3.8.56 \(\int \frac {1}{x+\sqrt {-3-2 x+x^2}} \, dx\) [756]

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

[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 = 0.02, 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 = {2141, 907} \begin {gather*} -\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 ) \end {gather*}

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 907

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 2141

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 \text {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 \text {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 = 0.19, size = 78, normalized size = 1.20 \begin {gather*} \frac {1}{2} \left (x-\sqrt {-3-2 x+x^2}-\log \left (-1-x+\sqrt {-3-2 x+x^2}\right )+4 \log \left (1+x+\sqrt {-3-2 x+x^2}\right )-3 \log \left (3+3 x+\sqrt {-3-2 x+x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 71, normalized size = 1.09

method	result	size
default	\(-\frac {\sqrt {4 \left (\frac {3}{2}+x \right )^{2}-20 x -21}}{4}+\frac {5 \ln \left (-1+x +\sqrt {\left (\frac {3}{2}+x \right )^{2}-5 x -\frac {21}{4}}\right )}{4}+\frac {3 \arctanh \left (\frac {-2-\frac {10 x}{3}}{\sqrt {4 \left (\frac {3}{2}+x \right )^{2}-20 x -21}}\right )}{4}+\frac {x}{2}-\frac {3 \ln \left (3+2 x \right )}{4}\)	\(71\)
trager	\(\frac {x}{2}-\frac {\sqrt {x^{2}-2 x -3}}{2}+\frac {\ln \left (\frac {\sqrt {x^{2}-2 x -3}\, x^{3}+x^{4}+3 \sqrt {x^{2}-2 x -3}\, x^{2}+2 x^{3}+\sqrt {x^{2}-2 x -3}\, x -4 x^{2}-3 \sqrt {x^{2}-2 x -3}-12 x -9}{\left (3+2 x \right )^{3}}\right )}{2}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.33, size = 77, normalized size = 1.18 \begin {gather*} \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 ) \end {gather*}

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

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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Giac [A]
time = 1.71, size = 81, normalized size = 1.25 \begin {gather*} \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 ) \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

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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