3.9 \(\int \frac{1}{\sqrt{-1+x^2} \left (\sqrt{x}+\sqrt{-1+x^2}\right )^2} \, dx\)

Optimal. Leaf size=220 \[ \frac{2-4 x}{5 \left (\sqrt{x^2-1}+\sqrt{x}\right )}-\frac{1}{50} \sqrt{50 \sqrt{5}-110} \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{5}-2} \sqrt{x^2-1}}{2-\left (1-\sqrt{5}\right ) x}\right )-\frac{1}{50} \sqrt{110+50 \sqrt{5}} \tanh ^{-1}\left (\frac{\sqrt{2+2 \sqrt{5}} \sqrt{x^2-1}}{-\sqrt{5} x-x+2}\right )+\frac{1}{25} \sqrt{50 \sqrt{5}-110} \tan ^{-1}\left (\frac{1}{2} \sqrt{2+2 \sqrt{5}} \sqrt{x}\right )-\frac{1}{25} \sqrt{110+50 \sqrt{5}} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2 \sqrt{5}-2} \sqrt{x}\right ) \]

[Out]

(2 - 4*x)/(5*(Sqrt[x] + Sqrt[-1 + x^2])) + (Sqrt[-110 + 50*Sqrt[5]]*ArcTan[(Sqrt
[2 + 2*Sqrt[5]]*Sqrt[x])/2])/25 - (Sqrt[-110 + 50*Sqrt[5]]*ArcTan[(Sqrt[-2 + 2*S
qrt[5]]*Sqrt[-1 + x^2])/(2 - (1 - Sqrt[5])*x)])/50 - (Sqrt[110 + 50*Sqrt[5]]*Arc
Tanh[(Sqrt[-2 + 2*Sqrt[5]]*Sqrt[x])/2])/25 - (Sqrt[110 + 50*Sqrt[5]]*ArcTanh[(Sq
rt[2 + 2*Sqrt[5]]*Sqrt[-1 + x^2])/(2 - x - Sqrt[5]*x)])/50

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Rubi [A]  time = 0.997216, antiderivative size = 365, normalized size of antiderivative = 1.66, number of steps used = 18, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ -\frac{2 \sqrt{x^2-1} (1-2 x)}{5 \left (-x^2+x+1\right )}+\frac{2 \sqrt{x} (1-2 x)}{5 \left (-x^2+x+1\right )}-\frac{2}{5} \sqrt{\frac{1}{5} \left (5 \sqrt{5}-2\right )} \tan ^{-1}\left (\frac{2-\left (1-\sqrt{5}\right ) x}{\sqrt{2 \left (\sqrt{5}-1\right )} \sqrt{x^2-1}}\right )+\sqrt{\frac{2}{5 \left (\sqrt{5}-1\right )}} \tan ^{-1}\left (\frac{2-\left (1-\sqrt{5}\right ) x}{\sqrt{2 \left (\sqrt{5}-1\right )} \sqrt{x^2-1}}\right )-\frac{2}{5} \sqrt{\frac{1}{5} \left (2+5 \sqrt{5}\right )} \tanh ^{-1}\left (\frac{2-\left (1+\sqrt{5}\right ) x}{\sqrt{2 \left (1+\sqrt{5}\right )} \sqrt{x^2-1}}\right )+\sqrt{\frac{2}{5 \left (1+\sqrt{5}\right )}} \tanh ^{-1}\left (\frac{2-\left (1+\sqrt{5}\right ) x}{\sqrt{2 \left (1+\sqrt{5}\right )} \sqrt{x^2-1}}\right )+\frac{1}{5} \sqrt{\frac{2}{5} \left (5 \sqrt{5}-11\right )} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x}\right )-\frac{1}{5} \sqrt{\frac{2}{5} \left (11+5 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Int[1/(Sqrt[-1 + x^2]*(Sqrt[x] + Sqrt[-1 + x^2])^2),x]

[Out]

(2*(1 - 2*x)*Sqrt[x])/(5*(1 + x - x^2)) - (2*(1 - 2*x)*Sqrt[-1 + x^2])/(5*(1 + x
- x^2)) + (Sqrt[(2*(-11 + 5*Sqrt[5]))/5]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[x]]
)/5 + Sqrt[2/(5*(-1 + Sqrt[5]))]*ArcTan[(2 - (1 - Sqrt[5])*x)/(Sqrt[2*(-1 + Sqrt
[5])]*Sqrt[-1 + x^2])] - (2*Sqrt[(-2 + 5*Sqrt[5])/5]*ArcTan[(2 - (1 - Sqrt[5])*x
)/(Sqrt[2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 - (Sqrt[(2*(11 + 5*Sqrt[5]))/5]*Ar
cTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]])/5 + Sqrt[2/(5*(1 + Sqrt[5]))]*ArcTanh[(2 -
(1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])] - (2*Sqrt[(2 + 5*Sqrt[
5])/5]*ArcTanh[(2 - (1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*Integral(x/((x + sqrt(x**4 - 1))**2*sqrt(x**4 - 1)), (x, sqrt(x)))

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Mathematica [A]  time = 4.93058, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-1+x^2} \left (\sqrt{x}+\sqrt{-1+x^2}\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/(Sqrt[-1 + x^2]*(Sqrt[x] + Sqrt[-1 + x^2])^2),x]

[Out]

Integrate[1/(Sqrt[-1 + x^2]*(Sqrt[x] + Sqrt[-1 + x^2])^2), x]

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Maple [B]  time = 0.177, size = 902, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/25*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2
)-1/2))/(2+2*5^(1/2))^(1/2)/(4*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2
)-1/2)+2+2*5^(1/2))^(1/2))-6/25*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)
+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2
+4*(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+2-2*5^(1/2))^(1/2))-1/5/(1/2+1/2*5^(1/2))/(x
-1/2*5^(1/2)-1/2)*((x-1/2*5^(1/2)-1/2)^2+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+1/2+1/2
*5^(1/2))^(1/2)+6/5/(1/2+1/2*5^(1/2))/(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(
5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))^(1/2)/(4*(x-1/2*5^(1/2)-1/2)^2+4*(
5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^(1/2))+2/5/(1/2+1/2*5^(1/2))/(2+2*5^
(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))
^(1/2)/(4*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^(
1/2))*5^(1/2)-1/5*5^(1/2)/(1/2+1/2*5^(1/2))/(x-1/2*5^(1/2)-1/2)*((x-1/2*5^(1/2)-
1/2)^2+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+1/2+1/2*5^(1/2))^(1/2)-1/5/(1/2-1/2*5^(1/
2))/(x+1/2*5^(1/2)-1/2)*((x+1/2*5^(1/2)-1/2)^2+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+
1/2-1/2*5^(1/2))^(1/2)+2/5/(1/2-1/2*5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^
(1/2)+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1
/2)^2+4*(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+2-2*5^(1/2))^(1/2))*5^(1/2)-6/5/(1/2-1/
2*5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)+(-5^(1/2)+1)*(x+1/2*5^(1/2)-
1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/2)+1)*(x+1/2*5^(1/2
)-1/2)+2-2*5^(1/2))^(1/2))+1/5*5^(1/2)/(1/2-1/2*5^(1/2))/(x+1/2*5^(1/2)-1/2)*((x
+1/2*5^(1/2)-1/2)^2+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+1/2-1/2*5^(1/2))^(1/2)+2/5*
x^(1/2)/(x+1/2*5^(1/2)-1/2)+4/5/(-2+2*5^(1/2))^(1/2)*arctan(2*x^(1/2)/(-2+2*5^(1
/2))^(1/2))-8/25/(-2+2*5^(1/2))^(1/2)*arctan(2*x^(1/2)/(-2+2*5^(1/2))^(1/2))*5^(
1/2)+2/5*x^(1/2)/(x-1/2*5^(1/2)-1/2)-4/5/(2+2*5^(1/2))^(1/2)*arctanh(2*x^(1/2)/(
2+2*5^(1/2))^(1/2))-8/25/(2+2*5^(1/2))^(1/2)*arctanh(2*x^(1/2)/(2+2*5^(1/2))^(1/
2))*5^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 1}{\left (\sqrt{x^{2} - 1} + \sqrt{x}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 1)*(sqrt(x^2 - 1) + sqrt(x))^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 1)*(sqrt(x^2 - 1) + sqrt(x))^2), x)

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Fricas [A]  time = 0.252677, size = 1185, normalized size = 5.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 1)*(sqrt(x^2 - 1) + sqrt(x))^2),x, algorithm="fricas")

[Out]

1/50*(40*x^3 + 40*x^2 - 4*(2*sqrt(2)*(x^3 - x^2 - x)*sqrt(x^2 - 1)*sqrt(-sqrt(5)
*(11*sqrt(5) - 25)) - sqrt(2)*(2*x^4 - 2*x^3 - 3*x^2 + x + 1)*sqrt(-sqrt(5)*(11*
sqrt(5) - 25)))*arctan(-1/2*sqrt(2)*sqrt(-sqrt(5)*(11*sqrt(5) - 25))*(sqrt(5) +
3)/(sqrt(5)*(2*x - 1) - 2*sqrt(5)*sqrt(x^2 - 1) - 2*sqrt(sqrt(5)*(sqrt(5)*(2*x^2
- x) - sqrt(x^2 - 1)*(sqrt(5)*(2*x - 1) + 5) + 5*x)) + 5)) + 4*(2*sqrt(2)*(x^3
- x^2 - x)*sqrt(x^2 - 1)*sqrt(-sqrt(5)*(11*sqrt(5) - 25)) - sqrt(2)*(2*x^4 - 2*x
^3 - 3*x^2 + x + 1)*sqrt(-sqrt(5)*(11*sqrt(5) - 25)))*arctan(1/2*sqrt(2)*sqrt(-s
qrt(5)*(11*sqrt(5) - 25))*(sqrt(5) + 3)/(sqrt(2)*sqrt(sqrt(5)*(sqrt(5)*(2*x - 1)
+ 5)) + 2*sqrt(5)*sqrt(x))) + (2*sqrt(2)*(x^3 - x^2 - x)*sqrt(x^2 - 1)*sqrt(sqr
t(5)*(11*sqrt(5) + 25)) - sqrt(2)*(2*x^4 - 2*x^3 - 3*x^2 + x + 1)*sqrt(sqrt(5)*(
11*sqrt(5) + 25)))*log(sqrt(2)*sqrt(sqrt(5)*(11*sqrt(5) + 25))*(sqrt(5) - 3) - 2
*sqrt(5)*(2*x - 1) + 4*sqrt(5)*sqrt(x^2 - 1) + 10) - (2*sqrt(2)*(x^3 - x^2 - x)*
sqrt(x^2 - 1)*sqrt(sqrt(5)*(11*sqrt(5) + 25)) - sqrt(2)*(2*x^4 - 2*x^3 - 3*x^2 +
x + 1)*sqrt(sqrt(5)*(11*sqrt(5) + 25)))*log(sqrt(2)*sqrt(sqrt(5)*(11*sqrt(5) +
25))*(sqrt(5) - 3) + 4*sqrt(5)*sqrt(x)) - (2*sqrt(2)*(x^3 - x^2 - x)*sqrt(x^2 -
1)*sqrt(sqrt(5)*(11*sqrt(5) + 25)) - sqrt(2)*(2*x^4 - 2*x^3 - 3*x^2 + x + 1)*sqr
t(sqrt(5)*(11*sqrt(5) + 25)))*log(-sqrt(2)*sqrt(sqrt(5)*(11*sqrt(5) + 25))*(sqrt
(5) - 3) - 2*sqrt(5)*(2*x - 1) + 4*sqrt(5)*sqrt(x^2 - 1) + 10) + (2*sqrt(2)*(x^3
- x^2 - x)*sqrt(x^2 - 1)*sqrt(sqrt(5)*(11*sqrt(5) + 25)) - sqrt(2)*(2*x^4 - 2*x
^3 - 3*x^2 + x + 1)*sqrt(sqrt(5)*(11*sqrt(5) + 25)))*log(-sqrt(2)*sqrt(sqrt(5)*(
11*sqrt(5) + 25))*(sqrt(5) - 3) + 4*sqrt(5)*sqrt(x)) - 20*(2*x^2 + 2*(2*x^2 - x)
*sqrt(x) + 2*x + 1)*sqrt(x^2 - 1) + 20*(4*x^3 - 2*x^2 - 2*x + 1)*sqrt(x) - 40)/(
2*x^4 - 2*x^3 - 3*x^2 - 2*(x^3 - x^2 - x)*sqrt(x^2 - 1) + x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt((x - 1)*(x + 1))*(sqrt(x) + sqrt(x**2 - 1))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 1)*(sqrt(x^2 - 1) + sqrt(x))^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 1)*(sqrt(x^2 - 1) + sqrt(x))^2), x)