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

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

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Rubi [A]  time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1033, 724, 206, 204} \begin {gather*} \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 ) \end {gather*}

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 1033

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[-(a*c), 2]}, Dist[h/2 + (c*g)/(2*q), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - (c*g)
/(2*q), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f
, 0] && PosQ[-(a*c)]

Rubi steps

\begin {align*} \int \frac {1+2 x}{\left (-1+x^2\right ) \sqrt {-1+x+x^2}} \, dx &=\frac {1}{2} \int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx+\frac {3}{2} \int \frac {1}{(-1+x) \sqrt {-1+x+x^2}} \, dx\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+3 x}{\sqrt {-1+x+x^2}}\right )\right )-\operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-x}{\sqrt {-1+x+x^2}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {-3-x}{2 \sqrt {-1+x+x^2}}\right )+\frac {3}{2} \tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {-1+x+x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.04 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {-x-3}{2 \sqrt {x^2+x-1}}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {3 x-1}{2 \sqrt {x^2+x-1}}\right ) \end {gather*}

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

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IntegrateAlgebraic [A]  time = 0.22, size = 37, normalized size = 0.79 \begin {gather*} -\tan ^{-1}\left (-\sqrt {x^2+x-1}+x+1\right )-3 \tanh ^{-1}\left (\sqrt {x^2+x-1}-x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 46, normalized size = 0.98 \begin {gather*} \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 ) \end {gather*}

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, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.25, size = 48, normalized size = 1.02 \begin {gather*} \arctan \left (-x + \sqrt {x^{2} + x - 1} - 1\right ) - \frac {3}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x - 1} + 2 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x - 1} \right |}\right ) \end {gather*}

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, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 46, normalized size = 0.98 \begin {gather*} -\frac {3 \arctanh \left (\frac {3 x -1}{2 \sqrt {3 x +\left (x -1\right )^{2}-2}}\right )}{2}+\frac {\arctan \left (\frac {-x -3}{2 \sqrt {-x +\left (x +1\right )^{2}-2}}\right )}{2} \end {gather*}

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]

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

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maxima [A]  time = 0.97, size = 65, normalized size = 1.38 \begin {gather*} -\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 ) \end {gather*}

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {2\,x+1}{\left (x^2-1\right )\,\sqrt {x^2+x-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x + 1}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{2} + x - 1}}\, dx \end {gather*}

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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