Integrand size = 16, antiderivative size = 75 \[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=\frac {1}{4} \left (4+\sqrt {\frac {1}{x}}\right ) \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} x+\frac {7 \text {arctanh}\left (\frac {4+\sqrt {\frac {1}{x}}}{2 \sqrt {2} \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}}}\right )}{8 \sqrt {2}} \]
7/16*arctanh(1/4*(4+(1/x)^(1/2))*2^(1/2)/(2+1/x+(1/x)^(1/2))^(1/2))*2^(1/2 )+1/4*x*(4+(1/x)^(1/2))*(2+1/x+(1/x)^(1/2))^(1/2)
Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=\frac {1}{8} \left (2 \left (4+\sqrt {\frac {1}{x}}\right ) \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} x+7 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}}-\sqrt {\frac {1}{x}}}{\sqrt {2}}\right )\right ) \]
(2*(4 + Sqrt[x^(-1)])*Sqrt[2 + Sqrt[x^(-1)] + x^(-1)]*x + 7*Sqrt[2]*ArcTan h[(Sqrt[2 + Sqrt[x^(-1)] + x^(-1)] - Sqrt[x^(-1)])/Sqrt[2]])/8
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2062, 1693, 1152, 1154, 219}
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.
\(\displaystyle \int \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} \, dx\) |
\(\Big \downarrow \) 2062 |
\(\displaystyle -\int \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle -2 \int \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x^2}+2} x^3d\sqrt {\frac {1}{x}}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle -2 \left (\frac {7}{16} \int \frac {x}{\sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x^2}+2}}d\sqrt {\frac {1}{x}}-\frac {1}{8} \left (\sqrt {\frac {1}{x}}+4\right ) \sqrt {\frac {1}{x^2}+\sqrt {\frac {1}{x}}+2} x^2\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle -2 \left (-\frac {7}{8} \int \frac {1}{8-\frac {1}{x^2}}d\frac {\sqrt {\frac {1}{x}}+4}{\sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x^2}+2}}-\frac {1}{8} \left (\sqrt {\frac {1}{x}}+4\right ) \sqrt {\frac {1}{x^2}+\sqrt {\frac {1}{x}}+2} x^2\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (-\frac {7 \text {arctanh}\left (\frac {\sqrt {\frac {1}{x}}+4}{2 \sqrt {2} \sqrt {\frac {1}{x^2}+\sqrt {\frac {1}{x}}+2}}\right )}{16 \sqrt {2}}-\frac {1}{8} \left (\sqrt {\frac {1}{x}}+4\right ) \sqrt {\frac {1}{x^2}+\sqrt {\frac {1}{x}}+2} x^2\right )\) |
-2*(-1/8*((4 + Sqrt[x^(-1)])*Sqrt[2 + Sqrt[x^(-1)] + x^(-2)]*x^2) - (7*Arc Tanh[(4 + Sqrt[x^(-1)])/(2*Sqrt[2]*Sqrt[2 + Sqrt[x^(-1)] + x^(-2)])])/(16* Sqrt[2]))
3.31.69.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Int[((a_.) + (c_.)*((d_.)/(x_))^(n2_.) + (b_.)*((d_.)/(x_))^(n_))^(p_), x_S ymbol] :> Simp[-d Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^2, x], x, d/x], x ] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, 2*n]
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(\frac {x \left (2+\frac {1}{x}+\sqrt {\frac {1}{x}}\right )^{\frac {3}{2}}}{2}-\frac {\left (2+\frac {1}{x}+\sqrt {\frac {1}{x}}\right )^{\frac {3}{2}}}{8 \sqrt {\frac {1}{x}}}-\frac {7 \sqrt {2+\frac {1}{x}+\sqrt {\frac {1}{x}}}}{16}+\frac {7 \,\operatorname {arctanh}\left (\frac {\left (4+\sqrt {\frac {1}{x}}\right ) \sqrt {2}}{4 \sqrt {2+\frac {1}{x}+\sqrt {\frac {1}{x}}}}\right ) \sqrt {2}}{16}+\frac {\left (2 \sqrt {\frac {1}{x}}+1\right ) \sqrt {2+\frac {1}{x}+\sqrt {\frac {1}{x}}}}{16}\) | \(103\) |
default | \(\frac {\sqrt {\frac {\sqrt {\frac {1}{x}}\, x +2 x +1}{x}}\, \sqrt {x}\, \left (4 \sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}\, \sqrt {\frac {1}{x}}\, \sqrt {x}+7 \sqrt {2}\, \ln \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {x}}{4}+\sqrt {2}\, \sqrt {x}+\sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}\right )+16 \sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}\, \sqrt {x}\right )}{16 \sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}}\) | \(123\) |
1/2*x*(2+1/x+(1/x)^(1/2))^(3/2)-1/8/(1/x)^(1/2)*(2+1/x+(1/x)^(1/2))^(3/2)- 7/16*(2+1/x+(1/x)^(1/2))^(1/2)+7/16*arctanh(1/4*(4+(1/x)^(1/2))*2^(1/2)/(2 +1/x+(1/x)^(1/2))^(1/2))*2^(1/2)+1/16*(2*(1/x)^(1/2)+1)*(2+1/x+(1/x)^(1/2) )^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (45) = 90\).
Time = 1.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25 \[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=\frac {1}{4} \, {\left (4 \, x + \sqrt {x}\right )} \sqrt {\frac {2 \, x + \sqrt {x} + 1}{x}} + \frac {7}{64} \, \sqrt {2} \log \left (-2048 \, x^{2} - 64 \, {\left (32 \, x + 9\right )} \sqrt {x} - 8 \, {\left (3 \, \sqrt {2} {\left (32 \, x + 3\right )} \sqrt {x} + 4 \, \sqrt {2} {\left (32 \, x^{2} + 13 \, x\right )}\right )} \sqrt {\frac {2 \, x + \sqrt {x} + 1}{x}} - 1664 \, x - 113\right ) \]
1/4*(4*x + sqrt(x))*sqrt((2*x + sqrt(x) + 1)/x) + 7/64*sqrt(2)*log(-2048*x ^2 - 64*(32*x + 9)*sqrt(x) - 8*(3*sqrt(2)*(32*x + 3)*sqrt(x) + 4*sqrt(2)*( 32*x^2 + 13*x))*sqrt((2*x + sqrt(x) + 1)/x) - 1664*x - 113)
\[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=\int \sqrt {\sqrt {\frac {1}{x}} + 2 + \frac {1}{x}}\, dx \]
\[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=\int { \sqrt {\frac {1}{\sqrt {x}} + \frac {1}{x} + 2} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=-\frac {1}{16} \, \sqrt {2} {\left (2 \, \sqrt {2} - 7 \, \log \left (2 \, \sqrt {2} - 1\right )\right )} + \frac {1}{4} \, \sqrt {2 \, x + \sqrt {x} + 1} {\left (4 \, \sqrt {x} + 1\right )} - \frac {7}{16} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} \sqrt {x} - \sqrt {2 \, x + \sqrt {x} + 1}\right )} - 1\right ) \]
-1/16*sqrt(2)*(2*sqrt(2) - 7*log(2*sqrt(2) - 1)) + 1/4*sqrt(2*x + sqrt(x) + 1)*(4*sqrt(x) + 1) - 7/16*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*sqrt(x) - sqrt (2*x + sqrt(x) + 1)) - 1)
Timed out. \[ \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx=\int \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} \,d x \]