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

Optimal. Leaf size=83 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{3}{2 \left (\sqrt{x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0330327, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{3}{2 \left (\sqrt{x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]))

Rubi steps

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

Mathematica [A]  time = 0.0270093, size = 79, normalized size = 0.95 \[ \frac{2}{\sqrt{x^2-2 x-3}+x-1}+\frac{3}{2 \left (\sqrt{x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 118, normalized size = 1.4 \begin{align*} -2\,\ln \left ( 3+2\,x \right ) +{\frac{x}{2}}-{\frac{9}{12+8\,x}}-{\frac{2}{3}\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}+2\,\ln \left ( -1+x+\sqrt{ \left ( x+3/2 \right ) ^{2}-5\,x-{\frac{21}{4}}} \right ) +2\,{\it Artanh} \left ( 2/3\,{\frac{-3-5\,x}{\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}} \right ) -{\frac{1}{3} \left ( \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{2\,x-2}{6}\sqrt{ \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(3+2*x)+1/2*x-9/4/(3+2*x)-2/3*(4*(x+3/2)^2-20*x-21)^(1/2)+2*ln(-1+x+((x+3/2)^2-5*x-21/4)^(1/2))+2*arctanh
(2/3*(-3-5*x)/(4*(x+3/2)^2-20*x-21)^(1/2))-1/3/(x+3/2)*((x+3/2)^2-5*x-21/4)^(3/2)+1/6*(2*x-2)*((x+3/2)^2-5*x-2
1/4)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x + \sqrt{x^{2} - 2 \, x - 3}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76576, size = 262, normalized size = 3.16 \begin{align*} \frac{4 \, x^{2} - 8 \,{\left (2 \, x + 3\right )} \log \left (x^{2} - \sqrt{x^{2} - 2 \, x - 3}{\left (x + 1\right )} - 3\right ) - 8 \,{\left (2 \, x + 3\right )} \log \left (2 \, x + 3\right ) + 8 \,{\left (2 \, x + 3\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3}\right ) - 4 \, \sqrt{x^{2} - 2 \, x - 3}{\left (x + 3\right )} + 2 \, x - 15}{4 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x^2 - 8*(2*x + 3)*log(x^2 - sqrt(x^2 - 2*x - 3)*(x + 1) - 3) - 8*(2*x + 3)*log(2*x + 3) + 8*(2*x + 3)*l
og(-x + sqrt(x^2 - 2*x - 3)) - 4*sqrt(x^2 - 2*x - 3)*(x + 3) + 2*x - 15)/(2*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x + \sqrt{x^{2} - 2 x - 3}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21153, size = 193, normalized size = 2.33 \begin{align*} \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - 2 \, x - 3} - \frac{3 \,{\left (5 \, x - 5 \, \sqrt{x^{2} - 2 \, x - 3} + 3\right )}}{4 \,{\left ({\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt{x^{2} - 2 \, x - 3}\right )}} - \frac{9}{4 \,{\left (2 \, x + 3\right )}} - 2 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 2 \, \log \left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 2 \, \log \left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} \right |}\right ) - 2 \, \log \left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 3/4*(5*x - 5*sqrt(x^2 - 2*x - 3) + 3)/((x - sqrt(x^2 - 2*x - 3))^2 + 3*x - 3
*sqrt(x^2 - 2*x - 3)) - 9/4/(2*x + 3) - 2*log(abs(2*x + 3)) - 2*log(abs(-x + sqrt(x^2 - 2*x - 3) + 1)) + 2*log
(abs(-x + sqrt(x^2 - 2*x - 3))) - 2*log(abs(-x + sqrt(x^2 - 2*x - 3) - 3))