\(\int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx\) [11]

Optimal result

Integrand size = 19, antiderivative size = 73 \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right ) \]

[Out]

2/5*ln(1+5^(1/2)-2*(1+(1+x)^(1/2))^(1/2))*(5-5^(1/2))+2/5*ln(1-5^(1/2)-2*(1+(1+x)^(1/2))^(1/2))*(5+5^(1/2))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {646, 31} \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \]

[In]

Int[(x - Sqrt[1 + Sqrt[1 + x]])^(-1),x]

[Out]

(2*(5 + Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/5 + (2*(5 - Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 +
Sqrt[1 + x]]])/5

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 646

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{-1+x^2-\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {-1+x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {2}{5} \left (-5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right ) \]

[In]

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

[Out]

(-2*(-5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/5 + (2*(5 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[
1 + Sqrt[1 + x]]])/5

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(2 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\)	\(46\)
default	\(\frac {\ln \left (x^{2}-x -1\right )}{2}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x -1\right ) \sqrt {5}}{5}\right )}{5}-\ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {\left (1+2 \sqrt {1+\sqrt {1+x}}\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}+\ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )+\frac {2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}+\frac {2 \left (7+3 \sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {2 \sqrt {5}\, \left (-7+3 \sqrt {5}\right ) \arctan \left (\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {\ln \left (x +\sqrt {1+x}\right )}{2}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (1+2 \sqrt {1+x}\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (x -\sqrt {1+x}\right )}{2}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+x}-1\right ) \sqrt {5}}{5}\right )}{5}-\frac {2 \left (3+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}+\frac {2 \left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {4 \left (2+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}+\frac {4 \left (-2+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}\)	\(419\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (51) = 102\).

Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} + \sqrt {5} {\left (3 \, x + 1\right )} + {\left (\sqrt {5} {\left (x + 2\right )} + 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} + 5\right )} \sqrt {x + 1} + 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.49 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=- \frac {2 \sqrt {5} \left (- \log {\left (\sqrt {\sqrt {x + 1} + 1} - \frac {1}{2} + \frac {\sqrt {5}}{2} \right )} + \log {\left (\sqrt {\sqrt {x + 1} + 1} - \frac {\sqrt {5}}{2} - \frac {1}{2} \right )}\right )}{5} + 2 \log {\left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right )} \]

[In]

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

[Out]

-2*sqrt(5)*(-log(sqrt(sqrt(x + 1) + 1) - 1/2 + sqrt(5)/2) + log(sqrt(sqrt(x + 1) + 1) - sqrt(5)/2 - 1/2))/5 +
2*log(sqrt(x + 1) - sqrt(sqrt(x + 1) + 1))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \]

[In]

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

[Out]

-2/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(sqrt(x + 1) + 1) + 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) - 1)) + 2*log(sqr
t(x + 1) - sqrt(sqrt(x + 1) + 1))

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {2}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}\right ) + 2 \, \log \left ({\left | \sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right |}\right ) \]

[In]

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

[Out]

-2/5*sqrt(5)*log(abs(-sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) - 1)/abs(sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) - 1)) + 2*l
og(abs(sqrt(x + 1) - sqrt(sqrt(x + 1) + 1)))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {1}{x-\sqrt {\sqrt {x+1}+1}} \,d x \]

[In]

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

[Out]

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