Integrand size = 16, antiderivative size = 101 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=-\frac {2}{1-x-\sqrt {-3-2 x+x^2}}+\frac {3}{4 \left (x+\sqrt {-3-2 x+x^2}\right )^2}+\frac {4}{x+\sqrt {-3-2 x+x^2}}+6 \log \left (1-x-\sqrt {-3-2 x+x^2}\right )-6 \log \left (x+\sqrt {-3-2 x+x^2}\right ) \] Output:
-2/(1-x-(x^2-2*x-3)^(1/2))+3/4/(x+(x^2-2*x-3)^(1/2))^2+4/(x+(x^2-2*x-3)^(1 /2))+6*ln(1-x-(x^2-2*x-3)^(1/2))-6*ln(x+(x^2-2*x-3)^(1/2))
Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=-\frac {189+108 x-48 x^2-16 x^3+4 \sqrt {-3-2 x+x^2} \left (33+31 x+4 x^2\right )+96 (3+2 x)^2 \text {arctanh}\left (\frac {1+x}{2+2 x+\sqrt {-3-2 x+x^2}}\right )}{8 (3+2 x)^2} \] Input:
Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-3),x]
Output:
-1/8*(189 + 108*x - 48*x^2 - 16*x^3 + 4*Sqrt[-3 - 2*x + x^2]*(33 + 31*x + 4*x^2) + 96*(3 + 2*x)^2*ArcTanh[(1 + x)/(2 + 2*x + Sqrt[-3 - 2*x + x^2])]) /(3 + 2*x)^2
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2541, 27, 1195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (\sqrt {x^2-2 x-3}+x\right )^3} \, dx\) |
\(\Big \downarrow \) 2541 |
\(\displaystyle 2 \int -\frac {-\left (x+\sqrt {x^2-2 x-3}\right )^2+2 \left (x+\sqrt {x^2-2 x-3}\right )+3}{4 \left (-x-\sqrt {x^2-2 x-3}+1\right )^2 \left (x+\sqrt {x^2-2 x-3}\right )^3}d\left (x+\sqrt {x^2-2 x-3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} \int \frac {-\left (x+\sqrt {x^2-2 x-3}\right )^2+2 \left (x+\sqrt {x^2-2 x-3}\right )+3}{\left (-x-\sqrt {x^2-2 x-3}+1\right )^2 \left (x+\sqrt {x^2-2 x-3}\right )^3}d\left (x+\sqrt {x^2-2 x-3}\right )\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {12}{x+\sqrt {x^2-2 x-3}}+\frac {8}{\left (x+\sqrt {x^2-2 x-3}\right )^2}+\frac {3}{\left (x+\sqrt {x^2-2 x-3}\right )^3}-\frac {12}{x+\sqrt {x^2-2 x-3}-1}+\frac {4}{\left (x+\sqrt {x^2-2 x-3}-1\right )^2}\right )d\left (x+\sqrt {x^2-2 x-3}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {4}{-\sqrt {x^2-2 x-3}-x+1}+\frac {8}{\sqrt {x^2-2 x-3}+x}+\frac {3}{2 \left (\sqrt {x^2-2 x-3}+x\right )^2}+12 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-12 \log \left (\sqrt {x^2-2 x-3}+x\right )\right )\) |
Input:
Int[(x + Sqrt[-3 - 2*x + x^2])^(-3),x]
Output:
(-4/(1 - x - Sqrt[-3 - 2*x + x^2]) + 3/(2*(x + Sqrt[-3 - 2*x + x^2])^2) + 8/(x + Sqrt[-3 - 2*x + x^2]) + 12*Log[1 - x - Sqrt[-3 - 2*x + x^2]] - 12*L og[x + Sqrt[-3 - 2*x + x^2]])/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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, m, n}, x ] && IGtQ[p, 0]
Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c _.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Simp[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]
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68
method | result | size |
trager | \(\frac {\left (4 x^{2}+33 x +36\right ) x}{2 \left (2 x +3\right )^{2}}-\frac {\left (4 x^{2}+31 x +33\right ) \sqrt {x^{2}-2 x -3}}{2 \left (2 x +3\right )^{2}}-6 \ln \left (3+x -\sqrt {x^{2}-2 x -3}\right )\) | \(69\) |
default | \(-\frac {9}{2 x +3}-3 \ln \left (2 x +3\right )+\frac {x}{2}+\frac {27}{8 \left (2 x +3\right )^{2}}+\frac {\left (\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}\right )^{\frac {3}{2}}}{4 \left (x +\frac {3}{2}\right )^{2}}-\frac {\left (\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}\right )^{\frac {3}{2}}}{2 \left (x +\frac {3}{2}\right )}-\sqrt {4 \left (x +\frac {3}{2}\right )^{2}-20 x -21}+3 \,\operatorname {arctanh}\left (\frac {-2-\frac {10 x}{3}}{\sqrt {4 \left (x +\frac {3}{2}\right )^{2}-20 x -21}}\right )+\frac {\left (2 x -2\right ) \sqrt {\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}}}{4}+3 \ln \left (-1+x +\sqrt {\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}}\right )\) | \(146\) |
Input:
int(1/(x+(x^2-2*x-3)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/2*(4*x^2+33*x+36)*x/(2*x+3)^2-1/2*(4*x^2+31*x+33)/(2*x+3)^2*(x^2-2*x-3)^ (1/2)-6*ln(3+x-(x^2-2*x-3)^(1/2))
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\frac {8 \, x^{3} - 10 \, x^{2} - 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} - \sqrt {x^{2} - 2 \, x - 3} {\left (x + 1\right )} - 3\right ) - 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (2 \, x + 3\right ) + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-x + \sqrt {x^{2} - 2 \, x - 3}\right ) - 2 \, {\left (4 \, x^{2} + 31 \, x + 33\right )} \sqrt {x^{2} - 2 \, x - 3} - 156 \, x - 171}{4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \] Input:
integrate(1/(x+(x^2-2*x-3)^(1/2))^3,x, algorithm="fricas")
Output:
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)
\[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\int \frac {1}{\left (x + \sqrt {x^{2} - 2 x - 3}\right )^{3}}\, dx \] Input:
integrate(1/(x+(x**2-2*x-3)**(1/2))**3,x)
Output:
Integral((x + sqrt(x**2 - 2*x - 3))**(-3), x)
\[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\int { \frac {1}{{\left (x + \sqrt {x^{2} - 2 \, x - 3}\right )}^{3}} \,d x } \] Input:
integrate(1/(x+(x^2-2*x-3)^(1/2))^3,x, algorithm="maxima")
Output:
integrate((x + sqrt(x^2 - 2*x - 3))^(-3), x)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (85) = 170\).
Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {104 \, {\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{3} + 315 \, {\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 162 \, x - 162 \, \sqrt {x^{2} - 2 \, x - 3} + 27}{8 \, {\left ({\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 2 \, x - 3}\right )}^{2}} - \frac {9 \, {\left (16 \, x + 21\right )}}{8 \, {\left (2 \, x + 3\right )}^{2}} - 3 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} \right |}\right ) - 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} - 3 \right |}\right ) \] Input:
integrate(1/(x+(x^2-2*x-3)^(1/2))^3,x, algorithm="giac")
Output:
1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 1/8*(104*(x - sqrt(x^2 - 2*x - 3))^3 + 3 15*(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 + 2 1)/(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))
Timed out. \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\int \frac {1}{{\left (x+\sqrt {x^2-2\,x-3}\right )}^3} \,d x \] Input:
int(1/(x + (x^2 - 2*x - 3)^(1/2))^3,x)
Output:
int(1/(x + (x^2 - 2*x - 3)^(1/2))^3, x)
Time = 0.16 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\frac {-8 \sqrt {x^{2}-2 x -3}\, x^{2}-62 \sqrt {x^{2}-2 x -3}\, x -66 \sqrt {x^{2}-2 x -3}-96 \,\mathrm {log}\left (\sqrt {x^{2}-2 x -3}+x \right ) x^{2}-288 \,\mathrm {log}\left (\sqrt {x^{2}-2 x -3}+x \right ) x -216 \,\mathrm {log}\left (\sqrt {x^{2}-2 x -3}+x \right )+96 \,\mathrm {log}\left (\frac {\sqrt {x^{2}-2 x -3}}{2}+\frac {x}{2}-\frac {1}{2}\right ) x^{2}+288 \,\mathrm {log}\left (\frac {\sqrt {x^{2}-2 x -3}}{2}+\frac {x}{2}-\frac {1}{2}\right ) x +216 \,\mathrm {log}\left (\frac {\sqrt {x^{2}-2 x -3}}{2}+\frac {x}{2}-\frac {1}{2}\right )+8 x^{3}+6 x^{2}-108 x -135}{16 x^{2}+48 x +36} \] Input:
int(1/(x+(x^2-2*x-3)^(1/2))^3,x)
Output:
( - 8*sqrt(x**2 - 2*x - 3)*x**2 - 62*sqrt(x**2 - 2*x - 3)*x - 66*sqrt(x**2 - 2*x - 3) - 96*log(sqrt(x**2 - 2*x - 3) + x)*x**2 - 288*log(sqrt(x**2 - 2*x - 3) + x)*x - 216*log(sqrt(x**2 - 2*x - 3) + x) + 96*log((sqrt(x**2 - 2*x - 3) + x - 1)/2)*x**2 + 288*log((sqrt(x**2 - 2*x - 3) + x - 1)/2)*x + 216*log((sqrt(x**2 - 2*x - 3) + x - 1)/2) + 8*x**3 + 6*x**2 - 108*x - 135) /(4*(4*x**2 + 12*x + 9))