3.436 \(\int \frac{-2 \cot ^2(x)+\sin (x)}{\left (1+5 \tan ^2(x)\right )^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{8} \cos (x) \sqrt{5 \tan ^2(x)+1}-\frac{\cos (x)}{4 \sqrt{5 \tan ^2(x)+1}}-\frac{1}{4} \tanh ^{-1}\left (\frac{2 \tan (x)}{\sqrt{5 \tan ^2(x)+1}}\right )+\frac{9}{2} \sqrt{5 \tan ^2(x)+1} \cot (x)-\frac{5 \cot (x)}{2 \sqrt{5 \tan ^2(x)+1}} \]

[Out]

-ArcTanh[(2*Tan[x])/Sqrt[1 + 5*Tan[x]^2]]/4 - Cos[x]/(4*Sqrt[1 + 5*Tan[x]^2]) -
(5*Cot[x])/(2*Sqrt[1 + 5*Tan[x]^2]) - (Cos[x]*Sqrt[1 + 5*Tan[x]^2])/8 + (9*Cot[x
]*Sqrt[1 + 5*Tan[x]^2])/2

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Rubi [A]  time = 0.352963, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{5 \sec (x)}{8 \sqrt{5 \sec ^2(x)-4}}+\frac{\cos (x)}{4 \sqrt{5 \sec ^2(x)-4}}-\frac{1}{4} \tanh ^{-1}\left (\frac{2 \tan (x)}{\sqrt{5 \tan ^2(x)+1}}\right )+\frac{9}{2} \sqrt{5 \tan ^2(x)+1} \cot (x)-\frac{5 \cot (x)}{2 \sqrt{5 \tan ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]  Int[(-2*Cot[x]^2 + Sin[x])/(1 + 5*Tan[x]^2)^(3/2),x]

[Out]

-ArcTanh[(2*Tan[x])/Sqrt[1 + 5*Tan[x]^2]]/4 + Cos[x]/(4*Sqrt[-4 + 5*Sec[x]^2]) -
(5*Sec[x])/(8*Sqrt[-4 + 5*Sec[x]^2]) - (5*Cot[x])/(2*Sqrt[1 + 5*Tan[x]^2]) + (9
*Cot[x]*Sqrt[1 + 5*Tan[x]^2])/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((-2*cot(x)**2+sin(x))/(1+5*tan(x)**2)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.515383, size = 75, normalized size = 0.8 \[ \frac{\csc (x) \sec (x) \left (-9 \sin (x)+\sin (3 x)-164 \cos (2 x)-4 \sin (x) \sqrt{2 \cos (2 x)-3} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2 \cos (2 x)-3}}\right )+196\right )}{16 \sqrt{2 \tan ^2(x)+3 \sec ^2(x)-2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-2*Cot[x]^2 + Sin[x])/(1 + 5*Tan[x]^2)^(3/2),x]

[Out]

(Csc[x]*Sec[x]*(196 - 164*Cos[2*x] - 9*Sin[x] - 4*ArcTan[(2*Sin[x])/Sqrt[-3 + 2*
Cos[2*x]]]*Sqrt[-3 + 2*Cos[2*x]]*Sin[x] + Sin[3*x]))/(16*Sqrt[-2 + 3*Sec[x]^2 +
2*Tan[x]^2])

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Maple [C]  time = 0.76, size = 975, normalized size = 10.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*cot(x)^2+sin(x))/(1+5*tan(x)^2)^(3/2),x)

[Out]

-1/8*I/(2+5^(1/2))^2/(-9+4*5^(1/2))^(1/2)/(-2+5^(1/2))^2/(4*cos(x)^2-5)^2*(3*I*5
^(1/2)*(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2)*arctanh(1/2*(-16)^(1/2)*cos(x)*(cos(
x)-1)/sin(x)^2/(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2))*sin(x)*cos(x)-4*I*2^(1/2)*(
(2*5^(1/2)*cos(x)-4*cos(x)-2*5^(1/2)+5)/(1+cos(x)))^(1/2)*(-2*(2*5^(1/2)*cos(x)-
2*5^(1/2)+4*cos(x)-5)/(1+cos(x)))^(1/2)*sin(x)*EllipticF(I*(-2+5^(1/2))*(cos(x)-
1)/sin(x),9+4*5^(1/2))+8*I*2^(1/2)*((2*5^(1/2)*cos(x)-4*cos(x)-2*5^(1/2)+5)/(1+c
os(x)))^(1/2)*(-2*(2*5^(1/2)*cos(x)-2*5^(1/2)+4*cos(x)-5)/(1+cos(x)))^(1/2)*sin(
x)*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))-6*I*(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2)*arcta
nh(1/2*(-16)^(1/2)*cos(x)*(cos(x)-1)/sin(x)^2/(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/
2))*sin(x)*cos(x)-4*I*2^(1/2)*((2*5^(1/2)*cos(x)-4*cos(x)-2*5^(1/2)+5)/(1+cos(x)
))^(1/2)*(-2*(2*5^(1/2)*cos(x)-2*5^(1/2)+4*cos(x)-5)/(1+cos(x)))^(1/2)*sin(x)*El
lipticF(I*(-2+5^(1/2))*(cos(x)-1)/sin(x),9+4*5^(1/2))*cos(x)+3*5^(1/2)*(-(4*cos(
x)^2-5)/(1+cos(x))^2)^(1/2)*arctan(2*cos(x)*(cos(x)-1)/sin(x)^2/(-(4*cos(x)^2-5)
/(1+cos(x))^2)^(1/2))*sin(x)*cos(x)-6*I*(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2)*sin
(x)*arctanh(1/2*(-16)^(1/2)*cos(x)*(cos(x)-1)/sin(x)^2/(-(4*cos(x)^2-5)/(1+cos(x
))^2)^(1/2))+3*I*5^(1/2)*(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2)*sin(x)*arctanh(1/2
*(-16)^(1/2)*cos(x)*(cos(x)-1)/sin(x)^2/(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2))+3*
5^(1/2)*(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2)*sin(x)*arctan(2*cos(x)*(cos(x)-1)/s
in(x)^2/(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2))-2*5^(1/2)*sin(x)*cos(x)^2-6*(-(4*c
os(x)^2-5)/(1+cos(x))^2)^(1/2)*sin(x)*arctan(2*cos(x)*(cos(x)-1)/sin(x)^2/(-(4*c
os(x)^2-5)/(1+cos(x))^2)^(1/2))*cos(x)+8*I*2^(1/2)*((2*5^(1/2)*cos(x)-4*cos(x)-2
*5^(1/2)+5)/(1+cos(x)))^(1/2)*(-2*(2*5^(1/2)*cos(x)-2*5^(1/2)+4*cos(x)-5)/(1+cos
(x)))^(1/2)*sin(x)*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))*cos(x)+164*5^(1/2)*cos(x)^2-6*
(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2)*sin(x)*arctan(2*cos(x)*(cos(x)-1)/sin(x)^2/
(-(4*cos(x)^2-5)/(1+cos(x))^2)^(1/2))+4*cos(x)^2*sin(x)+5*sin(x)*5^(1/2)-328*cos
(x)^2-180*5^(1/2)-10*sin(x)+360)*cos(x)^3*(-(4*cos(x)^2-5)/cos(x)^2)^(3/2)/sin(x
)

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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(-(2*cot(x)^2 - sin(x))/(5*tan(x)^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.296103, size = 131, normalized size = 1.39 \[ \frac{2 \,{\left (4 \, \cos \left (x\right )^{2} - 5\right )} \log \left (\sqrt{-\frac{4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}} \cos \left (x\right ) - 2 \, \sin \left (x\right )\right ) \sin \left (x\right ) +{\left (164 \, \cos \left (x\right )^{3} -{\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 180 \, \cos \left (x\right )\right )} \sqrt{-\frac{4 \, \cos \left (x\right )^{2} - 5}{\cos \left (x\right )^{2}}}}{8 \,{\left (4 \, \cos \left (x\right )^{2} - 5\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*cot(x)^2 - sin(x))/(5*tan(x)^2 + 1)^(3/2),x, algorithm="fricas")

[Out]

1/8*(2*(4*cos(x)^2 - 5)*log(sqrt(-(4*cos(x)^2 - 5)/cos(x)^2)*cos(x) - 2*sin(x))*
sin(x) + (164*cos(x)^3 - (2*cos(x)^3 - 5*cos(x))*sin(x) - 180*cos(x))*sqrt(-(4*c
os(x)^2 - 5)/cos(x)^2))/((4*cos(x)^2 - 5)*sin(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((-2*cot(x)**2+sin(x))/(1+5*tan(x)**2)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*cot(x)^2 - sin(x))/(5*tan(x)^2 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(-(2*cot(x)^2 - sin(x))/(5*tan(x)^2 + 1)^(3/2), x)