Integrand size = 23, antiderivative size = 105 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\sqrt {2} \arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right ) \]
2^(1/2)*arctan((-1/2*2^(1/2)+1/2*x*2^(1/2)+1/2*(x^2+1)^(1/2)*2^(1/2))/(x+( x^2+1)^(1/2))^(1/2))+2^(1/2)*arctanh((1/2*2^(1/2)+1/2*x*2^(1/2)+1/2*(x^2+1 )^(1/2)*2^(1/2))/(x+(x^2+1)^(1/2))^(1/2))
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\sqrt {2} \left (\arctan \left (\frac {-1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right )+\text {arctanh}\left (\frac {1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right )\right ) \]
Sqrt[2]*(ArcTan[(-1 + x + Sqrt[1 + x^2])/(Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]) ] + ArcTanh[(1 + x + Sqrt[1 + x^2])/(Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]])])
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.54, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2547, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (x^2+1\right ) \sqrt {\sqrt {x^2+1}+x}} \, dx\) |
| \(\Big \downarrow \) 2547 |
| \(\displaystyle 2 \int \frac {1}{\sqrt {x+\sqrt {x^2+1}} \left (\left (x+\sqrt {x^2+1}\right )^2+1\right )}d\left (x+\sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 4 \int \frac {1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 4 \left (\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {1}{2} \int \frac {x+\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {1}{2} \int \frac {1}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x-\sqrt {x^2+1}-1}d\left (1-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-\sqrt {x^2+1}-1}d\left (\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 4 \left (\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {x+\sqrt {x^2+1}}}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1\right )}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x+\sqrt {x^2+1}}}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1\right )}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x+\sqrt {x^2+1}}}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 4 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{2 \sqrt {2}}\right )\right )\) |
4*((-(ArcTan[1 - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2]) + ArcTan[1 + Sq rt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2])/2 + (-1/2*Log[1 + x + Sqrt[1 + x^2 ] - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2] + Log[1 + x + Sqrt[1 + x^2] + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/(2*Sqrt[2]))/2)
3.16.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_ .)*(x_)^2])^(n_.), x_Symbol] :> Simp[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegerQ[2*m] && (Intege rQ[m] || GtQ[i/c, 0])
\[\int \frac {1}{\left (x^{2}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}}d x\]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) \]
(1/2*I + 1/2)*sqrt(2)*log((I + 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1))) - ( 1/2*I - 1/2)*sqrt(2)*log(-(I - 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1))) + ( 1/2*I - 1/2)*sqrt(2)*log((I - 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1))) - (1 /2*I + 1/2)*sqrt(2)*log(-(I + 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1)))
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{\left (x^2+1\right )\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]