Integrand size = 13, antiderivative size = 58 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=-\frac {3 \sqrt {1+\sqrt {\frac {1}{x}}}}{2 \sqrt {\frac {1}{x}}}+\sqrt {1+\sqrt {\frac {1}{x}}} x+\frac {3}{2} \text {arctanh}\left (\sqrt {1+\sqrt {\frac {1}{x}}}\right ) \] Output:
-3/2*(1+(1/x)^(1/2))^(1/2)/(1/x)^(1/2)+(1+(1/x)^(1/2))^(1/2)*x+3/2*arctanh ((1+(1/x)^(1/2))^(1/2))
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=\frac {1}{2} \left (2-3 \sqrt {\frac {1}{x}}\right ) \sqrt {1+\sqrt {\frac {1}{x}}} x+\frac {3}{2} \text {arctanh}\left (\sqrt {1+\sqrt {\frac {1}{x}}}\right ) \] Input:
Integrate[1/Sqrt[1 + Sqrt[x^(-1)]],x]
Output:
((2 - 3*Sqrt[x^(-1)])*Sqrt[1 + Sqrt[x^(-1)]]*x)/2 + (3*ArcTanh[Sqrt[1 + Sq rt[x^(-1)]]])/2
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {787, 774, 798, 52, 52, 73, 220}
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.
\(\displaystyle \int \frac {1}{\sqrt {\sqrt {\frac {1}{x}}+1}} \, dx\) |
\(\Big \downarrow \) 787 |
\(\displaystyle \int \frac {1}{\sqrt {\frac {1}{\sqrt {x}}+1}}dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \frac {1}{\sqrt {\sqrt {\frac {1}{x}}+1} \sqrt {\frac {1}{x}}}d\frac {1}{\sqrt {\frac {1}{x}}}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -2 \int \frac {\left (\frac {1}{x}\right )^{3/2}}{\sqrt {\sqrt {\frac {1}{x}}+1}}d\sqrt {\frac {1}{x}}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -2 \left (-\frac {3}{4} \int \frac {1}{\sqrt {\sqrt {\frac {1}{x}}+1} x}d\sqrt {\frac {1}{x}}-\frac {\sqrt {\sqrt {\frac {1}{x}}+1}}{2 x}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -2 \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {\sqrt {\frac {1}{x}}}{\sqrt {\sqrt {\frac {1}{x}}+1}}d\sqrt {\frac {1}{x}}-\sqrt {\sqrt {\frac {1}{x}}+1} \sqrt {\frac {1}{x}}\right )-\frac {\sqrt {\sqrt {\frac {1}{x}}+1}}{2 x}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -2 \left (-\frac {3}{4} \left (-\int \frac {1}{x-1}d\sqrt {\sqrt {\frac {1}{x}}+1}-\sqrt {\sqrt {\frac {1}{x}}+1} \sqrt {\frac {1}{x}}\right )-\frac {\sqrt {\sqrt {\frac {1}{x}}+1}}{2 x}\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -2 \left (-\frac {3}{4} \left (\text {arctanh}\left (\sqrt {\sqrt {\frac {1}{x}}+1}\right )-\sqrt {\sqrt {\frac {1}{x}}+1} \sqrt {\frac {1}{x}}\right )-\frac {\sqrt {\sqrt {\frac {1}{x}}+1}}{2 x}\right )\) |
Input:
Int[1/Sqrt[1 + Sqrt[x^(-1)]],x]
Output:
-2*(-1/2*Sqrt[1 + Sqrt[x^(-1)]]/x - (3*(-(Sqrt[1 + Sqrt[x^(-1)]]*Sqrt[x^(- 1)]) + ArcTanh[Sqrt[1 + Sqrt[x^(-1)]]]))/4)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> With[{k = Den ominator[n]}, Subst[Int[(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/( c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, p, q}, x] && FractionQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(40)=80\).
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.55
method | result | size |
derivativedivides | \(\frac {1}{4 \left (\sqrt {1+\sqrt {\frac {1}{x}}}-1\right )^{2}}-\frac {3}{4 \left (\sqrt {1+\sqrt {\frac {1}{x}}}-1\right )}-\frac {3 \ln \left (\sqrt {1+\sqrt {\frac {1}{x}}}-1\right )}{4}-\frac {1}{4 \left (\sqrt {1+\sqrt {\frac {1}{x}}}+1\right )^{2}}-\frac {3}{4 \left (\sqrt {1+\sqrt {\frac {1}{x}}}+1\right )}+\frac {3 \ln \left (\sqrt {1+\sqrt {\frac {1}{x}}}+1\right )}{4}\) | \(90\) |
default | \(-\frac {\sqrt {1+\sqrt {\frac {1}{x}}}\, \sqrt {x}\, \left (6 \sqrt {x \sqrt {\frac {1}{x}}+x}\, \sqrt {\frac {1}{x}}\, \sqrt {x}-4 \sqrt {x \sqrt {\frac {1}{x}}+x}\, \sqrt {x}-3 \ln \left (\frac {\sqrt {\frac {1}{x}}\, \sqrt {x}}{2}+\sqrt {x}+\sqrt {x \sqrt {\frac {1}{x}}+x}\right )\right )}{4 \sqrt {x \left (1+\sqrt {\frac {1}{x}}\right )}}\) | \(92\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, x^{\frac {1}{4}} \left (\frac {1}{\sqrt {\frac {1}{x}}\, \sqrt {x}}\right )^{\frac {5}{2}} \left (-\frac {10}{\sqrt {\frac {1}{x}}}+15\right ) \sqrt {\frac {1}{\sqrt {\frac {1}{x}}}+1}}{10}+\frac {3 \sqrt {\pi }\, x^{\frac {5}{4}} \left (\frac {1}{\sqrt {\frac {1}{x}}\, \sqrt {x}}\right )^{\frac {5}{2}} \left (\frac {1}{x}\right )^{\frac {5}{4}} \operatorname {arcsinh}\left (\frac {1}{\left (\frac {1}{x}\right )^{\frac {1}{4}}}\right )}{2}}{\sqrt {\sqrt {\frac {1}{x}}\, \sqrt {x}}\, \sqrt {\frac {1}{\sqrt {\frac {1}{x}}\, \sqrt {x}}}\, \sqrt {\pi }}\) | \(96\) |
Input:
int(1/(1+(1/x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4/((1+(1/x)^(1/2))^(1/2)-1)^2-3/4/((1+(1/x)^(1/2))^(1/2)-1)-3/4*ln((1+(1 /x)^(1/2))^(1/2)-1)-1/4/((1+(1/x)^(1/2))^(1/2)+1)^2-3/4/((1+(1/x)^(1/2))^( 1/2)+1)+3/4*ln((1+(1/x)^(1/2))^(1/2)+1)
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=\frac {1}{2} \, {\left (2 \, x - 3 \, \sqrt {x}\right )} \sqrt {\frac {x + \sqrt {x}}{x}} + \frac {3}{4} \, \log \left (\sqrt {\frac {x + \sqrt {x}}{x}} + 1\right ) - \frac {3}{4} \, \log \left (\sqrt {\frac {x + \sqrt {x}}{x}} - 1\right ) \] Input:
integrate(1/(1+(1/x)^(1/2))^(1/2),x, algorithm="fricas")
Output:
1/2*(2*x - 3*sqrt(x))*sqrt((x + sqrt(x))/x) + 3/4*log(sqrt((x + sqrt(x))/x ) + 1) - 3/4*log(sqrt((x + sqrt(x))/x) - 1)
\[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=\int \frac {1}{\sqrt {\sqrt {\frac {1}{x}} + 1}}\, dx \] Input:
integrate(1/(1+(1/x)**(1/2))**(1/2),x)
Output:
Integral(1/sqrt(sqrt(1/x) + 1), x)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=-\frac {3 \, {\left (\frac {1}{\sqrt {x}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{\sqrt {x}} + 1}}{2 \, {\left ({\left (\frac {1}{\sqrt {x}} + 1\right )}^{2} - \frac {2}{\sqrt {x}} - 1\right )}} + \frac {3}{4} \, \log \left (\sqrt {\frac {1}{\sqrt {x}} + 1} + 1\right ) - \frac {3}{4} \, \log \left (\sqrt {\frac {1}{\sqrt {x}} + 1} - 1\right ) \] Input:
integrate(1/(1+(1/x)^(1/2))^(1/2),x, algorithm="maxima")
Output:
-1/2*(3*(1/sqrt(x) + 1)^(3/2) - 5*sqrt(1/sqrt(x) + 1))/((1/sqrt(x) + 1)^2 - 2/sqrt(x) - 1) + 3/4*log(sqrt(1/sqrt(x) + 1) + 1) - 3/4*log(sqrt(1/sqrt( x) + 1) - 1)
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=\frac {2 \, \sqrt {x + \sqrt {x}} {\left (2 \, \sqrt {x} - 3\right )} - 3 \, \log \left (-2 \, \sqrt {x + \sqrt {x}} + 2 \, \sqrt {x} + 1\right )}{4 \, \mathrm {sgn}\left (x\right )} \] Input:
integrate(1/(1+(1/x)^(1/2))^(1/2),x, algorithm="giac")
Output:
1/4*(2*sqrt(x + sqrt(x))*(2*sqrt(x) - 3) - 3*log(-2*sqrt(x + sqrt(x)) + 2* sqrt(x) + 1))/sgn(x)
Time = 23.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=\frac {3\,\mathrm {atanh}\left (\sqrt {\sqrt {\frac {1}{x}}+1}\right )}{2}+\frac {5\,x\,\sqrt {\sqrt {\frac {1}{x}}+1}}{2}-\frac {3\,x\,{\left (\sqrt {\frac {1}{x}}+1\right )}^{3/2}}{2} \] Input:
int(1/((1/x)^(1/2) + 1)^(1/2),x)
Output:
(3*atanh(((1/x)^(1/2) + 1)^(1/2)))/2 + (5*x*((1/x)^(1/2) + 1)^(1/2))/2 - ( 3*x*((1/x)^(1/2) + 1)^(3/2))/2
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt {1+\sqrt {\frac {1}{x}}}} \, dx=x^{\frac {3}{4}} \sqrt {\sqrt {x}+1}-\frac {3 x^{\frac {1}{4}} \sqrt {\sqrt {x}+1}}{2}+\frac {3 \,\mathrm {log}\left (\sqrt {\sqrt {x}+1}+x^{\frac {1}{4}}\right )}{2} \] Input:
int(1/(1+(1/x)^(1/2))^(1/2),x)
Output:
(2*x**(3/4)*sqrt(sqrt(x) + 1) - 3*x**(1/4)*sqrt(sqrt(x) + 1) + 3*log(sqrt( sqrt(x) + 1) + x**(1/4)))/2