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

Optimal. Leaf size=91 \[ \sqrt{\frac{2}{5} \left (\sqrt{5}-1\right )} \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{\sqrt{5}-2} \sqrt{x-1}}\right )-\cosh ^{-1}(x)+\sqrt{\frac{2}{5} \left (1+\sqrt{5}\right )} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2+\sqrt{5}} \sqrt{x-1}}\right ) \]

[Out]

-ArcCosh[x] + Sqrt[(2*(-1 + Sqrt[5]))/5]*ArcTan[Sqrt[1 + x]/(Sqrt[-2 + Sqrt[5]]*
Sqrt[-1 + x])] + Sqrt[(2*(1 + Sqrt[5]))/5]*ArcTanh[Sqrt[1 + x]/(Sqrt[2 + Sqrt[5]
]*Sqrt[-1 + x])]

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Rubi [B]  time = 0.297751, antiderivative size = 191, normalized size of antiderivative = 2.1, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \sqrt{x-1} \sqrt{x+1} \tan ^{-1}\left (\frac{2-\left (1-\sqrt{5}\right ) x}{\sqrt{2 \left (\sqrt{5}-1\right )} \sqrt{x^2-1}}\right )}{\sqrt{x^2-1}}-\frac{\sqrt{x-1} \sqrt{x+1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right )}{\sqrt{x^2-1}}-\frac{\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \sqrt{x-1} \sqrt{x+1} \tanh ^{-1}\left (\frac{2-\left (1+\sqrt{5}\right ) x}{\sqrt{2 \left (1+\sqrt{5}\right )} \sqrt{x^2-1}}\right )}{\sqrt{x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.4144, size = 187, normalized size = 2.05 \[ \frac{\sqrt{10} \sqrt{-1 + \sqrt{5}} \sqrt{x - 1} \sqrt{x + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x \left (-1 + \sqrt{5}\right ) + 2\right )}{2 \sqrt{-1 + \sqrt{5}} \sqrt{x^{2} - 1}} \right )}}{10 \sqrt{x^{2} - 1}} - \frac{\sqrt{x - 1} \sqrt{x + 1} \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 1}} \right )}}{\sqrt{x^{2} - 1}} - \frac{\sqrt{10} \sqrt{1 + \sqrt{5}} \sqrt{x - 1} \sqrt{x + 1} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (x \left (- \sqrt{5} - 1\right ) + 2\right )}{2 \sqrt{1 + \sqrt{5}} \sqrt{x^{2} - 1}} \right )}}{10 \sqrt{x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(10)*sqrt(-1 + sqrt(5))*sqrt(x - 1)*sqrt(x + 1)*atan(sqrt(2)*(x*(-1 + sqrt(5
)) + 2)/(2*sqrt(-1 + sqrt(5))*sqrt(x**2 - 1)))/(10*sqrt(x**2 - 1)) - sqrt(x - 1)
*sqrt(x + 1)*atanh(x/sqrt(x**2 - 1))/sqrt(x**2 - 1) - sqrt(10)*sqrt(1 + sqrt(5))
*sqrt(x - 1)*sqrt(x + 1)*atanh(sqrt(2)*(x*(-sqrt(5) - 1) + 2)/(2*sqrt(1 + sqrt(5
))*sqrt(x**2 - 1)))/(10*sqrt(x**2 - 1))

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Mathematica [A]  time = 0.775611, size = 143, normalized size = 1.57 \[ -2 \sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )-\frac{\sqrt{\frac{2}{x-1}+1} \sqrt{x-1} \left (\left (\sqrt{5}-3\right ) \sqrt{2+\sqrt{5}} \tan ^{-1}\left (\sqrt{2+\sqrt{5}} \sqrt{\frac{2}{x-1}+1}\right )-\sqrt{\sqrt{5}-2} \left (3+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\sqrt{5}-2} \sqrt{\frac{2}{x-1}+1}\right )\right )}{\sqrt{5} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

-2*ArcSinh[Sqrt[-1 + x]/Sqrt[2]] - (Sqrt[1 + 2/(-1 + x)]*Sqrt[-1 + x]*((-3 + Sqr
t[5])*Sqrt[2 + Sqrt[5]]*ArcTan[Sqrt[2 + Sqrt[5]]*Sqrt[1 + 2/(-1 + x)]] - Sqrt[-2
+ Sqrt[5]]*(3 + Sqrt[5])*ArcTanh[Sqrt[-2 + Sqrt[5]]*Sqrt[1 + 2/(-1 + x)]]))/(Sq
rt[5]*Sqrt[1 + x])

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Maple [B]  time = 0.094, size = 226, normalized size = 2.5 \[ -{\frac{1}{5\,\sqrt{2\,\sqrt{5}+2}\sqrt{-2+2\,\sqrt{5}}}\sqrt{-1+x}\sqrt{1+x} \left ( \arctan \left ({\frac{x\sqrt{5}-x+2}{\sqrt{-2+2\,\sqrt{5}}}{\frac{1}{\sqrt{{x}^{2}-1}}}} \right ) \sqrt{5}\sqrt{2\,\sqrt{5}+2}+5\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \sqrt{2\,\sqrt{5}+2}\sqrt{-2+2\,\sqrt{5}}-{\it Artanh} \left ({\frac{x\sqrt{5}+x-2}{\sqrt{2\,\sqrt{5}+2}}{\frac{1}{\sqrt{{x}^{2}-1}}}} \right ) \sqrt{5}\sqrt{-2+2\,\sqrt{5}}-5\,\arctan \left ({\frac{x\sqrt{5}-x+2}{\sqrt{-2+2\,\sqrt{5}}\sqrt{{x}^{2}-1}}} \right ) \sqrt{2\,\sqrt{5}+2}-5\,{\it Artanh} \left ({\frac{x\sqrt{5}+x-2}{\sqrt{2\,\sqrt{5}+2}\sqrt{{x}^{2}-1}}} \right ) \sqrt{-2+2\,\sqrt{5}} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(-1+x)^(1/2)*(1+x)^(1/2)*(arctan((x*5^(1/2)-x+2)/(-2+2*5^(1/2))^(1/2)/(x^2-
1)^(1/2))*5^(1/2)*(2*5^(1/2)+2)^(1/2)+5*ln(x+(x^2-1)^(1/2))*(2*5^(1/2)+2)^(1/2)*
(-2+2*5^(1/2))^(1/2)-arctanh((x*5^(1/2)+x-2)/(2*5^(1/2)+2)^(1/2)/(x^2-1)^(1/2))*
5^(1/2)*(-2+2*5^(1/2))^(1/2)-5*arctan((x*5^(1/2)-x+2)/(-2+2*5^(1/2))^(1/2)/(x^2-
1)^(1/2))*(2*5^(1/2)+2)^(1/2)-5*arctanh((x*5^(1/2)+x-2)/(2*5^(1/2)+2)^(1/2)/(x^2
-1)^(1/2))*(-2+2*5^(1/2))^(1/2))/(x^2-1)^(1/2)/(2*5^(1/2)+2)^(1/2)/(-2+2*5^(1/2)
)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.296825, size = 308, normalized size = 3.38 \[ \frac{2}{5} \, \sqrt{2} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{5} \sqrt{2} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}}{5 \,{\left (2 \, \sqrt{x + 1} \sqrt{x - 1} - 2 \, x - \sqrt{5} + \sqrt{-4 \,{\left (2 \, x + \sqrt{5} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} + 8 \, x^{2} + 4 \, \sqrt{5} x - 4 \, x} + 1\right )}}\right ) + \frac{1}{10} \, \sqrt{2} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (\frac{1}{5} \, \sqrt{5} \sqrt{2} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} + 2 \, \sqrt{x + 1} \sqrt{x - 1} - 2 \, x + \sqrt{5} + 1\right ) - \frac{1}{10} \, \sqrt{2} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (-\frac{1}{5} \, \sqrt{5} \sqrt{2} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} + 2 \, \sqrt{x + 1} \sqrt{x - 1} - 2 \, x + \sqrt{5} + 1\right ) + \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*sqrt(2)*sqrt(-sqrt(5)*(sqrt(5) - 5))*arctan(1/5*sqrt(5)*sqrt(2)*sqrt(-sqrt(5
)*(sqrt(5) - 5))/(2*sqrt(x + 1)*sqrt(x - 1) - 2*x - sqrt(5) + sqrt(-4*(2*x + sqr
t(5) - 1)*sqrt(x + 1)*sqrt(x - 1) + 8*x^2 + 4*sqrt(5)*x - 4*x) + 1)) + 1/10*sqrt
(2)*sqrt(sqrt(5)*(sqrt(5) + 5))*log(1/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5)*(sqrt(5) +
5)) + 2*sqrt(x + 1)*sqrt(x - 1) - 2*x + sqrt(5) + 1) - 1/10*sqrt(2)*sqrt(sqrt(5)
*(sqrt(5) + 5))*log(-1/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5)*(sqrt(5) + 5)) + 2*sqrt(x
+ 1)*sqrt(x - 1) - 2*x + sqrt(5) + 1) + log(sqrt(x + 1)*sqrt(x - 1) - x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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