Integrand size = 14, antiderivative size = 61 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {1}{5} \left (5-\sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {2+x}\right )+\frac {1}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {2+x}\right ) \] Output:
1/5*(5-5^(1/2))*ln(1-5^(1/2)-2*(2+x)^(1/2))+1/5*(5+5^(1/2))*ln(1+5^(1/2)-2 *(2+x)^(1/2))
Time = 0.45 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {1}{5} \left (\left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {2+x}\right )-\left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {2+x}\right )\right ) \] Input:
Integrate[(1 + x - Sqrt[2 + x])^(-1),x]
Output:
((5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[2 + x]] - (-5 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[2 + x]])/5
Time = 0.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7267, 25, 1141, 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.
\(\displaystyle \int \frac {1}{x-\sqrt {x+2}+1} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int -\frac {\sqrt {x+2}}{-x+\sqrt {x+2}-1}d\sqrt {x+2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\sqrt {x+2}}{-x+\sqrt {x+2}-1}d\sqrt {x+2}\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle 2 \int \left (-\frac {5+\sqrt {5}}{5 \left (-2 \sqrt {x+2}+\sqrt {5}+1\right )}-\frac {5-\sqrt {5}}{5 \left (-2 \sqrt {x+2}-\sqrt {5}+1\right )}\right )d\sqrt {x+2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{10} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}-\sqrt {5}+1\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {x+2}+\sqrt {5}+1\right )\right )\) |
Input:
Int[(1 + x - Sqrt[2 + x])^(-1),x]
Output:
2*(((5 - Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[2 + x]])/10 + ((5 + Sqrt[5])*Lo g[1 + Sqrt[5] - 2*Sqrt[2 + x]])/10)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(\ln \left (1+x -\sqrt {2+x}\right )-\frac {2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {2+x}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(33\) |
default | \(-\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (1+2 x \right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (x^{2}+x -1\right )}{2}-\frac {\ln \left (1+x +\sqrt {2+x}\right )}{2}-\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {2+x}+1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (1+x -\sqrt {2+x}\right )}{2}-\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {2+x}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(91\) |
trager | \(-\ln \left (-1-x +\sqrt {2+x}\right ) \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right )+\ln \left (150 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right )^{2} x +450 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right ) \sqrt {2+x}-445 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right ) x -244 \sqrt {2+x}-660 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right )+204 x +374\right ) \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right )+2 \ln \left (-1-x +\sqrt {2+x}\right )-\ln \left (150 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right )^{2} x +450 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right ) \sqrt {2+x}-445 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right ) x -244 \sqrt {2+x}-660 \operatorname {RootOf}\left (5 \textit {\_Z}^{2}-10 \textit {\_Z} +4\right )+204 x +374\right )\) | \(201\) |
Input:
int(1/(1+x-(2+x)^(1/2)),x,method=_RETURNVERBOSE)
Output:
ln(1+x-(2+x)^(1/2))-2/5*5^(1/2)*arctanh(1/5*(2*(2+x)^(1/2)-1)*5^(1/2))
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {1}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (x + 3\right )} - {\left (\sqrt {5} {\left (2 \, x + 1\right )} - 5\right )} \sqrt {x + 2} + 7 \, x + 3}{x^{2} + x - 1}\right ) + \log \left (x - \sqrt {x + 2} + 1\right ) \] Input:
integrate(1/(1+x-(2+x)^(1/2)),x, algorithm="fricas")
Output:
1/5*sqrt(5)*log((2*x^2 - sqrt(5)*(x + 3) - (sqrt(5)*(2*x + 1) - 5)*sqrt(x + 2) + 7*x + 3)/(x^2 + x - 1)) + log(x - sqrt(x + 2) + 1)
Time = 0.74 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {\sqrt {5} \left (- \log {\left (\sqrt {x + 2} - \frac {1}{2} + \frac {\sqrt {5}}{2} \right )} + \log {\left (\sqrt {x + 2} - \frac {\sqrt {5}}{2} - \frac {1}{2} \right )}\right )}{5} + \log {\left (x - \sqrt {x + 2} + 1 \right )} \] Input:
integrate(1/(1+x-(2+x)**(1/2)),x)
Output:
sqrt(5)*(-log(sqrt(x + 2) - 1/2 + sqrt(5)/2) + log(sqrt(x + 2) - sqrt(5)/2 - 1/2))/5 + log(x - sqrt(x + 2) + 1)
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {1}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {x + 2} + 1}{\sqrt {5} + 2 \, \sqrt {x + 2} - 1}\right ) + \log \left (x - \sqrt {x + 2} + 1\right ) \] Input:
integrate(1/(1+x-(2+x)^(1/2)),x, algorithm="maxima")
Output:
1/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(x + 2) + 1)/(sqrt(5) + 2*sqrt(x + 2) - 1)) + log(x - sqrt(x + 2) + 1)
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {1}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {x + 2} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {x + 2} - 1 \right |}}\right ) + \log \left ({\left | x - \sqrt {x + 2} + 1 \right |}\right ) \] Input:
integrate(1/(1+x-(2+x)^(1/2)),x, algorithm="giac")
Output:
1/5*sqrt(5)*log(abs(-sqrt(5) + 2*sqrt(x + 2) - 1)/abs(sqrt(5) + 2*sqrt(x + 2) - 1)) + log(abs(x - sqrt(x + 2) + 1))
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.16 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\ln \left (2\,\sqrt {x+2}-\left (\frac {\sqrt {5}}{5}+1\right )\,\left (2\,\sqrt {x+2}-1\right )\right )\,\left (\frac {\sqrt {5}}{5}+1\right )-\ln \left (2\,\sqrt {x+2}+\left (\frac {\sqrt {5}}{5}-1\right )\,\left (2\,\sqrt {x+2}-1\right )\right )\,\left (\frac {\sqrt {5}}{5}-1\right ) \] Input:
int(1/(x - (x + 2)^(1/2) + 1),x)
Output:
log(2*(x + 2)^(1/2) - (5^(1/2)/5 + 1)*(2*(x + 2)^(1/2) - 1))*(5^(1/2)/5 + 1) - log(2*(x + 2)^(1/2) + (5^(1/2)/5 - 1)*(2*(x + 2)^(1/2) - 1))*(5^(1/2) /5 - 1)
Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {1}{1+x-\sqrt {2+x}} \, dx=\frac {\sqrt {5}\, \mathrm {log}\left (2 \sqrt {x +2}-\sqrt {5}-1\right )}{5}-\frac {\sqrt {5}\, \mathrm {log}\left (2 \sqrt {x +2}+\sqrt {5}-1\right )}{5}+\mathrm {log}\left (2 \sqrt {x +2}-\sqrt {5}-1\right )+\mathrm {log}\left (2 \sqrt {x +2}+\sqrt {5}-1\right ) \] Input:
int(1/(1+x-(2+x)^(1/2)),x)
Output:
(sqrt(5)*log(2*sqrt(x + 2) - sqrt(5) - 1) - sqrt(5)*log(2*sqrt(x + 2) + sq rt(5) - 1) + 5*log(2*sqrt(x + 2) - sqrt(5) - 1) + 5*log(2*sqrt(x + 2) + sq rt(5) - 1))/5