[-5 + 4*Sin[x]^2]) + Sqrt[-5 + 4*Sin[x]^2]/8

Rule 4356

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]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\cos (x) \cos (2 x) \sin (3 x)}{\left (-5+4 \sin ^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{x \left (3-10 x^2+8 x^4\right )}{\left (-5+4 x^2\right )^{5/2}} \, dx,x,\sin (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{3-10 x+8 x^2}{(-5+4 x)^{5/2}} \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{3}{(-5+4 x)^{5/2}}+\frac{5}{2 (-5+4 x)^{3/2}}+\frac{1}{2 \sqrt{-5+4 x}}\right ) \, dx,x,\sin ^2(x)\right )\\ &=-\frac{1}{4 \left (-5+4 \sin ^2(x)\right )^{3/2}}-\frac{5}{8 \sqrt{-5+4 \sin ^2(x)}}+\frac{1}{8} \sqrt{-5+4 \sin ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0834154, size = 28, normalized size = 0.57 \[ \frac{11 \cos (2 x)+\cos (4 x)+12}{4 \left (4 \sin ^2(x)-5\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12 + 11*Cos[2*x] + Cos[4*x])/(4*(-5 + 4*Sin[x]^2)^(3/2))

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Maple [A]  time = 0.046, size = 46, normalized size = 0.9 \begin{align*} 2\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{ \left ( -4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{3/2}}}+{\frac{7\, \left ( \cos \left ( x \right ) \right ) ^{2}}{2} \left ( -4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{2} \left ( -4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*cos(x)^4/(-4*cos(x)^2-1)^(3/2)+7/2*cos(x)^2/(-4*cos(x)^2-1)^(3/2)+1/2/(-4*cos(x)^2-1)^(3/2)

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Maxima [B]  time = 1.02905, size = 259, normalized size = 5.29 \begin{align*} -\frac{{\left (\cos \left (11 \, x\right ) + 14 \, \cos \left (9 \, x\right ) + 58 \, \cos \left (7 \, x\right ) + 94 \, \cos \left (5 \, x\right ) + 58 \, \cos \left (3 \, x\right ) + 15 \, \cos \left (x\right )\right )} \cos \left (\frac{5}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 3 \, \sin \left (2 \, x\right ), -\cos \left (4 \, x\right ) - 3 \, \cos \left (2 \, x\right ) - 1\right )\right ) -{\left (\sin \left (11 \, x\right ) + 14 \, \sin \left (9 \, x\right ) + 58 \, \sin \left (7 \, x\right ) + 94 \, \sin \left (5 \, x\right ) + 58 \, \sin \left (3 \, x\right ) + 13 \, \sin \left (x\right )\right )} \sin \left (\frac{5}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 3 \, \sin \left (2 \, x\right ), -\cos \left (4 \, x\right ) - 3 \, \cos \left (2 \, x\right ) - 1\right )\right )}{8 \,{\left (2 \,{\left (3 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 9 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 6 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sin \left (2 \, x\right )^{2} + 6 \, \cos \left (2 \, x\right ) + 1\right )}^{\frac{5}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((cos(11*x) + 14*cos(9*x) + 58*cos(7*x) + 94*cos(5*x) + 58*cos(3*x) + 15*cos(x))*cos(5/2*arctan2(sin(4*x)
+ 3*sin(2*x), -cos(4*x) - 3*cos(2*x) - 1)) - (sin(11*x) + 14*sin(9*x) + 58*sin(7*x) + 94*sin(5*x) + 58*sin(3*
x) + 13*sin(x))*sin(5/2*arctan2(sin(4*x) + 3*sin(2*x), -cos(4*x) - 3*cos(2*x) - 1)))/(2*(3*cos(2*x) + 1)*cos(4
*x) + cos(4*x)^2 + 9*cos(2*x)^2 + sin(4*x)^2 + 6*sin(4*x)*sin(2*x) + 9*sin(2*x)^2 + 6*cos(2*x) + 1)^(5/4)

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Fricas [A]  time = 3.17695, size = 4, normalized size = 0.08 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

[Out]

Timed out

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Giac [A]  time = 1.09792, size = 45, normalized size = 0.92 \begin{align*} \frac{1}{8} \, \sqrt{4 \, \sin \left (x\right )^{2} - 5} - \frac{20 \, \sin \left (x\right )^{2} - 23}{8 \,{\left (4 \, \sin \left (x\right )^{2} - 5\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*sin(x)^2 - 5) - 1/8*(20*sin(x)^2 - 23)/(4*sin(x)^2 - 5)^(3/2)