3.8 \(\int \sqrt{2+2 \tan (x)+\tan ^2(x)} \, dx\)

Optimal. Leaf size=137 \[ -\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{5}-\left (5+\sqrt{5}\right ) \tan (x)}{\sqrt{10 \left (1+\sqrt{5}\right )} \sqrt{\tan ^2(x)+2 \tan (x)+2}}\right )-\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\left (5-\sqrt{5}\right ) \tan (x)+2 \sqrt{5}}{\sqrt{10 \left (\sqrt{5}-1\right )} \sqrt{\tan ^2(x)+2 \tan (x)+2}}\right )+\sinh ^{-1}(\tan (x)+1) \]

[Out]

ArcSinh[1 + Tan[x]] - Sqrt[(1 + Sqrt[5])/2]*ArcTan[(2*Sqrt[5] - (5 + Sqrt[5])*Ta
n[x])/(Sqrt[10*(1 + Sqrt[5])]*Sqrt[2 + 2*Tan[x] + Tan[x]^2])] - Sqrt[(-1 + Sqrt[
5])/2]*ArcTanh[(2*Sqrt[5] + (5 - Sqrt[5])*Tan[x])/(Sqrt[10*(-1 + Sqrt[5])]*Sqrt[
2 + 2*Tan[x] + Tan[x]^2])]

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Rubi [A]  time = 0.39316, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{5}-\left (5+\sqrt{5}\right ) \tan (x)}{\sqrt{10 \left (1+\sqrt{5}\right )} \sqrt{\tan ^2(x)+2 \tan (x)+2}}\right )-\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\left (5-\sqrt{5}\right ) \tan (x)+2 \sqrt{5}}{\sqrt{10 \left (\sqrt{5}-1\right )} \sqrt{\tan ^2(x)+2 \tan (x)+2}}\right )+\sinh ^{-1}(\tan (x)+1) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + 2*Tan[x] + Tan[x]^2],x]

[Out]

ArcSinh[1 + Tan[x]] - Sqrt[(1 + Sqrt[5])/2]*ArcTan[(2*Sqrt[5] - (5 + Sqrt[5])*Ta
n[x])/(Sqrt[10*(1 + Sqrt[5])]*Sqrt[2 + 2*Tan[x] + Tan[x]^2])] - Sqrt[(-1 + Sqrt[
5])/2]*ArcTanh[(2*Sqrt[5] + (5 - Sqrt[5])*Tan[x])/(Sqrt[10*(-1 + Sqrt[5])]*Sqrt[
2 + 2*Tan[x] + Tan[x]^2])]

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Rubi in Sympy [A]  time = 157.999, size = 178, normalized size = 1.3 \[ - \frac{\sqrt{10} \left (2 \sqrt{5} + 10\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\left (-5 - \sqrt{5}\right ) \tan{\left (x \right )} + 2 \sqrt{5}\right )}{10 \sqrt{1 + \sqrt{5}} \sqrt{\tan ^{2}{\left (x \right )} + 2 \tan{\left (x \right )} + 2}} \right )}}{20 \sqrt{1 + \sqrt{5}}} + \operatorname{atanh}{\left (\frac{2 \tan{\left (x \right )} + 2}{2 \sqrt{\tan ^{2}{\left (x \right )} + 2 \tan{\left (x \right )} + 2}} \right )} + \frac{\sqrt{10} \left (- 2 \sqrt{5} + 10\right ) \operatorname{atanh}{\left (\frac{\sqrt{10} \left (\left (-5 + \sqrt{5}\right ) \tan{\left (x \right )} - 2 \sqrt{5}\right )}{10 \sqrt{-1 + \sqrt{5}} \sqrt{\tan ^{2}{\left (x \right )} + 2 \tan{\left (x \right )} + 2}} \right )}}{20 \sqrt{-1 + \sqrt{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+2*tan(x)+tan(x)**2)**(1/2),x)

[Out]

-sqrt(10)*(2*sqrt(5) + 10)*atan(sqrt(10)*((-5 - sqrt(5))*tan(x) + 2*sqrt(5))/(10
*sqrt(1 + sqrt(5))*sqrt(tan(x)**2 + 2*tan(x) + 2)))/(20*sqrt(1 + sqrt(5))) + ata
nh((2*tan(x) + 2)/(2*sqrt(tan(x)**2 + 2*tan(x) + 2))) + sqrt(10)*(-2*sqrt(5) + 1
0)*atanh(sqrt(10)*((-5 + sqrt(5))*tan(x) - 2*sqrt(5))/(10*sqrt(-1 + sqrt(5))*sqr
t(tan(x)**2 + 2*tan(x) + 2)))/(20*sqrt(-1 + sqrt(5)))

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Mathematica [C]  time = 31.7642, size = 7376, normalized size = 53.84 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[2 + 2*Tan[x] + Tan[x]^2],x]

[Out]

Result too large to show

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Maple [B]  time = 0.25, size = 1604, normalized size = 11.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+2*tan(x)+tan(x)^2)^(1/2),x)

[Out]

arcsinh(1+tan(x))-1/10*(10*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))
^2-2*5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+10+2*5^(1/2
))^(1/2)*5^(1/2)*(3*(-10+10*5^(1/2))^(1/2)*arctan(1/80*(-1/2*5^(1/2)+1/2+tan(x))
/(-1/2*5^(1/2)-1/2-tan(x))*(5^(1/2)-5)*(-22+10*5^(1/2))^(1/2)*((5-5^(1/2))*(2*(-
1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+5^(1/2)+3))^(1/2)*(11*5^(1
/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+25*(-1/2*5^(1/2)+1/2
+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+4*5^(1/2)+10)/((-1/2*5^(1/2)+1/2+tan(x))^
4/(-1/2*5^(1/2)-1/2-tan(x))^4+3*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-ta
n(x))^2+1))*5^(1/2)*(-22+10*5^(1/2))^(1/2)+5*(-10+10*5^(1/2))^(1/2)*arctan(1/80*
(-1/2*5^(1/2)+1/2+tan(x))/(-1/2*5^(1/2)-1/2-tan(x))*(5^(1/2)-5)*(-22+10*5^(1/2))
^(1/2)*((5-5^(1/2))*(2*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+5
^(1/2)+3))^(1/2)*(11*5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x
))^2+25*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+4*5^(1/2)+10)/((
-1/2*5^(1/2)+1/2+tan(x))^4/(-1/2*5^(1/2)-1/2-tan(x))^4+3*(-1/2*5^(1/2)+1/2+tan(x
))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+1))*(-22+10*5^(1/2))^(1/2)-20*arctanh((10*(-1/2
*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-2*5^(1/2)*(-1/2*5^(1/2)+1/2+t
an(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+10+2*5^(1/2))^(1/2)/(-10+10*5^(1/2))^(1/2))
*5^(1/2)+60*arctanh((10*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-
2*5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+10+2*5^(1/2))^
(1/2)/(-10+10*5^(1/2))^(1/2)))/(-2*(5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^
(1/2)-1/2-tan(x))^2-5*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-5^
(1/2)-5)/(1+(-1/2*5^(1/2)+1/2+tan(x))/(-1/2*5^(1/2)-1/2-tan(x)))^2)^(1/2)/(1+(-1
/2*5^(1/2)+1/2+tan(x))/(-1/2*5^(1/2)-1/2-tan(x)))/(5^(1/2)-5)/(-10+10*5^(1/2))^(
1/2)-1/5*(10*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-2*5^(1/2)*(
-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+10+2*5^(1/2))^(1/2)*5^(1/
2)*((-10+10*5^(1/2))^(1/2)*arctan(1/80*(-1/2*5^(1/2)+1/2+tan(x))/(-1/2*5^(1/2)-1
/2-tan(x))*(5^(1/2)-5)*(-22+10*5^(1/2))^(1/2)*((5-5^(1/2))*(2*(-1/2*5^(1/2)+1/2+
tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+5^(1/2)+3))^(1/2)*(11*5^(1/2)*(-1/2*5^(1/2
)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+25*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2
*5^(1/2)-1/2-tan(x))^2+4*5^(1/2)+10)/((-1/2*5^(1/2)+1/2+tan(x))^4/(-1/2*5^(1/2)-
1/2-tan(x))^4+3*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+1))*5^(1
/2)*(-22+10*5^(1/2))^(1/2)+5*(-10+10*5^(1/2))^(1/2)*arctan(1/80*(-1/2*5^(1/2)+1/
2+tan(x))/(-1/2*5^(1/2)-1/2-tan(x))*(5^(1/2)-5)*(-22+10*5^(1/2))^(1/2)*((5-5^(1/
2))*(2*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+5^(1/2)+3))^(1/2)
*(11*5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+25*(-1/2*5^
(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+4*5^(1/2)+10)/((-1/2*5^(1/2)+1/2
+tan(x))^4/(-1/2*5^(1/2)-1/2-tan(x))^4+3*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/
2)-1/2-tan(x))^2+1))*(-22+10*5^(1/2))^(1/2)-20*arctanh((10*(-1/2*5^(1/2)+1/2+tan
(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-2*5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5
^(1/2)-1/2-tan(x))^2+10+2*5^(1/2))^(1/2)/(-10+10*5^(1/2))^(1/2))*5^(1/2)+20*arct
anh((10*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-2*5^(1/2)*(-1/2*
5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2+10+2*5^(1/2))^(1/2)/(-10+10*5^
(1/2))^(1/2)))/(-2*(5^(1/2)*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x)
)^2-5*(-1/2*5^(1/2)+1/2+tan(x))^2/(-1/2*5^(1/2)-1/2-tan(x))^2-5^(1/2)-5)/(1+(-1/
2*5^(1/2)+1/2+tan(x))/(-1/2*5^(1/2)-1/2-tan(x)))^2)^(1/2)/(1+(-1/2*5^(1/2)+1/2+t
an(x))/(-1/2*5^(1/2)-1/2-tan(x)))/(5^(1/2)-5)/(-10+10*5^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(tan(x)^2 + 2*tan(x) + 2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.564705, size = 1589, normalized size = 11.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(tan(x)^2 + 2*tan(x) + 2),x, algorithm="fricas")

[Out]

1/4*(2*(sqrt(5) - 1)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3))*log((cos(x)^2 + 4*cos(x)*
sin(x) + 2*(cos(x)^2 + cos(x)*sin(x))*sqrt((cos(x)^2 + 2*cos(x)*sin(x) + 1)/cos(
x)^2) + 2)/cos(x)^2) - 5^(1/4)*(sqrt(5) - 1)*log(125*((3*sqrt(5)*(2*sqrt(5) - 5)
+ 10*sqrt(5) - 25)*cos(x)^2 + 2*(3*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)*c
os(x)*sin(x) + (5^(1/4)*(sqrt(5) - 5)*cos(x)^2 - 5^(1/4)*(3*sqrt(5) - 5)*cos(x)*
sin(x))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3))*sqrt((cos(x)^2 + 2*cos(x)*sin(x) + 1)/
cos(x)^2) + 2*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)/(2*sqrt(5) - 5)) + 5^(1
/4)*(sqrt(5) - 1)*log(125*((3*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)*cos(x)^
2 + 2*(3*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)*cos(x)*sin(x) - (5^(1/4)*(sq
rt(5) - 5)*cos(x)^2 - 5^(1/4)*(3*sqrt(5) - 5)*cos(x)*sin(x))*sqrt((sqrt(5) - 5)/
(sqrt(5) - 3))*sqrt((cos(x)^2 + 2*cos(x)*sin(x) + 1)/cos(x)^2) + 2*sqrt(5)*(2*sq
rt(5) - 5) + 10*sqrt(5) - 25)/(2*sqrt(5) - 5)) + 8*5^(1/4)*arctan(-(2*5^(1/4)*(1
2*sqrt(5) - 19)*cos(x)^2 - 5^(1/4)*(7*sqrt(5) + 41)*cos(x)*sin(x) - (7*(sqrt(5)
- 1)*cos(x)^2 + 24*(sqrt(5) - 1)*cos(x)*sin(x))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)
)*sqrt((cos(x)^2 + 2*cos(x)*sin(x) + 1)/cos(x)^2) - 5^(1/4)*(9*sqrt(5) + 17))/(5
^(1/4)*(7*sqrt(5) + 41)*cos(x)^2 + 2*5^(1/4)*(12*sqrt(5) - 19)*cos(x)*sin(x) + 5
*sqrt(5)*(sqrt(5) - 1)*sqrt(((3*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)*cos(x
)^2 + 2*(3*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)*cos(x)*sin(x) + (5^(1/4)*(
sqrt(5) - 5)*cos(x)^2 - 5^(1/4)*(3*sqrt(5) - 5)*cos(x)*sin(x))*sqrt((sqrt(5) - 5
)/(sqrt(5) - 3))*sqrt((cos(x)^2 + 2*cos(x)*sin(x) + 1)/cos(x)^2) + 2*sqrt(5)*(2*
sqrt(5) - 5) + 10*sqrt(5) - 25)/(2*sqrt(5) - 5))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3
)) + (24*(sqrt(5) - 1)*cos(x)^2 - 7*(sqrt(5) - 1)*cos(x)*sin(x))*sqrt((sqrt(5) -
5)/(sqrt(5) - 3))*sqrt((cos(x)^2 + 2*cos(x)*sin(x) + 1)/cos(x)^2) + 5^(1/4)*(13
*sqrt(5) - 31))) - 8*5^(1/4)*arctan(-(2*5^(1/4)*(12*sqrt(5) - 19)*cos(x)^2 - 5^(
1/4)*(7*sqrt(5) + 41)*cos(x)*sin(x) + (7*(sqrt(5) - 1)*cos(x)^2 + 24*(sqrt(5) -
1)*cos(x)*sin(x))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3))*sqrt((cos(x)^2 + 2*cos(x)*si
n(x) + 1)/cos(x)^2) - 5^(1/4)*(9*sqrt(5) + 17))/(5^(1/4)*(7*sqrt(5) + 41)*cos(x)
^2 + 2*5^(1/4)*(12*sqrt(5) - 19)*cos(x)*sin(x) - 5*sqrt(5)*(sqrt(5) - 1)*sqrt(((
3*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25)*cos(x)^2 + 2*(3*sqrt(5)*(2*sqrt(5)
- 5) + 10*sqrt(5) - 25)*cos(x)*sin(x) - (5^(1/4)*(sqrt(5) - 5)*cos(x)^2 - 5^(1/4
)*(3*sqrt(5) - 5)*cos(x)*sin(x))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3))*sqrt((cos(x)^
2 + 2*cos(x)*sin(x) + 1)/cos(x)^2) + 2*sqrt(5)*(2*sqrt(5) - 5) + 10*sqrt(5) - 25
)/(2*sqrt(5) - 5))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) - (24*(sqrt(5) - 1)*cos(x)^
2 - 7*(sqrt(5) - 1)*cos(x)*sin(x))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3))*sqrt((cos(x
)^2 + 2*cos(x)*sin(x) + 1)/cos(x)^2) + 5^(1/4)*(13*sqrt(5) - 31))))/((sqrt(5) -
1)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\tan ^{2}{\left (x \right )} + 2 \tan{\left (x \right )} + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+2*tan(x)+tan(x)**2)**(1/2),x)

[Out]

Integral(sqrt(tan(x)**2 + 2*tan(x) + 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(tan(x)^2 + 2*tan(x) + 2),x, algorithm="giac")

[Out]

integrate(sqrt(tan(x)^2 + 2*tan(x) + 2), x)