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

Optimal. Leaf size=70 \[ \frac{\sqrt{x^2+x-1}}{2 \left (1-x^2\right )}-\frac{1}{8} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )-\frac{5}{8} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right ) \]

[Out]

Sqrt[-1 + x + x^2]/(2*(1 - x^2)) - ArcTan[(3 + x)/(2*Sqrt[-1 + x + x^2])]/8 - (5
*ArcTanh[(1 - 3*x)/(2*Sqrt[-1 + x + x^2])])/8

_______________________________________________________________________________________

Rubi [A]  time = 0.124569, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{\sqrt{x^2+x-1}}{2 \left (1-x^2\right )}-\frac{1}{8} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )-\frac{5}{8} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x + x^2]/(2*(1 - x^2)) - ArcTan[(3 + x)/(2*Sqrt[-1 + x + x^2])]/8 - (5
*ArcTanh[(1 - 3*x)/(2*Sqrt[-1 + x + x^2])])/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.1084, size = 60, normalized size = 0.86 \[ - \frac{\operatorname{atan}{\left (- \frac{- x - 3}{2 \sqrt{x^{2} + x - 1}} \right )}}{8} + \frac{5 \operatorname{atanh}{\left (\frac{3 x - 1}{2 \sqrt{x^{2} + x - 1}} \right )}}{8} + \frac{\sqrt{x^{2} + x - 1}}{2 \left (- x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(-(-x - 3)/(2*sqrt(x**2 + x - 1)))/8 + 5*atanh((3*x - 1)/(2*sqrt(x**2 + x -
1)))/8 + sqrt(x**2 + x - 1)/(2*(-x**2 + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0716154, size = 72, normalized size = 1.03 \[ \frac{1}{8} \left (-\frac{4 \sqrt{x^2+x-1}}{x^2-1}+5 \log \left (-2 \sqrt{x^2+x-1}-3 x+1\right )-\tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )-5 \log (1-x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-4*Sqrt[-1 + x + x^2])/(-1 + x^2) - ArcTan[(3 + x)/(2*Sqrt[-1 + x + x^2])] - 5
*Log[1 - x] + 5*Log[1 - 3*x - 2*Sqrt[-1 + x + x^2]])/8

_______________________________________________________________________________________

Maple [A]  time = 0.026, size = 84, normalized size = 1.2 \[{\frac{1}{4+4\,x}\sqrt{ \left ( 1+x \right ) ^{2}-2-x}}+{\frac{1}{8}\arctan \left ({\frac{-3-x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2-x}}}} \right ) }-{\frac{1}{-4+4\,x}\sqrt{ \left ( -1+x \right ) ^{2}+3\,x-2}}+{\frac{5}{8}{\it Artanh} \left ({\frac{-1+3\,x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+3\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/(1+x)*((1+x)^2-2-x)^(1/2)+1/8*arctan(1/2*(-3-x)/((1+x)^2-2-x)^(1/2))-1/4/(-1
+x)*((-1+x)^2+3*x-2)^(1/2)+5/8*arctanh(1/2*(-1+3*x)/((-1+x)^2+3*x-2)^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.272794, size = 348, normalized size = 4.97 \[ \frac{32 \, x^{3} + 48 \, x^{2} + 2 \,{\left (8 \, x^{4} + 8 \, x^{3} - 11 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2} - 2 \, x - 1\right )} \sqrt{x^{2} + x - 1} - 8 \, x + 3\right )} \arctan \left (-x + \sqrt{x^{2} + x - 1} - 1\right ) + 5 \,{\left (8 \, x^{4} + 8 \, x^{3} - 11 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2} - 2 \, x - 1\right )} \sqrt{x^{2} + x - 1} - 8 \, x + 3\right )} \log \left (-x + \sqrt{x^{2} + x - 1} + 2\right ) - 5 \,{\left (8 \, x^{4} + 8 \, x^{3} - 11 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2} - 2 \, x - 1\right )} \sqrt{x^{2} + x - 1} - 8 \, x + 3\right )} \log \left (-x + \sqrt{x^{2} + x - 1}\right ) - 4 \,{\left (8 \, x^{2} + 8 \, x - 3\right )} \sqrt{x^{2} + x - 1} - 16 \, x - 16}{8 \,{\left (8 \, x^{4} + 8 \, x^{3} - 11 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2} - 2 \, x - 1\right )} \sqrt{x^{2} + x - 1} - 8 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(32*x^3 + 48*x^2 + 2*(8*x^4 + 8*x^3 - 11*x^2 - 4*(2*x^3 + x^2 - 2*x - 1)*sqr
t(x^2 + x - 1) - 8*x + 3)*arctan(-x + sqrt(x^2 + x - 1) - 1) + 5*(8*x^4 + 8*x^3
- 11*x^2 - 4*(2*x^3 + x^2 - 2*x - 1)*sqrt(x^2 + x - 1) - 8*x + 3)*log(-x + sqrt(
x^2 + x - 1) + 2) - 5*(8*x^4 + 8*x^3 - 11*x^2 - 4*(2*x^3 + x^2 - 2*x - 1)*sqrt(x
^2 + x - 1) - 8*x + 3)*log(-x + sqrt(x^2 + x - 1)) - 4*(8*x^2 + 8*x - 3)*sqrt(x^
2 + x - 1) - 16*x - 16)/(8*x^4 + 8*x^3 - 11*x^2 - 4*(2*x^3 + x^2 - 2*x - 1)*sqrt
(x^2 + x - 1) - 8*x + 3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270882, size = 193, normalized size = 2.76 \[ \frac{2 \,{\left (x - \sqrt{x^{2} + x - 1}\right )}^{3} + 3 \,{\left (x - \sqrt{x^{2} + x - 1}\right )}^{2} - x + \sqrt{x^{2} + x - 1} - 1}{2 \,{\left ({\left (x - \sqrt{x^{2} + x - 1}\right )}^{4} - 2 \,{\left (x - \sqrt{x^{2} + x - 1}\right )}^{2} - 4 \, x + 4 \, \sqrt{x^{2} + x - 1}\right )}} + \frac{1}{4} \, \arctan \left (-x + \sqrt{x^{2} + x - 1} - 1\right ) + \frac{5}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x - 1} + 2 \right |}\right ) - \frac{5}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*(x - sqrt(x^2 + x - 1))^3 + 3*(x - sqrt(x^2 + x - 1))^2 - x + sqrt(x^2 +
x - 1) - 1)/((x - sqrt(x^2 + x - 1))^4 - 2*(x - sqrt(x^2 + x - 1))^2 - 4*x + 4*s
qrt(x^2 + x - 1)) + 1/4*arctan(-x + sqrt(x^2 + x - 1) - 1) + 5/8*ln(abs(-x + sqr
t(x^2 + x - 1) + 2)) - 5/8*ln(abs(-x + sqrt(x^2 + x - 1)))