3.11 \(\int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx\)

Optimal. Leaf size=73 \[ \frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \]

[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))

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Rubi [A]  time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {632, 31} \[ \frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \]

Antiderivative was successfully verified.

[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 632

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

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Mathematica [A]  time = 0.07, size = 71, normalized size = 0.97 \[ \frac {1}{5} \left (2 \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )-2 \left (\sqrt {5}-5\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right )\right ) \]

Antiderivative was successfully verified.

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

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fricas [B]  time = 0.41, size = 112, normalized size = 1.53 \[ \frac {2}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} + \sqrt {5} {\left (3 \, x + 1\right )} + {\left (\sqrt {5} {\left (x + 2\right )} + 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} + 5\right )} \sqrt {x + 1} + 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.81, size = 67, normalized size = 0.92 \[ -\frac {2}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}\right ) + 2 \, \log \left ({\left | \sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [B]  time = 0.17, size = 175, normalized size = 2.40 \[ \frac {2 \sqrt {5}\, \arctanh \left (\frac {\left (1+2 \sqrt {1+\sqrt {x +1}}\right ) \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x -1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {x +1}-1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {x +1}+1\right ) \sqrt {5}}{5}\right )}{5}+\frac {2 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {x +1}}-1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (x -\sqrt {x +1}\right )}{2}-\frac {\ln \left (x +\sqrt {x +1}\right )}{2}+\ln \left (\sqrt {x +1}-\sqrt {1+\sqrt {x +1}}\right )-\ln \left (\sqrt {x +1}+\sqrt {1+\sqrt {x +1}}\right )+\frac {\ln \left (x^{2}-x -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 63, normalized size = 0.86 \[ -\frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x-\sqrt {\sqrt {x+1}+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x - \sqrt {\sqrt {x + 1} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x - sqrt(sqrt(x + 1) + 1)), x)

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