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

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 = 0.17784, 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 \[ \frac{1}{8} \sqrt{4 \sin ^2(x)-5}-\frac{5}{8 \sqrt{4 \sin ^2(x)-5}}-\frac{1}{4 \left (4 \sin ^2(x)-5\right )^{3/2}} \]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*cos(2*x)*sin(3*x)/(-5+4*sin(x)**2)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.505549, size = 59, normalized size = 1.2 \[ -\frac{2 \cos (2 x) \left (\sqrt{6 \cos (2 x)+9}-11\right )+3 \left (\sqrt{6 \cos (2 x)+9}-8\right )-2 \cos (4 x)}{8 (-2 \cos (2 x)-3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Cos[2*x]*(-11 + Sqrt[9 + 6*Cos[2*x]]) + 3*(-8 + Sqrt[9 + 6*Cos[2*x]]) - 2*Co
s[4*x])/(8*(-3 - 2*Cos[2*x])^(3/2))

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Maple [A]  time = 0.058, size = 46, normalized size = 0.9 \[{\frac{1}{2} \left ( -4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}}}+{\frac{7\, \left ( \cos \left ( x \right ) \right ) ^{2}}{2} \left ( -4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{ \left ( -4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{3/2}}} \]

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]

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

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Maxima [A]  time = 1.4443, size = 259, normalized size = 5.29 \[ -\frac{{\left (\cos \left (11 \, x\right ) + 14 \, \cos \left (9 \, x\right ) + 58 \, \cos \left (7 \, x\right ) + 94 \, \cos \left (5 \, x\right ) + 58 \, \cos \left (3 \, x\right ) + 15 \, \cos \left (x\right )\right )} \cos \left (\frac{5}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 3 \, \sin \left (2 \, x\right ), -\cos \left (4 \, x\right ) - 3 \, \cos \left (2 \, x\right ) - 1\right )\right ) -{\left (\sin \left (11 \, x\right ) + 14 \, \sin \left (9 \, x\right ) + 58 \, \sin \left (7 \, x\right ) + 94 \, \sin \left (5 \, x\right ) + 58 \, \sin \left (3 \, x\right ) + 13 \, \sin \left (x\right )\right )} \sin \left (\frac{5}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 3 \, \sin \left (2 \, x\right ), -\cos \left (4 \, x\right ) - 3 \, \cos \left (2 \, x\right ) - 1\right )\right )}{8 \,{\left (2 \,{\left (3 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 9 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 6 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sin \left (2 \, x\right )^{2} + 6 \, \cos \left (2 \, x\right ) + 1\right )}^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(2*x)*cos(x)*sin(3*x)/(4*sin(x)^2 - 5)^(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*co
s(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 = 0.238004, size = 1, normalized size = 0.02 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

0

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.204251, size = 45, normalized size = 0.92 \[ \frac{1}{8} \, \sqrt{4 \, \sin \left (x\right )^{2} - 5} - \frac{20 \, \sin \left (x\right )^{2} - 23}{8 \,{\left (4 \, \sin \left (x\right )^{2} - 5\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(2*x)*cos(x)*sin(3*x)/(4*sin(x)^2 - 5)^(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)