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

Optimal. Leaf size=101 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left (\sqrt{x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

[Out]

-2/(1 - x - Sqrt[-3 - 2*x + x^2]) + 3/(4*(x + Sqrt[-3 - 2*x + x^2])^2) + 4/(x + Sqrt[-3 - 2*x + x^2]) + 6*Log[
1 - x - Sqrt[-3 - 2*x + x^2]] - 6*Log[x + Sqrt[-3 - 2*x + x^2]]

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Rubi [A]  time = 0.037457, antiderivative size = 101, 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} \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left (\sqrt{x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(1 - x - Sqrt[-3 - 2*x + x^2]) + 3/(4*(x + Sqrt[-3 - 2*x + x^2])^2) + 4/(x + Sqrt[-3 - 2*x + x^2]) + 6*Log[
1 - x - Sqrt[-3 - 2*x + x^2]] - 6*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 )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{-3-2 x+x^2}{x^3 (-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{3}{-1+x}-\frac{3}{4 x^3}-\frac{2}{x^2}-\frac{3}{x}\right ) \, dx,x,x+\sqrt{-3-2 x+x^2}\right )\\ &=-\frac{2}{1-x-\sqrt{-3-2 x+x^2}}+\frac{3}{4 \left (x+\sqrt{-3-2 x+x^2}\right )^2}+\frac{4}{x+\sqrt{-3-2 x+x^2}}+6 \log \left (1-x-\sqrt{-3-2 x+x^2}\right )-6 \log \left (x+\sqrt{-3-2 x+x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0381865, size = 97, normalized size = 0.96 \[ \frac{2}{\sqrt{x^2-2 x-3}+x-1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left (\sqrt{x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(-1 + x + Sqrt[-3 - 2*x + x^2]) + 3/(4*(x + Sqrt[-3 - 2*x + x^2])^2) + 4/(x + Sqrt[-3 - 2*x + x^2]) + 6*Log[
1 - x - Sqrt[-3 - 2*x + x^2]] - 6*Log[x + Sqrt[-3 - 2*x + x^2]]

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Maple [A]  time = 0.022, size = 146, normalized size = 1.5 \begin{align*} -9\, \left ( 3+2\,x \right ) ^{-1}-3\,\ln \left ( 3+2\,x \right ) +{\frac{x}{2}}+{\frac{27}{8\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{1}{2} \left ( \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}+3\,{\it Artanh} \left ( 2/3\,{\frac{-3-5\,x}{\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}} \right ) +{\frac{2\,x-2}{4}\sqrt{ \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}}}}+3\,\ln \left ( -1+x+\sqrt{ \left ( x+3/2 \right ) ^{2}-5\,x-{\frac{21}{4}}} \right ) +{\frac{1}{4} \left ( \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/(3+2*x)-3*ln(3+2*x)+1/2*x+27/8/(3+2*x)^2-1/2/(x+3/2)*((x+3/2)^2-5*x-21/4)^(3/2)-(4*(x+3/2)^2-20*x-21)^(1/2)
+3*arctanh(2/3*(-3-5*x)/(4*(x+3/2)^2-20*x-21)^(1/2))+1/4*(2*x-2)*((x+3/2)^2-5*x-21/4)^(1/2)+3*ln(-1+x+((x+3/2)
^2-5*x-21/4)^(1/2))+1/4/(x+3/2)^2*((x+3/2)^2-5*x-21/4)^(3/2)

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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 )}^{3}}\,{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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71949, size = 347, normalized size = 3.44 \begin{align*} \frac{8 \, x^{3} - 10 \, x^{2} - 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} - \sqrt{x^{2} - 2 \, x - 3}{\left (x + 1\right )} - 3\right ) - 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (2 \, x + 3\right ) + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3}\right ) - 2 \,{\left (4 \, x^{2} + 31 \, x + 33\right )} \sqrt{x^{2} - 2 \, x - 3} - 156 \, x - 171}{4 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*x^3 - 10*x^2 - 12*(4*x^2 + 12*x + 9)*log(x^2 - sqrt(x^2 - 2*x - 3)*(x + 1) - 3) - 12*(4*x^2 + 12*x + 9)
*log(2*x + 3) + 12*(4*x^2 + 12*x + 9)*log(-x + sqrt(x^2 - 2*x - 3)) - 2*(4*x^2 + 31*x + 33)*sqrt(x^2 - 2*x - 3
) - 156*x - 171)/(4*x^2 + 12*x + 9)

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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 )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.14482, size = 248, normalized size = 2.46 \begin{align*} \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - 2 \, x - 3} - \frac{104 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{3} + 315 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{2} + 162 \, x - 162 \, \sqrt{x^{2} - 2 \, x - 3} + 27}{8 \,{\left ({\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt{x^{2} - 2 \, x - 3}\right )}^{2}} - \frac{9 \,{\left (16 \, x + 21\right )}}{8 \,{\left (2 \, x + 3\right )}^{2}} - 3 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 3 \, \log \left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 3 \, \log \left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} \right |}\right ) - 3 \, \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))^3,x, algorithm="giac")

[Out]

1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 1/8*(104*(x - sqrt(x^2 - 2*x - 3))^3 + 315*(x - sqrt(x^2 - 2*x - 3))^2 + 162
*x - 162*sqrt(x^2 - 2*x - 3) + 27)/((x - sqrt(x^2 - 2*x - 3))^2 + 3*x - 3*sqrt(x^2 - 2*x - 3))^2 - 9/8*(16*x +
21)/(2*x + 3)^2 - 3*log(abs(2*x + 3)) - 3*log(abs(-x + sqrt(x^2 - 2*x - 3) + 1)) + 3*log(abs(-x + sqrt(x^2 -
2*x - 3))) - 3*log(abs(-x + sqrt(x^2 - 2*x - 3) - 3))