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

Optimal. Leaf size=47 \[ \frac{3}{2} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0929122, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{3}{2} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.2647, size = 44, normalized size = 0.94 \[ - \frac{\operatorname{atan}{\left (- \frac{- x - 3}{2 \sqrt{x^{2} + x - 1}} \right )}}{2} - \frac{3 \operatorname{atanh}{\left (\frac{3 x - 1}{2 \sqrt{x^{2} + x - 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0264789, size = 55, normalized size = 1.17 \[ -\frac{3}{2} \log \left (-2 \sqrt{x^2+x-1}-3 x+1\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )+\frac{3}{2} \log (1-x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 46, normalized size = 1. \[{\frac{1}{2}\arctan \left ({\frac{-3-x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2-x}}}} \right ) }-{\frac{3}{2}{\it Artanh} \left ({\frac{-1+3\,x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+3\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.756138, size = 88, normalized size = 1.87 \[ -\frac{1}{2} \, \arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | 2 \, x + 2 \right |}} + \frac{6 \, \sqrt{5}}{5 \,{\left | 2 \, x + 2 \right |}}\right ) - \frac{3}{2} \, \log \left (\frac{2 \, \sqrt{x^{2} + x - 1}}{{\left | 2 \, x - 2 \right |}} + \frac{2}{{\left | 2 \, x - 2 \right |}} + \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*arcsin(2/5*sqrt(5)*x/abs(2*x + 2) + 6/5*sqrt(5)/abs(2*x + 2)) - 3/2*log(2*s
qrt(x^2 + x - 1)/abs(2*x - 2) + 2/abs(2*x - 2) + 3/2)

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Fricas [A]  time = 0.285837, size = 62, normalized size = 1.32 \[ \arctan \left (-x + \sqrt{x^{2} + x - 1} - 1\right ) - \frac{3}{2} \, \log \left (-x + \sqrt{x^{2} + x - 1} + 2\right ) + \frac{3}{2} \, \log \left (-x + \sqrt{x^{2} + x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(-x + sqrt(x^2 + x - 1) - 1) - 3/2*log(-x + sqrt(x^2 + x - 1) + 2) + 3/2*l
og(-x + sqrt(x^2 + x - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269572, size = 65, normalized size = 1.38 \[ \arctan \left (-x + \sqrt{x^{2} + x - 1} - 1\right ) - \frac{3}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x - 1} + 2 \right |}\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(-x + sqrt(x^2 + x - 1) - 1) - 3/2*ln(abs(-x + sqrt(x^2 + x - 1) + 2)) + 3
/2*ln(abs(-x + sqrt(x^2 + x - 1)))