3.438 \(\int \frac{\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2}{15 \sqrt{5-4 \sec ^2(x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{5-4 \sec ^2(x)}}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{5-4 \sec ^2(x)}}{\sqrt{5}}\right )}{5 \sqrt{5}} \]

[Out]

-ArcTanh[Sqrt[5 - 4*Sec[x]^2]/Sqrt[3]]/(6*Sqrt[3]) - ArcTanh[Sqrt[5 - 4*Sec[x]^2
]/Sqrt[5]]/(5*Sqrt[5]) - 2/(15*Sqrt[5 - 4*Sec[x]^2])

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Rubi [A]  time = 2.21203, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387 \[ -\frac{2}{15 \sqrt{5-4 \sec ^2(x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{5-4 \sec ^2(x)}}{\sqrt{3}}\right )}{6 \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{5-4 \sec ^2(x)}}{\sqrt{5}}\right )}{5 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((3 + Sin[x]^2)*Tan[x]^3)/((-2 + Cos[x]^2)*(5 - 4*Sec[x]^2)^(3/2)),x]

[Out]

-ArcTanh[Sqrt[5 - 4*Sec[x]^2]/Sqrt[3]]/(6*Sqrt[3]) - ArcTanh[Sqrt[5 - 4*Sec[x]^2
]/Sqrt[5]]/(5*Sqrt[5]) - 2/(15*Sqrt[5 - 4*Sec[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((3+sin(x)**2)*tan(x)**3/(-2+cos(x)**2)/(5-4*sec(x)**2)**(3/2),x)

[Out]

Timed out

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Mathematica [B]  time = 2.17114, size = 234, normalized size = 3.21 \[ \frac{\sec ^2(x) \left (-20 \cos (2 x)+\frac{\sqrt{2} (5 \cos (2 x)-3)^{3/2} \left (15 \sqrt{3} \sin ^2(x) \tanh ^{-1}\left (\frac{\sqrt{5 \cos (2 x)-3}}{\sqrt{6} \sqrt{\cos ^2(x)}}\right )-18 \sqrt{5} \sin ^2(x) \left (\log \left (10 \sin ^2(x)\right )-\log \left (5 \left (\sqrt{10} \sqrt{\sin ^2(x)} \sqrt{\sin ^2(2 x)}+\sqrt{5 \cos (2 x)-3} \cos (2 x)-\sqrt{5 \cos (2 x)-3}\right )\right )\right )-20 \sqrt{3} \sqrt{\sin ^2(x)} \sqrt{\sin ^2(2 x)} \sec (x) \tanh ^{-1}\left (\frac{\sqrt{6} \cos (x)}{\sqrt{5 \cos (2 x)-3}}\right )\right )}{15 \sqrt{\sin ^2(x)} \sqrt{\sin ^2(2 x)}}+12\right )}{60 \left (5-4 \sec ^2(x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sec[x]^2*(12 - 20*Cos[2*x] + (Sqrt[2]*(-3 + 5*Cos[2*x])^(3/2)*(15*Sqrt[3]*ArcTa
nh[Sqrt[-3 + 5*Cos[2*x]]/(Sqrt[6]*Sqrt[Cos[x]^2])]*Sin[x]^2 - 18*Sqrt[5]*(Log[10
*Sin[x]^2] - Log[5*(-Sqrt[-3 + 5*Cos[2*x]] + Cos[2*x]*Sqrt[-3 + 5*Cos[2*x]] + Sq
rt[10]*Sqrt[Sin[x]^2]*Sqrt[Sin[2*x]^2])])*Sin[x]^2 - 20*Sqrt[3]*ArcTanh[(Sqrt[6]
*Cos[x])/Sqrt[-3 + 5*Cos[2*x]]]*Sec[x]*Sqrt[Sin[x]^2]*Sqrt[Sin[2*x]^2]))/(15*Sqr
t[Sin[x]^2]*Sqrt[Sin[2*x]^2])))/(60*(5 - 4*Sec[x]^2)^(3/2))

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Maple [B]  time = 0.345, size = 1597, normalized size = 21.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+sin(x)^2)*tan(x)^3/(cos(x)^2-2)/(5-4*sec(x)^2)^(3/2),x)

[Out]

3/5/(5+2*5^(1/2))/(-5+2*5^(1/2))/(-6+2*5^(1/2)+2^(1/2))/(-6+2*5^(1/2)-2^(1/2))/(
6+2*5^(1/2)-2^(1/2))/(-2*3^(1/2)+6^(1/2))/(6+2*5^(1/2)+2^(1/2))/(2*3^(1/2)+6^(1/
2))*(-4+5*cos(x)^2)*(-25*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(-2*3^
(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x
)^2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2))*6^(1/2)*cos(x)*2^(1/2)-50*((-4+5*cos(x
)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(-2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*
cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1
/2))*cos(x)*2^(1/2)*3^(1/2)-25*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/
(2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/
sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2))*6^(1/2)*cos(x)*2^(1/2)+50*((-4+5*
cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)
*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2
)^(1/2))*cos(x)*2^(1/2)*3^(1/2)+72*5^(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*
arctanh(1/2*5^(1/2)*cos(x)*4^(1/2)*(cos(x)-1)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x
))^2)^(1/2))*cos(x)+50*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(-2*3^(1
/2)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x)^
2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2))*6^(1/2)*cos(x)-25*arctanh(1/2/(-2*3^(1/2
)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x)^2/
((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2))*6^(1/2)*2^(1/2)*((-4+5*cos(x)^2)/(1+cos(x)
)^2)^(1/2)+100*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(-2*3^(1/2)+6^(1
/2))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x)^2/((-4+5
*cos(x)^2)/(1+cos(x))^2)^(1/2))*cos(x)*3^(1/2)-50*arctanh(1/2/(-2*3^(1/2)+6^(1/2
))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x)^2/((-4+5*c
os(x)^2)/(1+cos(x))^2)^(1/2))*2^(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*3^(1/
2)-50*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(2*3^(1/2)+6^(1/2))*4^(1/
2)*(cos(x)-1)*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*cos(x)^2)
/(1+cos(x))^2)^(1/2))*6^(1/2)*cos(x)-25*arctanh(1/2/(2*3^(1/2)+6^(1/2))*4^(1/2)*
(cos(x)-1)*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*cos(x)^2)/(1
+cos(x))^2)^(1/2))*6^(1/2)*2^(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)+100*((-4
+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*arctanh(1/2/(2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)
-1)*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x)
)^2)^(1/2))*cos(x)*3^(1/2)+50*arctanh(1/2/(2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)
*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2
)^(1/2))*2^(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*3^(1/2)+72*arctanh(1/2*5^(
1/2)*cos(x)*4^(1/2)*(cos(x)-1)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2))*5^
(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)+50*arctanh(1/2/(-2*3^(1/2)+6^(1/2))*4
^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(1/2)-4)/sin(x)^2/((-4+5*cos(x
)^2)/(1+cos(x))^2)^(1/2))*6^(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)+100*arcta
nh(1/2/(-2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)-10*cos(x)+4*2^(
1/2)-4)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2))*((-4+5*cos(x)^2)/(1+cos(x
))^2)^(1/2)*3^(1/2)-50*arctanh(1/2/(2*3^(1/2)+6^(1/2))*4^(1/2)*(cos(x)-1)*(5*cos
(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)
)*6^(1/2)*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)+100*arctanh(1/2/(2*3^(1/2)+6^(1/2
))*4^(1/2)*(cos(x)-1)*(5*cos(x)*2^(1/2)+10*cos(x)+4*2^(1/2)+4)/sin(x)^2/((-4+5*c
os(x)^2)/(1+cos(x))^2)^(1/2))*((-4+5*cos(x)^2)/(1+cos(x))^2)^(1/2)*3^(1/2)-240*c
os(x))/cos(x)^3/((-4+5*cos(x)^2)/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((sin(x)^2 + 3)*tan(x)^3/((cos(x)^2 - 2)*(-4*sec(x)^2 + 5)^(3/2)),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.314283, size = 358, normalized size = 4.9 \[ -\frac{480 \, \sqrt{\frac{5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}} \cos \left (x\right )^{2} - 18 \,{\left (5 \, \sqrt{5} \cos \left (x\right )^{2} - 4 \, \sqrt{5}\right )} \log \left (625 \, \sqrt{5} \cos \left (x\right )^{8} - 1000 \, \sqrt{5} \cos \left (x\right )^{6} + 500 \, \sqrt{5} \cos \left (x\right )^{4} - 80 \, \sqrt{5} \cos \left (x\right )^{2} - 5 \,{\left (125 \, \cos \left (x\right )^{8} - 150 \, \cos \left (x\right )^{6} + 50 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2}\right )} \sqrt{\frac{5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}} + 2 \, \sqrt{5}\right ) - 25 \,{\left (5 \, \sqrt{3} \cos \left (x\right )^{2} - 4 \, \sqrt{3}\right )} \log \left (\frac{1921 \, \sqrt{3} \cos \left (x\right )^{8} - 3464 \, \sqrt{3} \cos \left (x\right )^{6} + 2040 \, \sqrt{3} \cos \left (x\right )^{4} - 416 \, \sqrt{3} \cos \left (x\right )^{2} - 24 \,{\left (62 \, \cos \left (x\right )^{8} - 87 \, \cos \left (x\right )^{6} + 36 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2}\right )} \sqrt{\frac{5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}} + 16 \, \sqrt{3}}{\cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + 24 \, \cos \left (x\right )^{4} - 32 \, \cos \left (x\right )^{2} + 16}\right )}{3600 \,{\left (5 \, \cos \left (x\right )^{2} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3600*(480*sqrt((5*cos(x)^2 - 4)/cos(x)^2)*cos(x)^2 - 18*(5*sqrt(5)*cos(x)^2 -
4*sqrt(5))*log(625*sqrt(5)*cos(x)^8 - 1000*sqrt(5)*cos(x)^6 + 500*sqrt(5)*cos(x
)^4 - 80*sqrt(5)*cos(x)^2 - 5*(125*cos(x)^8 - 150*cos(x)^6 + 50*cos(x)^4 - 4*cos
(x)^2)*sqrt((5*cos(x)^2 - 4)/cos(x)^2) + 2*sqrt(5)) - 25*(5*sqrt(3)*cos(x)^2 - 4
*sqrt(3))*log((1921*sqrt(3)*cos(x)^8 - 3464*sqrt(3)*cos(x)^6 + 2040*sqrt(3)*cos(
x)^4 - 416*sqrt(3)*cos(x)^2 - 24*(62*cos(x)^8 - 87*cos(x)^6 + 36*cos(x)^4 - 4*co
s(x)^2)*sqrt((5*cos(x)^2 - 4)/cos(x)^2) + 16*sqrt(3))/(cos(x)^8 - 8*cos(x)^6 + 2
4*cos(x)^4 - 32*cos(x)^2 + 16)))/(5*cos(x)^2 - 4)

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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((3+sin(x)**2)*tan(x)**3/(-2+cos(x)**2)/(5-4*sec(x)**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.324906, size = 231, normalized size = 3.16 \[ -\frac{1}{4500} \, \sqrt{15} \sqrt{5}{\left (6 i \, \sqrt{15} \pi + 6 \, \sqrt{15}{\rm ln}\left (4\right ) - 25 \,{\rm ln}\left (-\frac{\sqrt{15} + 5}{\sqrt{15} - 5}\right )\right )}{\rm sign}\left (\cos \left (x\right )\right ) - \frac{\sqrt{15} \sqrt{5}{\rm ln}\left (-\frac{2 \,{\left ({\left (\sqrt{5} \cos \left (x\right ) - \sqrt{5 \, \cos \left (x\right )^{2} - 4}\right )}^{2} - 4 \, \sqrt{15} - 16\right )}}{{\left | 2 \,{\left (\sqrt{5} \cos \left (x\right ) - \sqrt{5 \, \cos \left (x\right )^{2} - 4}\right )}^{2} + 8 \, \sqrt{15} - 32 \right |}}\right )}{180 \,{\rm sign}\left (\cos \left (x\right )\right )} + \frac{\sqrt{5}{\rm ln}\left ({\left (\sqrt{5} \cos \left (x\right ) - \sqrt{5 \, \cos \left (x\right )^{2} - 4}\right )}^{2}\right )}{50 \,{\rm sign}\left (\cos \left (x\right )\right )} - \frac{2 \, \cos \left (x\right )}{15 \, \sqrt{5 \, \cos \left (x\right )^{2} - 4}{\rm sign}\left (\cos \left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4500*sqrt(15)*sqrt(5)*(6*I*sqrt(15)*pi + 6*sqrt(15)*ln(4) - 25*ln(-(sqrt(15)
+ 5)/(sqrt(15) - 5)))*sign(cos(x)) - 1/180*sqrt(15)*sqrt(5)*ln(-2*((sqrt(5)*cos(
x) - sqrt(5*cos(x)^2 - 4))^2 - 4*sqrt(15) - 16)/abs(2*(sqrt(5)*cos(x) - sqrt(5*c
os(x)^2 - 4))^2 + 8*sqrt(15) - 32))/sign(cos(x)) + 1/50*sqrt(5)*ln((sqrt(5)*cos(
x) - sqrt(5*cos(x)^2 - 4))^2)/sign(cos(x)) - 2/15*cos(x)/(sqrt(5*cos(x)^2 - 4)*s
ign(cos(x)))