3.421 \(\int \cos (x) \left (-\cos ^2(x)-5 \sin ^2(x)\right )^{3/2} \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{4} \sin (x) \left (-4 \sin ^2(x)-1\right )^{3/2}-\frac{3}{8} \sin (x) \sqrt{-4 \sin ^2(x)-1}+\frac{3}{16} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{-4 \sin ^2(x)-1}}\right ) \]

[Out]

(3*ArcTan[(2*Sin[x])/Sqrt[-1 - 4*Sin[x]^2]])/16 - (3*Sin[x]*Sqrt[-1 - 4*Sin[x]^2
])/8 + (Sin[x]*(-1 - 4*Sin[x]^2)^(3/2))/4

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Rubi [A]  time = 0.0830833, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{4} \sin (x) \left (-4 \sin ^2(x)-1\right )^{3/2}-\frac{3}{8} \sin (x) \sqrt{-4 \sin ^2(x)-1}+\frac{3}{16} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{-4 \sin ^2(x)-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*ArcTan[(2*Sin[x])/Sqrt[-1 - 4*Sin[x]^2]])/16 - (3*Sin[x]*Sqrt[-1 - 4*Sin[x]^2
])/8 + (Sin[x]*(-1 - 4*Sin[x]^2)^(3/2))/4

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Rubi in Sympy [A]  time = 163.063, size = 61, normalized size = 1.05 \[ \frac{\left (- 4 \sin ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}} \sin{\left (x \right )}}{4} - \frac{3 \sqrt{- 4 \sin ^{2}{\left (x \right )} - 1} \sin{\left (x \right )}}{8} + \frac{3 \operatorname{atan}{\left (\frac{2 \sin{\left (x \right )}}{\sqrt{- 4 \sin ^{2}{\left (x \right )} - 1}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-4*sin(x)**2 - 1)**(3/2)*sin(x)/4 - 3*sqrt(-4*sin(x)**2 - 1)*sin(x)/8 + 3*atan(
2*sin(x)/sqrt(-4*sin(x)**2 - 1))/16

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Mathematica [A]  time = 0.107873, size = 50, normalized size = 0.86 \[ \left (\frac{1}{4} \sin (3 x)-\frac{11 \sin (x)}{8}\right ) \sqrt{2 \cos (2 x)-3}+\frac{3}{16} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2 \cos (2 x)-3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*ArcTan[(2*Sin[x])/Sqrt[-3 + 2*Cos[2*x]]])/16 + Sqrt[-3 + 2*Cos[2*x]]*((-11*Si
n[x])/8 + Sin[3*x]/4)

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Maple [A]  time = 0.135, size = 82, normalized size = 1.4 \[ -{\frac{1}{32\,\sin \left ( x \right ) }\sqrt{ \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-5 \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 32\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}- \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{2}+20\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}- \left ( \sin \left ( x \right ) \right ) ^{2}}-3\,\arcsin \left ( 8\, \left ( \sin \left ( x \right ) \right ) ^{2}+1 \right ) \right ){\frac{1}{\sqrt{4\, \left ( \cos \left ( x \right ) \right ) ^{2}-5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*(-cos(x)^2-5*sin(x)^2)^(3/2),x)

[Out]

-1/32*((4*cos(x)^2-5)*sin(x)^2)^(1/2)*(32*(-4*sin(x)^4-sin(x)^2)^(1/2)*sin(x)^2+
20*(-4*sin(x)^4-sin(x)^2)^(1/2)-3*arcsin(8*sin(x)^2+1))/sin(x)/(4*cos(x)^2-5)^(1
/2)

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Maxima [A]  time = 1.5958, size = 49, normalized size = 0.84 \[ \frac{1}{4} \,{\left (-4 \, \sin \left (x\right )^{2} - 1\right )}^{\frac{3}{2}} \sin \left (x\right ) - \frac{3}{8} \, \sqrt{-4 \, \sin \left (x\right )^{2} - 1} \sin \left (x\right ) - \frac{3}{16} i \, \operatorname{arsinh}\left (2 \, \sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(-4*sin(x)^2 - 1)^(3/2)*sin(x) - 3/8*sqrt(-4*sin(x)^2 - 1)*sin(x) - 3/16*I*a
rcsinh(2*sin(x))

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Fricas [A]  time = 0.220021, size = 528, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(((1536*I*e^(10*I*x) - 6912*I*e^(8*I*x) + 9408*I*e^(6*I*x) - 3744*I*e^(4*I
*x))*sqrt(e^(4*I*x) - 3*e^(2*I*x) + 1) - 1536*I*e^(12*I*x) + 9216*I*e^(10*I*x) -
18816*I*e^(8*I*x) + 14976*I*e^(6*I*x) - 3756*I*e^(4*I*x))*log(-1/2*sqrt(e^(4*I*
x) - 3*e^(2*I*x) + 1)*(4*e^(2*I*x) - 5) + 2*e^(4*I*x) - 11/2*e^(2*I*x) + 5/2) +
((-1536*I*e^(10*I*x) + 6912*I*e^(8*I*x) - 9408*I*e^(6*I*x) + 3744*I*e^(4*I*x))*s
qrt(e^(4*I*x) - 3*e^(2*I*x) + 1) + 1536*I*e^(12*I*x) - 9216*I*e^(10*I*x) + 18816
*I*e^(8*I*x) - 14976*I*e^(6*I*x) + 3756*I*e^(4*I*x))*log(sqrt(e^(4*I*x) - 3*e^(2
*I*x) + 1) - e^(2*I*x) - 1) + (2048*I*e^(14*I*x) - 23552*I*e^(12*I*x) + 85376*I*
e^(10*I*x) - 144064*I*e^(8*I*x) + 151424*I*e^(6*I*x) - 117216*I*e^(4*I*x) + 4751
2*I*e^(2*I*x) - 5008*I)*sqrt(e^(4*I*x) - 3*e^(2*I*x) + 1) - 2048*I*e^(16*I*x) +
26624*I*e^(14*I*x) - 119424*I*e^(12*I*x) + 259328*I*e^(10*I*x) - 332960*I*e^(8*I
*x) + 302752*I*e^(6*I*x) - 185271*I*e^(4*I*x) + 54976*I*e^(2*I*x) - 4992*I)/(8*(
16*e^(10*I*x) - 72*e^(8*I*x) + 98*e^(6*I*x) - 39*e^(4*I*x))*sqrt(e^(4*I*x) - 3*e
^(2*I*x) + 1) - 128*e^(12*I*x) + 768*e^(10*I*x) - 1568*e^(8*I*x) + 1248*e^(6*I*x
) - 313*e^(4*I*x))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.211122, size = 41, normalized size = 0.71 \[ -\frac{1}{8} i \,{\left (8 \, \sin \left (x\right )^{2} + 5\right )} \sqrt{4 \, \sin \left (x\right )^{2} + 1} \sin \left (x\right ) - \frac{3}{16} i \, \arcsin \left (2 i \, \sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*I*(8*sin(x)^2 + 5)*sqrt(4*sin(x)^2 + 1)*sin(x) - 3/16*I*arcsin(2*I*sin(x))