3.424 \(\int \frac{\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{295 \cos (x)}{243 \sqrt{9-4 \cos ^2(x)}}-\frac{55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}-\frac{1}{2} \sin ^{-1}\left (\frac{2 \cos (x)}{3}\right ) \]

[Out]

-ArcSin[(2*Cos[x])/3]/2 - (55*Cos[x])/(27*(9 - 4*Cos[x]^2)^(3/2)) + (295*Cos[x])
/(243*Sqrt[9 - 4*Cos[x]^2])

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Rubi [A]  time = 0.123762, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{295 \cos (x)}{243 \sqrt{9-4 \cos ^2(x)}}-\frac{55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}-\frac{1}{2} \sin ^{-1}\left (\frac{2 \cos (x)}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sin[5*x]/(5*Cos[x]^2 + 9*Sin[x]^2)^(5/2),x]

[Out]

-ArcSin[(2*Cos[x])/3]/2 - (55*Cos[x])/(27*(9 - 4*Cos[x]^2)^(3/2)) + (295*Cos[x])
/(243*Sqrt[9 - 4*Cos[x]^2])

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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(sin(5*x)/(5*cos(x)**2+9*sin(x)**2)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.412996, size = 63, normalized size = 1.31 \[ \frac{2550 \cos (x)-590 \cos (3 x)+243 i (7-2 \cos (2 x))^{3/2} \log \left (\sqrt{7-2 \cos (2 x)}+2 i \cos (x)\right )}{486 (7-2 \cos (2 x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[5*x]/(5*Cos[x]^2 + 9*Sin[x]^2)^(5/2),x]

[Out]

(2550*Cos[x] - 590*Cos[3*x] + (243*I)*(7 - 2*Cos[2*x])^(3/2)*Log[(2*I)*Cos[x] +
Sqrt[7 - 2*Cos[2*x]]])/(486*(7 - 2*Cos[2*x])^(3/2))

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Maple [A]  time = 0.106, size = 53, normalized size = 1.1 \[{\frac{26\,\cos \left ( x \right ) }{27} \left ( 9-4\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{214\,\cos \left ( x \right ) }{243}{\frac{1}{\sqrt{9-4\, \left ( \cos \left ( x \right ) \right ) ^{2}}}}}-{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{3}}{3} \left ( 9-4\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}\arcsin \left ({\frac{2\,\cos \left ( x \right ) }{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(5*x)/(5*cos(x)^2+9*sin(x)^2)^(5/2),x)

[Out]

26/27*cos(x)/(9-4*cos(x)^2)^(3/2)+214/243*cos(x)/(9-4*cos(x)^2)^(1/2)-4/3*cos(x)
^3/(9-4*cos(x)^2)^(3/2)-1/2*arcsin(2/3*cos(x))

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Maxima [A]  time = 1.52975, size = 93, normalized size = 1.94 \[ -2 \,{\left (\frac{2 \, \cos \left (x\right )^{2}}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac{3}{2}}}\right )} \cos \left (x\right ) + \frac{52 \, \cos \left (x\right )}{243 \, \sqrt{-4 \, \cos \left (x\right )^{2} + 9}} + \frac{26 \, \cos \left (x\right )}{27 \,{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac{3}{2}}} - \frac{1}{2} \, \arcsin \left (\frac{2}{3} \, \cos \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-2*(2*cos(x)^2/(-4*cos(x)^2 + 9)^(3/2) - 3/(-4*cos(x)^2 + 9)^(3/2))*cos(x) + 52/
243*cos(x)/sqrt(-4*cos(x)^2 + 9) + 26/27*cos(x)/(-4*cos(x)^2 + 9)^(3/2) - 1/2*ar
csin(2/3*cos(x))

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Fricas [A]  time = 0.251594, size = 217, normalized size = 4.52 \[ -\frac{243 \,{\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (-\frac{2 \,{\left (16 \, \cos \left (x\right )^{3} - 27 \, \cos \left (x\right )\right )} \sin \left (x\right ) +{\left (16 \, \cos \left (x\right )^{3} - 17 \, \cos \left (x\right )\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9}}{32 \, \cos \left (x\right )^{4} -{\left (16 \, \cos \left (x\right )^{2} - 9\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9} \sin \left (x\right ) - 70 \, \cos \left (x\right )^{2} + 18}\right ) + 243 \,{\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) + 80 \,{\left (59 \, \cos \left (x\right )^{3} - 108 \, \cos \left (x\right )\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9}}{972 \,{\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x, algorithm="fricas")

[Out]

-1/972*(243*(16*cos(x)^4 - 72*cos(x)^2 + 81)*arctan(-(2*(16*cos(x)^3 - 27*cos(x)
)*sin(x) + (16*cos(x)^3 - 17*cos(x))*sqrt(-4*cos(x)^2 + 9))/(32*cos(x)^4 - (16*c
os(x)^2 - 9)*sqrt(-4*cos(x)^2 + 9)*sin(x) - 70*cos(x)^2 + 18)) + 243*(16*cos(x)^
4 - 72*cos(x)^2 + 81)*arctan(sin(x)/cos(x)) + 80*(59*cos(x)^3 - 108*cos(x))*sqrt
(-4*cos(x)^2 + 9))/(16*cos(x)^4 - 72*cos(x)^2 + 81)

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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(sin(5*x)/(5*cos(x)**2+9*sin(x)**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212165, size = 54, normalized size = 1.12 \[ -\frac{20 \,{\left (59 \, \cos \left (x\right )^{2} - 108\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9} \cos \left (x\right )}{243 \,{\left (4 \, \cos \left (x\right )^{2} - 9\right )}^{2}} - \frac{1}{2} \, \arcsin \left (\frac{2}{3} \, \cos \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x, algorithm="giac")

[Out]

-20/243*(59*cos(x)^2 - 108)*sqrt(-4*cos(x)^2 + 9)*cos(x)/(4*cos(x)^2 - 9)^2 - 1/
2*arcsin(2/3*cos(x))