3.756 \(\int \frac {1}{x+\sqrt {-3-2 x+x^2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+2 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-\frac {3}{2} \log \left (\sqrt {x^2-2 x-3}+x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2116, 893} \[ -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+2 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-\frac {3}{2} \log \left (\sqrt {x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[-3 - 2*x + x^2])^(-1),x]

[Out]

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

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2116

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol]
:> Dist[2, Subst[Int[((g + h*x^n)^p*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2))/(-2*d*e + b*f^2 +
2*e*x)^2, x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && EqQ[e^2 -
c*f^2, 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 59, normalized size = 0.91 \[ 2 \left (\frac {1}{\sqrt {x^2-2 x-3}+x-1}+\log \left (-\sqrt {x^2-2 x-3}-x+1\right )-\frac {3}{4} \log \left (\sqrt {x^2-2 x-3}+x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-1),x]

[Out]

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

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fricas [A]  time = 0.43, size = 77, normalized size = 1.18 \[ \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {3}{4} \, \log \left (2 \, x + 3\right ) - \frac {5}{4} \, \log \left (-x + \sqrt {x^{2} - 2 \, x - 3} + 1\right ) + \frac {3}{4} \, \log \left (-x + \sqrt {x^{2} - 2 \, x - 3}\right ) - \frac {3}{4} \, \log \left (-x + \sqrt {x^{2} - 2 \, x - 3} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(x^2-2*x-3)^(1/2)),x, algorithm="fricas")

[Out]

1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 3/4*log(2*x + 3) - 5/4*log(-x + sqrt(x^2 - 2*x - 3) + 1) + 3/4*log(-x + sqrt
(x^2 - 2*x - 3)) - 3/4*log(-x + sqrt(x^2 - 2*x - 3) - 3)

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giac [A]  time = 0.37, size = 81, normalized size = 1.25 \[ \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {3}{4} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {5}{4} \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} + 1 \right |}\right ) + \frac {3}{4} \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} \right |}\right ) - \frac {3}{4} \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(x^2-2*x-3)^(1/2)),x, algorithm="giac")

[Out]

1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 3/4*log(abs(2*x + 3)) - 5/4*log(abs(-x + sqrt(x^2 - 2*x - 3) + 1)) + 3/4*log
(abs(-x + sqrt(x^2 - 2*x - 3))) - 3/4*log(abs(-x + sqrt(x^2 - 2*x - 3) - 3))

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maple [A]  time = 0.01, size = 71, normalized size = 1.09 \[ \frac {x}{2}+\frac {3 \arctanh \left (\frac {-\frac {10 x}{3}-2}{\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}\right )}{4}-\frac {3 \ln \left (2 x +3\right )}{4}+\frac {5 \ln \left (x -1+\sqrt {-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}}\right )}{4}-\frac {\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(x+3/2)^2-20*x-21)^(1/2)+5/4*ln(-1+x+((x+3/2)^2-5*x-21/4)^(1/2))+3/4*arctanh(2/3*(-3-5*x)/(4*(x+3/2)^2
-20*x-21)^(1/2))+1/2*x-3/4*ln(2*x+3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x + \sqrt {x^{2} - 2 \, x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(x^2-2*x-3)^(1/2)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \frac {x}{2}-\frac {3\,\ln \left (x+\frac {3}{2}\right )}{4}-\int \frac {\sqrt {x^2-2\,x-3}}{2\,x+3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + (x^2 - 2*x - 3)^(1/2)),x)

[Out]

x/2 - (3*log(x + 3/2))/4 - int((x^2 - 2*x - 3)^(1/2)/(2*x + 3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x + \sqrt {x^{2} - 2 x - 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(x**2-2*x-3)**(1/2)),x)

[Out]

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

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