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 + 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

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Rubi [A]  time = 0.108042, antiderivative size = 73, normalized size of antiderivative = 1., 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 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]

Rule 31

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

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*}

Mathematica [A]  time = 0.0581307, 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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Maple [B]  time = 0.104, size = 175, normalized size = 2.4 \begin{align*}{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{5}}{5}} \right ) }+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+x}+1 \right ) } \right ) }+{\frac{2\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 1+2\,\sqrt{1+\sqrt{1+x}} \right ) } \right ) }+{\frac{\ln \left ({x}^{2}-x-1 \right ) }{2}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+x}-1 \right ) } \right ) }+{\frac{2\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+\sqrt{1+x}}-1 \right ) } \right ) }+\ln \left ( \sqrt{1+x}-\sqrt{1+\sqrt{1+x}} \right ) -\ln \left ( \sqrt{1+x}+\sqrt{1+\sqrt{1+x}} \right ) +{\frac{1}{2}\ln \left ( x-\sqrt{1+x} \right ) }-{\frac{1}{2}\ln \left ( x+\sqrt{1+x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45819, size = 85, normalized size = 1.16 \begin{align*} -\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 ) \end{align*}

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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Fricas [B]  time = 1.09437, size = 323, normalized size = 4.42 \begin{align*} \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 ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x - \sqrt{\sqrt{x + 1} + 1}}\, dx \end{align*}

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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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out