Integrand size = 25, antiderivative size = 126 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) x}{\sqrt {10 \left (-1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right ) \]
-1/2*arctanh((x*(5-5^(1/2))+2*5^(1/2))/(x^2+2*x+2)^(1/2)/(-10+10*5^(1/2))^ (1/2))*(-2+2*5^(1/2))^(1/2)-1/2*arctan((2*5^(1/2)-x*(5+5^(1/2)))/(x^2+2*x+ 2)^(1/2)/(10+10*5^(1/2))^(1/2))*(2+2*5^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.77 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\text {RootSum}\left [8-8 \text {$\#$1}+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+\text {$\#$1}^3}\&\right ] \]
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) & ]
Time = 0.33 (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 |
| \(\displaystyle \frac {\left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\left (5-\sqrt {5}\right ) x+2 \sqrt {5}}{\sqrt {10 \left (\sqrt {5}-1\right )} \sqrt {x^2+2 x+2}}\right )}{\sqrt {2 \left (\sqrt {5}-1\right )}}-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {x^2+2 x+2}}\right )\) |
-(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])]
3.10.98.3.1 Defintions of rubi rules used
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])
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])
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.67 (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 +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 )+96 \sqrt {x^{2}+2 x +2}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}-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 -10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+2\right )+38 \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 )}{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 +48 \sqrt {x^{2}+2 x +2}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+80 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{3}+25 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right ) x +5 \sqrt {x^{2}+2 x +2}+50 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )}{4 x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+5\right )^{2}+x -2}\right )\) | \(430\) |
| default | \(\text {Expression too large to display}\) | \(753\) |
-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+80*RootOf(16*_Z^4 +8*_Z^2+5)^2*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+2)+96*(x^2+2*x+2)^(1 /2)*RootOf(16*_Z^4+8*_Z^2+5)^2-7*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+ 2)*x-10*RootOf(_Z^2+4*RootOf(16*_Z^4+8*_Z^2+5)^2+2)+38*(x^2+2*x+2)^(1/2))/ (4*x*RootOf(16*_Z^4+8*_Z^2+5)^2+x+2))-RootOf(16*_Z^4+8*_Z^2+5)*ln(-(-32*Ro otOf(16*_Z^4+8*_Z^2+5)^5*x+20*RootOf(16*_Z^4+8*_Z^2+5)^3*x+48*(x^2+2*x+2)^ (1/2)*RootOf(16*_Z^4+8*_Z^2+5)^2+80*RootOf(16*_Z^4+8*_Z^2+5)^3+25*RootOf(1 6*_Z^4+8*_Z^2+5)*x+5*(x^2+2*x+2)^(1/2)+50*RootOf(16*_Z^4+8*_Z^2+5))/(4*x*R ootOf(16*_Z^4+8*_Z^2+5)^2+x-2))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\frac {1}{2} \, \sqrt {2 i - 1} \log \left (-x + i \, \sqrt {2 i - 1} + \sqrt {x^{2} + 2 \, x + 2} - i\right ) - \frac {1}{2} \, \sqrt {2 i - 1} \log \left (-x - i \, \sqrt {2 i - 1} + \sqrt {x^{2} + 2 \, x + 2} - i\right ) - \frac {1}{2} \, \sqrt {-2 i - 1} \log \left (-x + i \, \sqrt {-2 i - 1} + \sqrt {x^{2} + 2 \, x + 2} + i\right ) + \frac {1}{2} \, \sqrt {-2 i - 1} \log \left (-x - i \, \sqrt {-2 i - 1} + \sqrt {x^{2} + 2 \, x + 2} + i\right ) \]
1/2*sqrt(2*I - 1)*log(-x + I*sqrt(2*I - 1) + sqrt(x^2 + 2*x + 2) - I) - 1/ 2*sqrt(2*I - 1)*log(-x - I*sqrt(2*I - 1) + sqrt(x^2 + 2*x + 2) - I) - 1/2* sqrt(-2*I - 1)*log(-x + I*sqrt(-2*I - 1) + sqrt(x^2 + 2*x + 2) + I) + 1/2* sqrt(-2*I - 1)*log(-x - I*sqrt(-2*I - 1) + sqrt(x^2 + 2*x + 2) + I)
\[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\int \frac {2 x + 1}{\left (x^{2} + 1\right ) \sqrt {x^{2} + 2 x + 2}}\, dx \]
\[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\int { \frac {2 \, x + 1}{\sqrt {x^{2} + 2 \, x + 2} {\left (x^{2} + 1\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (93) = 186\).
Time = 0.38 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.52 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x + \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} - 2 \, \sqrt {\sqrt {5} - 2} - 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} + \sqrt {\sqrt {5} - 2} + 2\right )}^{2}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} + 2 \, \sqrt {\sqrt {5} - 2} - 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} - \sqrt {\sqrt {5} - 2} + 2\right )}^{2}\right ) + \frac {{\left (\pi + 4 \, \arctan \left (\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} + 3\right )} + \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} + \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} - \frac {{\left (\pi + 4 \, \arctan \left (-\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} - \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} - 3\right )} - \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} - \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} \]
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)
Timed out. \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx=\int \frac {2\,x+1}{\left (x^2+1\right )\,\sqrt {x^2+2\,x+2}} \,d x \]