3.447 \(\int \frac{\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx\)

Optimal. Leaf size=100 \[ \frac{2 \tan ^3(x)}{3 (\tan (x) \tan (2 x))^{3/2}}+\frac{3 \tan (x)}{4 \sqrt{\tan (x) \tan (2 x)}}+\frac{\tan (x)}{2 (\tan (x) \tan (2 x))^{3/2}}+2 \tanh ^{-1}\left (\frac{\tan (x)}{\sqrt{\tan (x) \tan (2 x)}}\right )-\frac{11 \tanh ^{-1}\left (\frac{\sqrt{2} \tan (x)}{\sqrt{\tan (x) \tan (2 x)}}\right )}{4 \sqrt{2}} \]

[Out]

2*ArcTanh[Tan[x]/Sqrt[Tan[x]*Tan[2*x]]] - (11*ArcTanh[(Sqrt[2]*Tan[x])/Sqrt[Tan[
x]*Tan[2*x]]])/(4*Sqrt[2]) + Tan[x]/(2*(Tan[x]*Tan[2*x])^(3/2)) + (2*Tan[x]^3)/(
3*(Tan[x]*Tan[2*x])^(3/2)) + (3*Tan[x])/(4*Sqrt[Tan[x]*Tan[2*x]])

_______________________________________________________________________________________

Rubi [B]  time = 1.92717, antiderivative size = 208, normalized size of antiderivative = 2.08, number of steps used = 21, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345 \[ \frac{\left (1-\tan ^2(x)\right ) \tan (x)}{3 \sqrt{2} \sqrt{\frac{\tan ^2(x)}{1-\tan ^2(x)}}}+\frac{3 \tan (x)}{4 \sqrt{2} \sqrt{\frac{\tan ^2(x)}{1-\tan ^2(x)}}}-\frac{11 \tan ^{-1}\left (\sqrt{\tan ^2(x)-1}\right ) \tan (x)}{4 \sqrt{2} \sqrt{\frac{\tan ^2(x)}{1-\tan ^2(x)}} \sqrt{\tan ^2(x)-1}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\tan ^2(x)-1}}{\sqrt{2}}\right ) \tan (x)}{\sqrt{\frac{\tan ^2(x)}{1-\tan ^2(x)}} \sqrt{\tan ^2(x)-1}}+\frac{\left (1-\tan ^2(x)\right ) \cot (x)}{4 \sqrt{2} \sqrt{\frac{\tan ^2(x)}{1-\tan ^2(x)}}} \]

Antiderivative was successfully verified.

[In]  Int[(Sec[x]^2*(-Cos[2*x] + 2*Tan[x]^2))/(Tan[x]*Tan[2*x])^(3/2),x]

[Out]

(3*Tan[x])/(4*Sqrt[2]*Sqrt[Tan[x]^2/(1 - Tan[x]^2)]) + (Cot[x]*(1 - Tan[x]^2))/(
4*Sqrt[2]*Sqrt[Tan[x]^2/(1 - Tan[x]^2)]) + (Tan[x]*(1 - Tan[x]^2))/(3*Sqrt[2]*Sq
rt[Tan[x]^2/(1 - Tan[x]^2)]) - (11*ArcTan[Sqrt[-1 + Tan[x]^2]]*Tan[x])/(4*Sqrt[2
]*Sqrt[Tan[x]^2/(1 - Tan[x]^2)]*Sqrt[-1 + Tan[x]^2]) + (2*ArcTan[Sqrt[-1 + Tan[x
]^2]/Sqrt[2]]*Tan[x])/(Sqrt[Tan[x]^2/(1 - Tan[x]^2)]*Sqrt[-1 + Tan[x]^2])

_______________________________________________________________________________________

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(2*x)+2*tan(x)**2)/cos(x)**2/(tan(x)*tan(2*x))**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 2.5532, size = 207, normalized size = 2.07 \[ \frac{\tan ^2(2 x) \left (2 \tan ^2(x)-\cos (2 x)\right ) \left (-\frac{72 \sin ^2(x) \cos (2 x) \tan (x) F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};\cot ^2(x),-\cot ^2(x)\right )}{2 F_1\left (\frac{3}{2};-\frac{1}{2},2;\frac{5}{2};\cot ^2(x),-\cot ^2(x)\right )+F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};\cot ^2(x),-\cot ^2(x)\right )-3 \tan ^2(x) F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};\cot ^2(x),-\cot ^2(x)\right )}-4 \tan ^3(x)-3 \cot (x)+18 \sin ^2(x) \tan (x)-4 \sin (x) \cos (x)-9 \sin (x) \cos (x) \tan ^{-1}\left (\sqrt{\tan ^2(x)-1}\right ) \sqrt{\tan ^2(x)-1}\right )}{6 (6 \cos (2 x)+\cos (4 x)-3) (\tan (x) \tan (2 x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(Sec[x]^2*(-Cos[2*x] + 2*Tan[x]^2))/(Tan[x]*Tan[2*x])^(3/2),x]

[Out]

((-Cos[2*x] + 2*Tan[x]^2)*(-3*Cot[x] - 4*Cos[x]*Sin[x] + 18*Sin[x]^2*Tan[x] - 4*
Tan[x]^3 - 9*ArcTan[Sqrt[-1 + Tan[x]^2]]*Cos[x]*Sin[x]*Sqrt[-1 + Tan[x]^2] - (72
*AppellF1[1/2, -1/2, 1, 3/2, Cot[x]^2, -Cot[x]^2]*Cos[2*x]*Sin[x]^2*Tan[x])/(2*A
ppellF1[3/2, -1/2, 2, 5/2, Cot[x]^2, -Cot[x]^2] + AppellF1[3/2, 1/2, 1, 5/2, Cot
[x]^2, -Cot[x]^2] - 3*AppellF1[1/2, -1/2, 1, 3/2, Cot[x]^2, -Cot[x]^2]*Tan[x]^2)
)*Tan[2*x]^2)/(6*(-3 + 6*Cos[2*x] + Cos[4*x])*(Tan[x]*Tan[2*x])^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.748, size = 559, normalized size = 5.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-cos(2*x)+2*tan(x)^2)/cos(x)^2/(tan(x)*tan(2*x))^(3/2),x)

[Out]

-1/96*2^(1/2)*4^(1/2)*(cos(x)-1)^2*(-48*cos(x)^4*2^(1/2)*arctanh(1/2*2^(1/2)*cos
(x)*4^(1/2)*(cos(x)-1)/sin(x)^2/((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2))+22*cos(x)^4
*((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2)+33*cos(x)^4*arctanh(1/2*4^(1/2)*(2*cos(x)^2
-3*cos(x)+1)/sin(x)^2/((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2))-168*cos(x)^4*ln(-4*(c
os(x)^2*((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2)-2*cos(x)^2+cos(x)-((2*cos(x)^2-1)/(1
+cos(x))^2)^(1/2)+1)/sin(x)^2)+201*cos(x)^4*ln(-2*(cos(x)^2*((2*cos(x)^2-1)/(1+c
os(x))^2)^(1/2)-2*cos(x)^2+cos(x)-((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2)+1)/sin(x)^
2)+48*cos(x)^3*2^(1/2)*arctanh(1/2*2^(1/2)*cos(x)*4^(1/2)*(cos(x)-1)/sin(x)^2/((
2*cos(x)^2-1)/(1+cos(x))^2)^(1/2))-33*cos(x)^3*arctanh(1/2*4^(1/2)*(2*cos(x)^2-3
*cos(x)+1)/sin(x)^2/((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2))+168*cos(x)^3*ln(-4*(cos
(x)^2*((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2)-2*cos(x)^2+cos(x)-((2*cos(x)^2-1)/(1+c
os(x))^2)^(1/2)+1)/sin(x)^2)-201*cos(x)^3*ln(-2*(cos(x)^2*((2*cos(x)^2-1)/(1+cos
(x))^2)^(1/2)-2*cos(x)^2+cos(x)-((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2)+1)/sin(x)^2)
-36*cos(x)^2*((2*cos(x)^2-1)/(1+cos(x))^2)^(1/2)+8*((2*cos(x)^2-1)/(1+cos(x))^2)
^(1/2))/sin(x)^3/cos(x)^3/((2*cos(x)^2-1)/(1+cos(x))^2)^(3/2)/(sin(x)^2/(2*cos(x
)^2-1))^(3/2)

_______________________________________________________________________________________

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*tan(x)^2 - cos(2*x))/((tan(2*x)*tan(x))^(3/2)*cos(x)^2),x, algorithm="maxima")

[Out]

Timed out

_______________________________________________________________________________________

Fricas [A]  time = 0.336954, size = 498, normalized size = 4.98 \[ -\frac{24 \,{\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \log \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (32 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} + 1\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} - 8 \,{\left (4 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right )\right )}}{2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}}}\right ) \sin \left (x\right ) - 33 \,{\left (\sqrt{2} \cos \left (x\right )^{5} - \sqrt{2} \cos \left (x\right )^{3}\right )} \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (32 \, \sqrt{2} \cos \left (x\right )^{6} - 16 \,{\left (3 \, \sqrt{2} - 2\right )} \cos \left (x\right )^{4} +{\left (25 \, \sqrt{2} - 28\right )} \cos \left (x\right )^{2} - 3 \, \sqrt{2} + 4\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} - 4 \,{\left (8 \, \sqrt{2} \cos \left (x\right )^{5} - 2 \,{\left (5 \, \sqrt{2} - 4\right )} \cos \left (x\right )^{3} +{\left (4 \, \sqrt{2} - 5\right )} \cos \left (x\right )\right )} \sin \left (x\right )\right )}}{\sqrt{2}{\left (32 \, \cos \left (x\right )^{6} - 48 \, \cos \left (x\right )^{4} + 17 \, \cos \left (x\right )^{2} - 1\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} - 8 \,{\left (4 \, \cos \left (x\right )^{5} - 5 \, \cos \left (x\right )^{3} + \cos \left (x\right )\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 2 \, \sqrt{2}{\left (22 \, \cos \left (x\right )^{6} - 47 \, \cos \left (x\right )^{4} + 26 \, \cos \left (x\right )^{2} - 4\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} - 44 \,{\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right )}{48 \,{\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(24*(cos(x)^5 - cos(x)^3)*log(1/2*sqrt(2)*(sqrt(2)*(32*cos(x)^4 - 16*cos(x
)^2 + 1)*sqrt(-(cos(x)^2 - 1)/(2*cos(x)^2 - 1)) - 8*(4*cos(x)^3 - cos(x))*sin(x)
)/sqrt(-(cos(x)^2 - 1)/(2*cos(x)^2 - 1)))*sin(x) - 33*(sqrt(2)*cos(x)^5 - sqrt(2
)*cos(x)^3)*log(-2*(sqrt(2)*(32*sqrt(2)*cos(x)^6 - 16*(3*sqrt(2) - 2)*cos(x)^4 +
(25*sqrt(2) - 28)*cos(x)^2 - 3*sqrt(2) + 4)*sqrt(-(cos(x)^2 - 1)/(2*cos(x)^2 -
1)) - 4*(8*sqrt(2)*cos(x)^5 - 2*(5*sqrt(2) - 4)*cos(x)^3 + (4*sqrt(2) - 5)*cos(x
))*sin(x))/(sqrt(2)*(32*cos(x)^6 - 48*cos(x)^4 + 17*cos(x)^2 - 1)*sqrt(-(cos(x)^
2 - 1)/(2*cos(x)^2 - 1)) - 8*(4*cos(x)^5 - 5*cos(x)^3 + cos(x))*sin(x)))*sin(x)
- 2*sqrt(2)*(22*cos(x)^6 - 47*cos(x)^4 + 26*cos(x)^2 - 4)*sqrt(-(cos(x)^2 - 1)/(
2*cos(x)^2 - 1)) - 44*(cos(x)^5 - cos(x)^3)*sin(x))/((cos(x)^5 - cos(x)^3)*sin(x
))

_______________________________________________________________________________________

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(2*x)+2*tan(x)**2)/cos(x)**2/(tan(x)*tan(2*x))**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.336899, size = 265, normalized size = 2.65 \[ \frac{11 \, \sqrt{2}{\rm ln}\left (\sqrt{-\tan \left (x\right )^{2} + 1} + 1\right )}{16 \,{\rm sign}\left (\tan \left (x\right )^{2} - 1\right ){\rm sign}\left (\tan \left (x\right )\right )} - \frac{11 \, \sqrt{2}{\rm ln}\left (-\sqrt{-\tan \left (x\right )^{2} + 1} + 1\right )}{16 \,{\rm sign}\left (\tan \left (x\right )^{2} - 1\right ){\rm sign}\left (\tan \left (x\right )\right )} - \frac{2 \, \sqrt{2}{\left (-\tan \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{2} \sqrt{-\tan \left (x\right )^{2} + 1}}{12 \,{\rm sign}\left (\tan \left (x\right )^{2} - 1\right ){\rm sign}\left (\tan \left (x\right )\right )} + \frac{{\rm ln}\left (\frac{\sqrt{2} - \sqrt{-\tan \left (x\right )^{2} + 1}}{\sqrt{2} + \sqrt{-\tan \left (x\right )^{2} + 1}}\right )}{{\rm sign}\left (\tan \left (x\right )^{2} - 1\right ){\rm sign}\left (\tan \left (x\right )\right )} - \frac{\sqrt{2} \sqrt{-\tan \left (x\right )^{2} + 1}}{8 \,{\rm sign}\left (\tan \left (x\right )^{2} - 1\right ){\rm sign}\left (\tan \left (x\right )\right ) \tan \left (x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/16*sqrt(2)*ln(sqrt(-tan(x)^2 + 1) + 1)/(sign(tan(x)^2 - 1)*sign(tan(x))) - 11
/16*sqrt(2)*ln(-sqrt(-tan(x)^2 + 1) + 1)/(sign(tan(x)^2 - 1)*sign(tan(x))) - 1/1
2*(2*sqrt(2)*(-tan(x)^2 + 1)^(3/2) + 3*sqrt(2)*sqrt(-tan(x)^2 + 1))/(sign(tan(x)
^2 - 1)*sign(tan(x))) + ln((sqrt(2) - sqrt(-tan(x)^2 + 1))/(sqrt(2) + sqrt(-tan(
x)^2 + 1)))/(sign(tan(x)^2 - 1)*sign(tan(x))) - 1/8*sqrt(2)*sqrt(-tan(x)^2 + 1)/
(sign(tan(x)^2 - 1)*sign(tan(x))*tan(x)^2)