\(\int \frac {\sqrt {1+x^2}}{x (1-x^3)} \, dx\) [57]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=-\frac {1}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {1+x^2}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )-\text {arctanh}\left (\sqrt {1+x^2}\right ) \] Output:

-1/3*arctan((1+x)/(x^2+1)^(1/2))+1/3*arctanh(1/3*(1-x)*3^(1/2)/(x^2+1)^(1/ 
2))*3^(1/2)+1/3*2^(1/2)*arctanh(1/2*(1+x)*2^(1/2)/(x^2+1)^(1/2))-arctanh(( 
x^2+1)^(1/2))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=\frac {1}{3} \left (6 \text {arctanh}\left (x-\sqrt {1+x^2}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )-\text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right )-4 \log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {1+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right ) \] Input:

Integrate[Sqrt[1 + x^2]/(x*(1 - x^3)),x]
 

Output:

(6*ArcTanh[x - Sqrt[1 + x^2]] + 2*Sqrt[2]*ArcTanh[(1 - x + Sqrt[1 + x^2])/ 
Sqrt[2]] - RootSum[1 + 2*#1 + 2*#1^2 - 2*#1^3 + #1^4 & , (-Log[-x + Sqrt[1 
 + x^2] - #1] - 4*Log[-x + Sqrt[1 + x^2] - #1]*#1 + Log[-x + Sqrt[1 + x^2] 
 - #1]*#1^2)/(1 + 2*#1 - 3*#1^2 + 2*#1^3) & ])/3
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {\sqrt {x^2+1}}{x \left (1-x^3\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {\sqrt {x^2+1} (-2 x-1)}{3 \left (x^2+x+1\right )}-\frac {\sqrt {x^2+1}}{3 (x-1)}+\frac {\sqrt {x^2+1}}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{3} \arctan \left (\frac {x+1}{\sqrt {x^2+1}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2+1}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )-\text {arctanh}\left (\sqrt {x^2+1}\right )\)

Input:

Int[Sqrt[1 + x^2]/(x*(1 - x^3)),x]
 

Output:

-1/3*ArcTan[(1 + x)/Sqrt[1 + x^2]] + ArcTanh[(1 - x)/(Sqrt[3]*Sqrt[1 + x^2 
])]/Sqrt[3] + (Sqrt[2]*ArcTanh[(1 + x)/(Sqrt[2]*Sqrt[1 + x^2])])/3 - ArcTa 
nh[Sqrt[1 + x^2]]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs. \(2(69)=138\).

Time = 0.62 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.79

method	result	size
default	\(-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {x^{2}+1}}{3}-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\)	\(326\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+2 \sqrt {x^{2}+1}}{x -1}\right )}{3}-\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-1053 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5} x +2106 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5}+333 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x +72 \sqrt {x^{2}+1}\, \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-315 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3}-22 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) x -3 \sqrt {x^{2}+1}+11 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-18 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-3 x -1}\right )-9 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {-162 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5} x +324 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5}-117 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x +72 \sqrt {x^{2}+1}\, \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+180 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3}-11 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) x -5 \sqrt {x^{2}+1}-11 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-18 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+2 x +3}\right )+\operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-162 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5} x +324 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{5}-117 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3} x +72 \sqrt {x^{2}+1}\, \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+180 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{3}-11 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right ) x -5 \sqrt {x^{2}+1}-11 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{9 x \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}-18 \operatorname {RootOf}\left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )^{2}+2 x +3}\right )-\ln \left (\frac {\sqrt {x^{2}+1}+1}{x}\right )\)	\(641\)

Input:

int((x^2+1)^(1/2)/x/(-x^3+1),x,method=_RETURNVERBOSE)
 

Output:

-1/3*((x-1)^2+2*x)^(1/2)+1/3*2^(1/2)*arctanh(1/4*(2*x+2)*2^(1/2)/((x-1)^2+ 
2*x)^(1/2))+1/3*(x^2+1)^(1/2)-arctanh(1/(x^2+1)^(1/2))-1/3*2^(1/2)/(((x+1) 
^2/(1-x)^2+1)/((x+1)/(1-x)+1)^2)^(1/2)/((x+1)/(1-x)+1)*(2*(x+1)^2/(1-x)^2+ 
2)^(1/2)*arctan(1/((x+1)^2/(1-x)^2+1)*(2*(x+1)^2/(1-x)^2+2)^(1/2)*(x+1)/(1 
-x))+1/6*2^(1/2)*(2*(x+1)^2/(1-x)^2+2)^(1/2)*(3^(1/2)*arctanh(1/2*(2*(x+1) 
^2/(1-x)^2+2)^(1/2)*3^(1/2))+arctan(1/((x+1)^2/(1-x)^2+1)*(2*(x+1)^2/(1-x) 
^2+2)^(1/2)*(x+1)/(1-x)))/(((x+1)^2/(1-x)^2+1)/((x+1)/(1-x)+1)^2)^(1/2)/(( 
x+1)/(1-x)+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (67) = 134\).

Time = 0.08 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x + \sqrt {3} + 1\right )} + \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{6} \, \sqrt {3} \log \left (2 \, x^{2} - \sqrt {x^{2} + 1} {\left (2 \, x - \sqrt {3} + 1\right )} - \sqrt {3} {\left (x + 1\right )} + x + 3\right ) + \frac {1}{3} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} + 1\right )} - x + 1\right ) + \frac {1}{3} \, \arctan \left (-\sqrt {3} x + \sqrt {x^{2} + 1} {\left (\sqrt {3} - 1\right )} + x - 1\right ) - \log \left (-x + \sqrt {x^{2} + 1} + 1\right ) + \log \left (-x + \sqrt {x^{2} + 1} - 1\right ) \] Input:

integrate((x^2+1)^(1/2)/x/(-x^3+1),x, algorithm="fricas")
 

Output:

-1/6*sqrt(3)*log(2*x^2 - sqrt(x^2 + 1)*(2*x + sqrt(3) + 1) + sqrt(3)*(x + 
1) + x + 3) + 1/6*sqrt(3)*log(2*x^2 - sqrt(x^2 + 1)*(2*x - sqrt(3) + 1) - 
sqrt(3)*(x + 1) + x + 3) + 1/3*sqrt(2)*log(-(sqrt(2)*(x + 1) + sqrt(x^2 + 
1)*(sqrt(2) + 2) + x + 1)/(x - 1)) - 1/3*arctan(-sqrt(3)*x + sqrt(x^2 + 1) 
*(sqrt(3) + 1) - x + 1) + 1/3*arctan(-sqrt(3)*x + sqrt(x^2 + 1)*(sqrt(3) - 
 1) + x - 1) - log(-x + sqrt(x^2 + 1) + 1) + log(-x + sqrt(x^2 + 1) - 1)
 

Sympy [F]

\[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=- \int \frac {\sqrt {x^{2} + 1}}{x^{4} - x}\, dx \] Input:

integrate((x**2+1)**(1/2)/x/(-x**3+1),x)
                                                                                    
                                                                                    
 

Output:

-Integral(sqrt(x**2 + 1)/(x**4 - x), x)
 

Maxima [F]

\[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=\int { -\frac {\sqrt {x^{2} + 1}}{{\left (x^{3} - 1\right )} x} \,d x } \] Input:

integrate((x^2+1)^(1/2)/x/(-x^3+1),x, algorithm="maxima")
 

Output:

-integrate(sqrt(x^2 + 1)/((x^3 - 1)*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (67) = 134\).

Time = 0.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=-\frac {1}{6} \, \pi - \frac {1}{6} \, \sqrt {3} \log \left ({\left (x + \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) + \frac {1}{6} \, \sqrt {3} \log \left ({\left (x - \sqrt {3} - \sqrt {x^{2} + 1} + 1\right )}^{2} + {\left (x - \sqrt {x^{2} + 1}\right )}^{2}\right ) - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) - \frac {1}{3} \, \arctan \left (-{\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} + 1\right )} + 1\right ) - \frac {1}{3} \, \arctan \left ({\left (x - \sqrt {x^{2} + 1}\right )} {\left (\sqrt {3} - 1\right )} + 1\right ) - \log \left ({\left | -x + \sqrt {x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + 1} - 1 \right |}\right ) \] Input:

integrate((x^2+1)^(1/2)/x/(-x^3+1),x, algorithm="giac")
 

Output:

-1/6*pi - 1/6*sqrt(3)*log((x + sqrt(3) - sqrt(x^2 + 1) + 1)^2 + (x - sqrt( 
x^2 + 1))^2) + 1/6*sqrt(3)*log((x - sqrt(3) - sqrt(x^2 + 1) + 1)^2 + (x - 
sqrt(x^2 + 1))^2) - 1/3*sqrt(2)*log(abs(-2*x - 2*sqrt(2) + 2*sqrt(x^2 + 1) 
 + 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x^2 + 1) + 2)) - 1/3*arctan(-(x - sqrt 
(x^2 + 1))*(sqrt(3) + 1) + 1) - 1/3*arctan((x - sqrt(x^2 + 1))*(sqrt(3) - 
1) + 1) - log(abs(-x + sqrt(x^2 + 1) + 1)) + log(abs(-x + sqrt(x^2 + 1) - 
1))
 

Mupad [B] (verification not implemented)

Time = 23.61 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.20 \[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=\ln \left (x\right )-\ln \left (\sqrt {x^2+1}+1\right )-\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{3}-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}{4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-1}+\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}{4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+1} \] Input:

int(-(x^2 + 1)^(1/2)/(x*(x^3 - 1)),x)
 

Output:

log(x) - log((x^2 + 1)^(1/2) + 1) - (2^(1/2)*(log(x - 1) - log(x + 2^(1/2) 
*(x^2 + 1)^(1/2) + 1)))/3 - ((log(x - (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2) 
/2 - 1i/2)*(x^2 + 1)^(1/2) - x/2 + (3^(1/2)*x*1i)/2 + 1))*(((3^(1/2)*1i)/2 
 - 1/2)^2 + 1)^(1/2))/(4*((3^(1/2)*1i)/2 - 1/2)^3 - 1) + ((log(x + (3^(1/2 
)*1i)/2 + 1/2) - log((3^(1/2)/2 + 1i/2)*(x^2 + 1)^(1/2) - x/2 - (3^(1/2)*x 
*1i)/2 + 1))*(((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2))/(4*((3^(1/2)*1i)/2 + 1/ 
2)^3 + 1)
 

Reduce [F]

\[ \int \frac {\sqrt {1+x^2}}{x \left (1-x^3\right )} \, dx=-\frac {\sqrt {x^{2}+1}}{3}+\frac {\sqrt {2}\, \mathrm {log}\left (-\sqrt {x^{2}+1}\, \sqrt {2}-x -1\right )}{3}-\frac {\sqrt {2}\, \mathrm {log}\left (x -1\right )}{3}-\frac {2 \left (\int \frac {\sqrt {x^{2}+1}}{x^{4}+x^{3}+2 x^{2}+x +1}d x \right )}{3}+\frac {\left (\int \frac {\sqrt {x^{2}+1}\, x^{3}}{x^{4}+x^{3}+2 x^{2}+x +1}d x \right )}{3}+\frac {\left (\int \frac {\sqrt {x^{2}+1}\, x^{2}}{x^{4}+x^{3}+2 x^{2}+x +1}d x \right )}{3}+\frac {\mathrm {log}\left (\sqrt {x^{2}+1}-1\right )}{2}-\frac {\mathrm {log}\left (\sqrt {x^{2}+1}+1\right )}{2} \] Input:

int((x^2+1)^(1/2)/x/(-x^3+1),x)
 

Output:

( - 2*sqrt(x**2 + 1) + 2*sqrt(2)*log( - sqrt(x**2 + 1)*sqrt(2) - x - 1) - 
2*sqrt(2)*log(x - 1) - 4*int(sqrt(x**2 + 1)/(x**4 + x**3 + 2*x**2 + x + 1) 
,x) + 2*int((sqrt(x**2 + 1)*x**3)/(x**4 + x**3 + 2*x**2 + x + 1),x) + 2*in 
t((sqrt(x**2 + 1)*x**2)/(x**4 + x**3 + 2*x**2 + x + 1),x) + 3*log(sqrt(x** 
2 + 1) - 1) - 3*log(sqrt(x**2 + 1) + 1))/6