3.435 \(\int \frac{1}{\left (4-5 \sec ^2(x)\right )^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{8} \tan ^{-1}\left (\frac{2 \tan (x)}{\sqrt{-5 \tan ^2(x)-1}}\right )-\frac{5 \tan (x)}{4 \sqrt{-5 \tan ^2(x)-1}} \]

[Out]

ArcTan[(2*Tan[x])/Sqrt[-1 - 5*Tan[x]^2]]/8 - (5*Tan[x])/(4*Sqrt[-1 - 5*Tan[x]^2]
)

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Rubi [A]  time = 0.0635397, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{8} \tan ^{-1}\left (\frac{2 \tan (x)}{\sqrt{-5 \tan ^2(x)-1}}\right )-\frac{5 \tan (x)}{4 \sqrt{-5 \tan ^2(x)-1}} \]

Antiderivative was successfully verified.

[In]  Int[(4 - 5*Sec[x]^2)^(-3/2),x]

[Out]

ArcTan[(2*Tan[x])/Sqrt[-1 - 5*Tan[x]^2]]/8 - (5*Tan[x])/(4*Sqrt[-1 - 5*Tan[x]^2]
)

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Rubi in Sympy [A]  time = 4.07887, size = 41, normalized size = 1.02 \[ \frac{\operatorname{atan}{\left (\frac{2 \tan{\left (x \right )}}{\sqrt{- 5 \tan ^{2}{\left (x \right )} - 1}} \right )}}{8} - \frac{5 \tan{\left (x \right )}}{4 \sqrt{- 5 \tan ^{2}{\left (x \right )} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4-5*sec(x)**2)**(3/2),x)

[Out]

atan(2*tan(x)/sqrt(-5*tan(x)**2 - 1))/8 - 5*tan(x)/(4*sqrt(-5*tan(x)**2 - 1))

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Mathematica [A]  time = 0.118732, size = 64, normalized size = 1.6 \[ -\frac{(2 \cos (2 x)-3) \sec ^3(x) \left (10 \sin (x)-\sqrt{2 \cos (2 x)-3} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2 \cos (2 x)-3}}\right )\right )}{8 \left (4-5 \sec ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((-3 + 2*Cos[2*x])*Sec[x]^3*(-(ArcTan[(2*Sin[x])/Sqrt[-3 + 2*Cos[2*x]]]*Sqrt[-3
+ 2*Cos[2*x]]) + 10*Sin[x]))/(8*(4 - 5*Sec[x]^2)^(3/2))

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Maple [C]  time = 0.316, size = 473, normalized size = 11.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4-5*sec(x)^2)^(3/2),x)

[Out]

1/4*I/(-9-4*5^(1/2))^(1/2)/(2+5^(1/2))*(4*cos(x)^2-5)*(I*5^(1/2)*EllipticF(I*(2+
5^(1/2))*(cos(x)-1)/sin(x),9-4*5^(1/2))*sin(x)*(-2*(2*5^(1/2)*cos(x)-2*5^(1/2)+4
*cos(x)-5)/(1+cos(x)))^(1/2)*((2*5^(1/2)*cos(x)-4*cos(x)-2*5^(1/2)+5)/(1+cos(x))
)^(1/2)*2^(1/2)-2*I*5^(1/2)*(-2*(2*5^(1/2)*cos(x)-2*5^(1/2)+4*cos(x)-5)/(1+cos(x
)))^(1/2)*2^(1/2)*((2*5^(1/2)*cos(x)-4*cos(x)-2*5^(1/2)+5)/(1+cos(x)))^(1/2)*Ell
ipticPi((-9-4*5^(1/2))^(1/2)*(cos(x)-1)/sin(x),1/(9+4*5^(1/2)),(-9+4*5^(1/2))^(1
/2)/(-9-4*5^(1/2))^(1/2))*sin(x)+2*I*EllipticF(I*(2+5^(1/2))*(cos(x)-1)/sin(x),9
-4*5^(1/2))*sin(x)*(-2*(2*5^(1/2)*cos(x)-2*5^(1/2)+4*cos(x)-5)/(1+cos(x)))^(1/2)
*((2*5^(1/2)*cos(x)-4*cos(x)-2*5^(1/2)+5)/(1+cos(x)))^(1/2)*2^(1/2)-4*I*(-2*(2*5
^(1/2)*cos(x)-2*5^(1/2)+4*cos(x)-5)/(1+cos(x)))^(1/2)*2^(1/2)*((2*5^(1/2)*cos(x)
-4*cos(x)-2*5^(1/2)+5)/(1+cos(x)))^(1/2)*EllipticPi((-9-4*5^(1/2))^(1/2)*(cos(x)
-1)/sin(x),1/(9+4*5^(1/2)),(-9+4*5^(1/2))^(1/2)/(-9-4*5^(1/2))^(1/2))*sin(x)-20*
5^(1/2)*cos(x)+20*5^(1/2)-45*cos(x)+45)*sin(x)/(cos(x)-1)/cos(x)^3/((4*cos(x)^2-
5)/cos(x)^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-5*sec(x)^2 + 4)^(-3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.257376, size = 207, normalized size = 5.18 \[ -\frac{20 \, \sqrt{\frac{4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \cos \left (x\right ) \sin \left (x\right ) -{\left (4 \, \cos \left (x\right )^{2} - 5\right )} \arctan \left (\frac{{\left (16 \, \cos \left (x\right )^{3} - 9 \, \cos \left (x\right )\right )} \sqrt{\frac{4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \sin \left (x\right ) + 2 \,{\left (16 \, \cos \left (x\right )^{3} - 19 \, \cos \left (x\right )\right )} \sin \left (x\right )}{32 \, \cos \left (x\right )^{4} - 54 \, \cos \left (x\right )^{2} +{\left (16 \, \cos \left (x\right )^{4} - 17 \, \cos \left (x\right )^{2}\right )} \sqrt{\frac{4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} + 20}\right ) +{\left (4 \, \cos \left (x\right )^{2} - 5\right )} \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right )}{16 \,{\left (4 \, \cos \left (x\right )^{2} - 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(20*sqrt((4*cos(x)^2 - 5)/cos(x)^2)*cos(x)*sin(x) - (4*cos(x)^2 - 5)*arcta
n(((16*cos(x)^3 - 9*cos(x))*sqrt((4*cos(x)^2 - 5)/cos(x)^2)*sin(x) + 2*(16*cos(x
)^3 - 19*cos(x))*sin(x))/(32*cos(x)^4 - 54*cos(x)^2 + (16*cos(x)^4 - 17*cos(x)^2
)*sqrt((4*cos(x)^2 - 5)/cos(x)^2) + 20)) + (4*cos(x)^2 - 5)*arctan(sin(x)/cos(x)
))/(4*cos(x)^2 - 5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 5 \sec ^{2}{\left (x \right )} + 4\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4-5*sec(x)**2)**(3/2),x)

[Out]

Integral((-5*sec(x)**2 + 4)**(-3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-5 \, \sec \left (x\right )^{2} + 4\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-5*sec(x)^2 + 4)^(-3/2),x, algorithm="giac")

[Out]

integrate((-5*sec(x)^2 + 4)^(-3/2), x)