\(\int \frac {1+2 x}{(1+x^2) \sqrt {2+2 x+x^2}} \, dx\) [364]

Optimal result

Integrand size = 25, antiderivative size = 126 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) x}{\sqrt {10 \left (-1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right ) \] Output:

-1/2*(2+2*5^(1/2))^(1/2)*arctan((2*5^(1/2)-(5+5^(1/2))*x)/(10+10*5^(1/2))^ 
(1/2)/(x^2+2*x+2)^(1/2))-1/2*(-2+2*5^(1/2))^(1/2)*arctanh((2*5^(1/2)+(5-5^ 
(1/2))*x)/(-10+10*5^(1/2))^(1/2)/(x^2+2*x+2)^(1/2))
 

Mathematica [C] (verified)

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

Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.77 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\text {RootSum}\left [8-8 \text {$\#$1}+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+\text {$\#$1}^3}\&\right ] \] Input:

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

Output:

RootSum[8 - 8*#1 + #1^4 & , (-Log[-x + Sqrt[2 + 2*x + x^2] - #1] - Log[-x 
+ Sqrt[2 + 2*x + x^2] - #1]*#1 + Log[-x + Sqrt[2 + 2*x + x^2] - #1]*#1^2)/ 
(-2 + #1^3) & ]
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1369, 25, 1363, 216, 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.

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

\(\Big \downarrow \) 1369

\(\displaystyle \frac {\int -\frac {-2 \sqrt {5} x-\sqrt {5}+5}{\left (x^2+1\right ) \sqrt {x^2+2 x+2}}dx}{2 \sqrt {5}}-\frac {\int -\frac {2 \sqrt {5} x+\sqrt {5}+5}{\left (x^2+1\right ) \sqrt {x^2+2 x+2}}dx}{2 \sqrt {5}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 \sqrt {5} x+\sqrt {5}+5}{\left (x^2+1\right ) \sqrt {x^2+2 x+2}}dx}{2 \sqrt {5}}-\frac {\int \frac {-2 \sqrt {5} x-\sqrt {5}+5}{\left (x^2+1\right ) \sqrt {x^2+2 x+2}}dx}{2 \sqrt {5}}\)

\(\Big \downarrow \) 1363

\(\displaystyle 2 \sqrt {5} \left (1-\sqrt {5}\right ) \int \frac {1}{\frac {2 \left (\left (5-\sqrt {5}\right ) x+2 \sqrt {5}\right )^2}{x^2+2 x+2}+20 \left (1-\sqrt {5}\right )}d\left (-\frac {\left (5-\sqrt {5}\right ) x+2 \sqrt {5}}{\sqrt {x^2+2 x+2}}\right )-2 \sqrt {5} \left (1+\sqrt {5}\right ) \int \frac {1}{\frac {2 \left (2 \sqrt {5}-\left (5+\sqrt {5}\right ) x\right )^2}{x^2+2 x+2}+20 \left (1+\sqrt {5}\right )}d\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {x^2+2 x+2}}\)

\(\Big \downarrow \) 216

\(\displaystyle 2 \sqrt {5} \left (1-\sqrt {5}\right ) \int \frac {1}{\frac {2 \left (\left (5-\sqrt {5}\right ) x+2 \sqrt {5}\right )^2}{x^2+2 x+2}+20 \left (1-\sqrt {5}\right )}d\left (-\frac {\left (5-\sqrt {5}\right ) x+2 \sqrt {5}}{\sqrt {x^2+2 x+2}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {x^2+2 x+2}}\right )\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {\left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\left (5-\sqrt {5}\right ) x+2 \sqrt {5}}{\sqrt {10 \left (\sqrt {5}-1\right )} \sqrt {x^2+2 x+2}}\right )}{\sqrt {2 \left (\sqrt {5}-1\right )}}-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {x^2+2 x+2}}\right )\)

Input:

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

Output:

-(Sqrt[(1 + Sqrt[5])/2]*ArcTan[(2*Sqrt[5] - (5 + Sqrt[5])*x)/(Sqrt[10*(1 + 
 Sqrt[5])]*Sqrt[2 + 2*x + x^2])]) + ((1 - Sqrt[5])*ArcTanh[(2*Sqrt[5] + (5 
 - Sqrt[5])*x)/(Sqrt[10*(-1 + Sqrt[5])]*Sqrt[2 + 2*x + x^2])])/Sqrt[2*(-1 
+ Sqrt[5])]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 220

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])
 

rule 1363

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f 
_.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h   Subst[Int[1/Simp[2*a^2*g*h*c + a 
*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ 
[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
 

rule 1369

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp 
[1/(2*q)   Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c 
*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q)   Int[ 
Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + 
 c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] 
&& NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
 

Maple [C] (warning: unable to verify)

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

Time = 0.85 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.41

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \ln \left (-\frac {32 x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right )+52 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2} x -96 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2} \sqrt {x^{2}+2 x +2}+80 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right )-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right ) x -38 \sqrt {x^{2}+2 x +2}-10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right )}{4 x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+x +2}\right )}{2}-\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right ) \ln \left (\frac {32 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{5} x -20 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{3} x -80 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{3}-48 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2} \sqrt {x^{2}+2 x +2}-25 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right ) x -50 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )-5 \sqrt {x^{2}+2 x +2}}{4 x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+x -2}\right )\)	\(430\)
default	\(\text {Expression too large to display}\)	\(753\)

Input:

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

Output:

1/2*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+2)*ln(-(32*x*RootOf(16*_Z^4+8 
*_Z^2+5)^4*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+2)+52*RootOf(_Z^2+4*Ro 
otOf(16*_Z^4+8*_Z^2+5)^2+2)*RootOf(16*_Z^4+8*_Z^2+5)^2*x-96*RootOf(16*_Z^4 
+8*_Z^2+5)^2*(x^2+2*x+2)^(1/2)+80*RootOf(16*_Z^4+8*_Z^2+5)^2*RootOf(_Z^2+4 
*RootOf(16*_Z^4+8*_Z^2+5)^2+2)-7*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+ 
2)*x-38*(x^2+2*x+2)^(1/2)-10*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+2))/ 
(4*x*RootOf(16*_Z^4+8*_Z^2+5)^2+x+2))-RootOf(16*_Z^4+8*_Z^2+5)*ln((32*Root 
Of(16*_Z^4+8*_Z^2+5)^5*x-20*RootOf(16*_Z^4+8*_Z^2+5)^3*x-80*RootOf(16*_Z^4 
+8*_Z^2+5)^3-48*RootOf(16*_Z^4+8*_Z^2+5)^2*(x^2+2*x+2)^(1/2)-25*RootOf(16* 
_Z^4+8*_Z^2+5)*x-50*RootOf(16*_Z^4+8*_Z^2+5)-5*(x^2+2*x+2)^(1/2))/(4*x*Roo 
tOf(16*_Z^4+8*_Z^2+5)^2+x-2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (93) = 186\).

Time = 0.09 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.59 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=-\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (-\frac {1}{2} \, {\left (\sqrt {5} {\left (x + 1\right )} - \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 1\right )} + {\left (\sqrt {5} {\left (x + 1\right )} - \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 3\right )} + 3 \, x + 3\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + x + 3\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} {\left (x + 1\right )} - \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 1\right )} - {\left (\sqrt {5} {\left (x + 1\right )} - \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 3\right )} + 3 \, x + 3\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + x + 3\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (2 \, x^{2} - 2 \, \sqrt {x^{2} + 2 \, x + 2} x + {\left (\sqrt {5} x - \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 1\right )} + x - 2\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + 2 \, x + \sqrt {5} + 3\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (2 \, x^{2} - 2 \, \sqrt {x^{2} + 2 \, x + 2} x - {\left (\sqrt {5} x - \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 1\right )} + x - 2\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + 2 \, x + \sqrt {5} + 3\right ) \] Input:

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

Output:

-sqrt(1/2*sqrt(5) + 1/2)*arctan(-1/2*(sqrt(5)*(x + 1) - sqrt(x^2 + 2*x + 2 
)*(sqrt(5) + 1) + (sqrt(5)*(x + 1) - sqrt(x^2 + 2*x + 2)*(sqrt(5) + 3) + 3 
*x + 3)*sqrt(1/2*sqrt(5) - 1/2) + x + 3)*sqrt(1/2*sqrt(5) + 1/2)) + sqrt(1 
/2*sqrt(5) + 1/2)*arctan(1/2*(sqrt(5)*(x + 1) - sqrt(x^2 + 2*x + 2)*(sqrt( 
5) + 1) - (sqrt(5)*(x + 1) - sqrt(x^2 + 2*x + 2)*(sqrt(5) + 3) + 3*x + 3)* 
sqrt(1/2*sqrt(5) - 1/2) + x + 3)*sqrt(1/2*sqrt(5) + 1/2)) + 1/2*sqrt(1/2*s 
qrt(5) - 1/2)*log(2*x^2 - 2*sqrt(x^2 + 2*x + 2)*x + (sqrt(5)*x - sqrt(x^2 
+ 2*x + 2)*(sqrt(5) + 1) + x - 2)*sqrt(1/2*sqrt(5) - 1/2) + 2*x + sqrt(5) 
+ 3) - 1/2*sqrt(1/2*sqrt(5) - 1/2)*log(2*x^2 - 2*sqrt(x^2 + 2*x + 2)*x - ( 
sqrt(5)*x - sqrt(x^2 + 2*x + 2)*(sqrt(5) + 1) + x - 2)*sqrt(1/2*sqrt(5) - 
1/2) + 2*x + sqrt(5) + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (93) = 186\).

Time = 0.18 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.52 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x + \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} - 2 \, \sqrt {\sqrt {5} - 2} - 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} + \sqrt {\sqrt {5} - 2} + 2\right )}^{2}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} + 2 \, \sqrt {\sqrt {5} - 2} - 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} - \sqrt {\sqrt {5} - 2} + 2\right )}^{2}\right ) + \frac {{\left (\pi + 4 \, \arctan \left (\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} + 3\right )} + \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} + \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} - \frac {{\left (\pi + 4 \, \arctan \left (-\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} - \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} - 3\right )} - \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} - \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} \] Input:

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

Output:

1/4*sqrt(2*sqrt(5) - 2)*log(256*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x + 
 sqrt(5)*sqrt(sqrt(5) - 2) + sqrt(5) + 2*sqrt(x^2 + 2*x + 2) - 2*sqrt(sqrt 
(5) - 2) - 2)^2 + 256*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x - sqrt(5) + 
 2*sqrt(x^2 + 2*x + 2) + sqrt(sqrt(5) - 2) + 2)^2) - 1/4*sqrt(2*sqrt(5) - 
2)*log(256*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x - sqrt(5)*sqrt(sqrt(5) 
 - 2) + sqrt(5) + 2*sqrt(x^2 + 2*x + 2) + 2*sqrt(sqrt(5) - 2) - 2)^2 + 256 
*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x - sqrt(5) + 2*sqrt(x^2 + 2*x + 2 
) - sqrt(sqrt(5) - 2) + 2)^2) + 1/4*(pi + 4*arctan(1/2*(x - sqrt(x^2 + 2*x 
 + 2))*(2*sqrt(5)*sqrt(sqrt(5) - 2) + sqrt(5) + 4*sqrt(sqrt(5) - 2) + 3) + 
 3/2*sqrt(5)*sqrt(sqrt(5) - 2) + 1/2*sqrt(5) + 7/2*sqrt(sqrt(5) - 2) + 3/2 
))*sqrt(2*sqrt(5) - 2)/(sqrt(5) - 1) - 1/4*(pi + 4*arctan(-1/2*(x - sqrt(x 
^2 + 2*x + 2))*(2*sqrt(5)*sqrt(sqrt(5) - 2) - sqrt(5) + 4*sqrt(sqrt(5) - 2 
) - 3) - 3/2*sqrt(5)*sqrt(sqrt(5) - 2) + 1/2*sqrt(5) - 7/2*sqrt(sqrt(5) - 
2) + 3/2))*sqrt(2*sqrt(5) - 2)/(sqrt(5) - 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

2*int(x/(sqrt(x**2 + 2*x + 2)*x**2 + sqrt(x**2 + 2*x + 2)),x) + int(1/(sqr 
t(x**2 + 2*x + 2)*x**2 + sqrt(x**2 + 2*x + 2)),x)