Integrand size = 32, antiderivative size = 81 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=-\frac {1}{2} \sqrt {1-i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \]
-1/2*arctanh((1+I*x)/(1-I)^(1/2)/(1-I*x^2)^(1/2))*(1-I)^(1/2)-1/2*arctanh( (1-I*x)/(1+I)^(1/2)/(1+I*x^2)^(1/2))*(1+I)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(81)=162\).
Time = 1.07 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {\sqrt {-1+\sqrt {2}} \left (\arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )-\arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
(Sqrt[-1 + Sqrt[2]]*(ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] - ArcTan[(Sqrt[2*(-1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sq rt[1 + x^4])]) - Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + S qrt[1 + x^4]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[2*(1 + Sqrt[2])]*x*Sqrt[x ^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/Sqrt[2]
Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2558, 488, 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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{(x+1) \sqrt {x^4+1}} \, dx\) |
\(\Big \downarrow \) 2558 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(x+1) \sqrt {1-i x^2}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(x+1) \sqrt {i x^2+1}}dx\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1-i)-\frac {(i x+1)^2}{1-i x^2}}d\frac {i x+1}{\sqrt {1-i x^2}}-\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1+i)-\frac {(1-i x)^2}{i x^2+1}}d\frac {1-i x}{\sqrt {i x^2+1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{2} \sqrt {1-i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )\) |
-1/2*(Sqrt[1 - I]*ArcTanh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])]) - (Sqr t[1 + I]*ArcTanh[(1 - I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/2
3.1.13.3.1 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[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^ 4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_Symbol] :> Simp[(1 - I)/2 Int[(c + d*x) ^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Simp[(1 + I)/2 Int[(c + d*x)^m/Sqrt[ Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && G tQ[a, 0]
\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (1+x \right ) \sqrt {x^{4}+1}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (49) = 98\).
Time = 3.27 (sec) , antiderivative size = 504, normalized size of antiderivative = 6.22 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {1}{8} \, \sqrt {-2 \, \sqrt {2} + 2} \log \left (-\frac {\sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 2} + {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {-2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{8} \, \sqrt {-2 \, \sqrt {2} + 2} \log \left (\frac {\sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 2} - {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {-2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) \]
1/8*sqrt(-2*sqrt(2) + 2)*log(-(sqrt(x^4 + 1)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 2) + (2*x^3 + sqrt(2)*(x^3 - x^2 - x - 1) - sqrt(x^4 + 1)*(sqrt(2)*(x - 1) + 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (x^2 + sqrt(2)*(x^2 + 1) + 1)* sqrt(-2*sqrt(2) + 2))/(x^2 + 2*x + 1)) - 1/8*sqrt(-2*sqrt(2) + 2)*log((sqr t(x^4 + 1)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 2) - (2*x^3 + sqrt(2)*(x^3 - x^ 2 - x - 1) - sqrt(x^4 + 1)*(sqrt(2)*(x - 1) + 2*x) - 2)*sqrt(x^2 + sqrt(x^ 4 + 1)) - (x^2 + sqrt(2)*(x^2 + 1) + 1)*sqrt(-2*sqrt(2) + 2))/(x^2 + 2*x + 1)) - 1/8*sqrt(2*sqrt(2) + 2)*log(-((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) + ( x^2 - sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1)) + 1/8*sqrt(2*sqrt(2) + 2)*log(-((2*x^3 - sqrt(2)*(x ^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (x^2 - sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1))
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\left (x + 1\right ) \sqrt {x^{4} + 1}}\, dx \]
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x + 1\right )}} \,d x } \]
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}\,\left (x+1\right )} \,d x \]
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{5}+x^{4}+x +1}d x \]