Integrand size = 49, antiderivative size = 337 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=-x^2+\frac {1}{2} \left (2+\sqrt {2}\right ) \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}\right )-2 \log \left (-2 x^2+\sqrt {2} x \sqrt {1+2 x^2}\right )+\frac {1}{2} \text {RootSum}\left [1+3 \text {$\#$1}^2-2 \sqrt {2} \text {$\#$1}^2+3 \text {$\#$1}^4+2 \sqrt {2} \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right )+4 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+4 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^5}{3 \text {$\#$1}-2 \sqrt {2} \text {$\#$1}+6 \text {$\#$1}^3+4 \sqrt {2} \text {$\#$1}^3+3 \text {$\#$1}^5}\&\right ] \]
Time = 0.79 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.97 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=-x^2+\left (1+\frac {1}{\sqrt {2}}\right ) \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}\right )-2 \log \left (x \left (-2 x+\sqrt {2+4 x^2}\right )\right )+\frac {1}{2} \text {RootSum}\left [1+3 \text {$\#$1}^2-2 \sqrt {2} \text {$\#$1}^2+3 \text {$\#$1}^4+2 \sqrt {2} \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right )+4 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+4 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^5}{3 \text {$\#$1}-2 \sqrt {2} \text {$\#$1}+6 \text {$\#$1}^3+4 \sqrt {2} \text {$\#$1}^3+3 \text {$\#$1}^5}\&\right ] \]
-x^2 + (1 + 1/Sqrt[2])*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2]] - 2*Log[x*(-2*x + Sqrt[2 + 4*x^2])] + RootSum[1 + 3*#1^2 - 2*Sqrt[2]*#1^2 + 3*#1^4 + 2*Sq rt[2]*#1^4 + #1^6 & , (-Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1] + 4*Log[- (Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1 + 2*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x ^2] - #1]*#1^3 + 4*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^3 + Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^4 + 2*Log[-(Sqrt[2]*x) + Sqrt [1 + 2*x^2] - #1]*#1^5)/(3*#1 - 2*Sqrt[2]*#1 + 6*#1^3 + 4*Sqrt[2]*#1^3 + 3 *#1^5) & ]/2
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {7293, 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.
| \(\displaystyle \int \frac {\left (2 x^2+1\right )^{5/2}+\sqrt {2 x^2+1}-x^2}{x^2-x \left (2 x^2+1\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2+1\right )^{5/2}}{x \left (2 \sqrt {2 x^2+1} x^2+\sqrt {2 x^2+1}-x\right )}-\frac {\sqrt {2 x^2+1}}{x \left (2 \sqrt {2 x^2+1} x^2+\sqrt {2 x^2+1}-x\right )}-\frac {x}{x-\left (2 x^2+1\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arcsinh}\left (\sqrt {2} x\right )}{\sqrt {2}}-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {2 x^2+1}}\right )+\frac {1}{5} \arctan \left (\sqrt {2 x^2+1}\right )-\frac {2}{5} \arctan \left (4 x^2+1\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {2 x^2+1}}{\sqrt {\sqrt {2}-1}}\right )-\frac {3 \arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {2 x^2+1}}{\sqrt {\sqrt {2}-1}}\right )}{20 \sqrt {\sqrt {2}-1}}-\frac {1}{10} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {2 \sqrt {2 x^2+1}+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {3 \arctan \left (\frac {2 \sqrt {2 x^2+1}+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{20 \sqrt {\sqrt {2}-1}}-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {5 \sqrt {2}-1} \arctan \left (-2 \sqrt {2 \left (\sqrt {2}-1\right )} x-\sqrt {2}+1\right )+\frac {1}{20} \sqrt {5 \sqrt {2}-1} \arctan \left (2 \sqrt {2 \left (\sqrt {2}-1\right )} x-\sqrt {2}+1\right )+\frac {2}{5} \text {arctanh}\left (\frac {x}{\sqrt {2 x^2+1}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {2 x^2+1}}\right )-x^2+\frac {1}{5} \log \left (x^2+1\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (4 x^2-2 \sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (4 x^2+2 \sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )+\frac {\log \left (2 \left (2 x^2+1\right )-2 \sqrt {1+\sqrt {2}} \sqrt {2 x^2+1}+\sqrt {2}\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {3 \log \left (2 \left (2 x^2+1\right )-2 \sqrt {1+\sqrt {2}} \sqrt {2 x^2+1}+\sqrt {2}\right )}{40 \sqrt {1+\sqrt {2}}}-\frac {\log \left (\sqrt {2} \left (2 x^2+1\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {2 x^2+1}+1\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {3 \log \left (\sqrt {2} \left (2 x^2+1\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {2 x^2+1}+1\right )}{40 \sqrt {1+\sqrt {2}}}+\frac {3}{20} \log \left (8 x^4+4 x^2+1\right )-2 \log (x)\) |
-x^2 - ArcSinh[Sqrt[2]*x]/Sqrt[2] - ArcTan[x]/5 - (Sqrt[-1 + 5*Sqrt[2]]*Ar cTan[1 - Sqrt[2] - 2*Sqrt[2*(-1 + Sqrt[2])]*x])/20 + (Sqrt[-1 + 5*Sqrt[2]] *ArcTan[1 - Sqrt[2] + 2*Sqrt[2*(-1 + Sqrt[2])]*x])/20 - (2/5 - (3*I)/10)*A rcTan[((1 + I)*x)/Sqrt[1 + 2*x^2]] + ArcTan[Sqrt[1 + 2*x^2]]/5 - (2*ArcTan [1 + 4*x^2])/5 - (3*ArcTan[(Sqrt[1 + Sqrt[2]] - 2*Sqrt[1 + 2*x^2])/Sqrt[-1 + Sqrt[2]]])/(20*Sqrt[-1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqr t[1 + Sqrt[2]] - 2*Sqrt[1 + 2*x^2])/Sqrt[-1 + Sqrt[2]]])/10 + (3*ArcTan[(S qrt[1 + Sqrt[2]] + 2*Sqrt[1 + 2*x^2])/Sqrt[-1 + Sqrt[2]]])/(20*Sqrt[-1 + S qrt[2]]) - (Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[1 + Sqrt[2]] + 2*Sqrt[1 + 2 *x^2])/Sqrt[-1 + Sqrt[2]]])/10 + (2*ArcTanh[x/Sqrt[1 + 2*x^2]])/5 - (3/10 - (2*I)/5)*ArcTanh[((1 + I)*x)/Sqrt[1 + 2*x^2]] - 2*Log[x] + Log[1 + x^2]/ 5 + (Sqrt[1 + 5*Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[2*(-1 + Sqrt[2])]*x + 4*x^2] )/40 - (Sqrt[1 + 5*Sqrt[2]]*Log[Sqrt[2] + 2*Sqrt[2*(-1 + Sqrt[2])]*x + 4*x ^2])/40 + (3*Log[1 + 4*x^2 + 8*x^4])/20 + (3*Log[Sqrt[2] - 2*Sqrt[1 + Sqrt [2]]*Sqrt[1 + 2*x^2] + 2*(1 + 2*x^2)])/(40*Sqrt[1 + Sqrt[2]]) + Log[Sqrt[2 ] - 2*Sqrt[1 + Sqrt[2]]*Sqrt[1 + 2*x^2] + 2*(1 + 2*x^2)]/(20*Sqrt[2*(1 + S qrt[2])]) - (3*Log[1 + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + 2*x^2] + Sqrt[2]*(1 + 2*x^2)])/(40*Sqrt[1 + Sqrt[2]]) - Log[1 + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + 2*x^2] + Sqrt[2]*(1 + 2*x^2)]/(20*Sqrt[2*(1 + Sqrt[2])])
3.30.26.3.1 Defintions of rubi rules used
Timed out.
hanged
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.08 (sec) , antiderivative size = 2052, normalized size of antiderivative = 6.09 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
integrate((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x , algorithm="fricas")
-x^2 - 1/20*(10*sqrt(7/200*I + 1/200) + 4*I - 3)*log(17/25*(10*sqrt(7/200* I + 1/200) + 4*I - 3)^3 - 11/25*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 1 67*x - 54*sqrt(7/200*I + 1/200) - 108/5*I - 254/5) - 1/20*(10*sqrt(-7/200* I + 1/200) - 4*I - 3)*log(-17/25*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^3 - 2/25*(85*sqrt(7/200*I + 1/200) + 34*I + 82)*(10*sqrt(-7/200*I + 1/200) - 4 *I - 3)^2 - 204/25*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/25*(17*(10*s qrt(7/200*I + 1/200) + 4*I - 3)^2 + 2040*sqrt(7/200*I + 1/200) + 816*I + 9 68)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 167*x - 578*sqrt(7/200*I + 1/2 00) - 1156/5*I - 658/5) + 1/20*(5*sqrt(7/200*I + 1/200) + 5*sqrt(-7/200*I + 1/200) - sqrt(-3/4*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt (7/200*I + 1/200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*( 10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200) - 24*I - 31) + 3)*log(1/25*(85*sqrt(7/200*I + 1/200) + 34*I + 82)*(10*sqrt(-7/200* I + 1/200) - 4*I - 3)^2 + 43/10*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 1 /50*(17*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 2040*sqrt(7/200*I + 1/200 ) + 816*I + 968)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 1/25*sqrt(-3/4*(1 0*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/200*I + 1/20 0) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200) - 24*I - 31)*(2*(85*sqrt(7/200* I + 1/200) + 34*I + 82)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 2150*sq...
Not integrable
Time = 102.66 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.53 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {x^{2}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\right )\, dx - \int \frac {2 \sqrt {2 x^{2} + 1}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\, dx - \int \frac {4 x^{2} \sqrt {2 x^{2} + 1}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\, dx - \int \frac {4 x^{4} \sqrt {2 x^{2} + 1}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\, dx \]
-Integral(-x**2/(2*x**3*sqrt(2*x**2 + 1) - x**2 + x*sqrt(2*x**2 + 1)), x) - Integral(2*sqrt(2*x**2 + 1)/(2*x**3*sqrt(2*x**2 + 1) - x**2 + x*sqrt(2*x **2 + 1)), x) - Integral(4*x**2*sqrt(2*x**2 + 1)/(2*x**3*sqrt(2*x**2 + 1) - x**2 + x*sqrt(2*x**2 + 1)), x) - Integral(4*x**4*sqrt(2*x**2 + 1)/(2*x** 3*sqrt(2*x**2 + 1) - x**2 + x*sqrt(2*x**2 + 1)), x)
Not integrable
Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.72 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\int { -\frac {{\left (2 \, x^{2} + 1\right )}^{\frac {5}{2}} - x^{2} + \sqrt {2 \, x^{2} + 1}}{{\left (2 \, x^{2} + 1\right )}^{\frac {3}{2}} x - x^{2}} \,d x } \]
integrate((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x , algorithm="maxima")
-x^2 - integrate(-(256*x^16 - 128*x^15 + 1024*x^14 - 448*x^13 + 1792*x^12 - 640*x^11 + 1824*x^10 - 496*x^9 + 1188*x^8 - 234*x^7 + 508*x^6 - 69*x^5 + 142*x^4 - 12*x^3 + 24*x^2 - x + 2)/(384*x^15 + 1344*x^13 + 2176*x^11 + 20 00*x^9 + 1086*x^7 + 335*x^5 + 52*x^3 - (256*x^16 + 1024*x^14 + 1984*x^12 + 2272*x^10 + 1636*x^8 + 724*x^6 + 181*x^4 + 22*x^2 + 1)*sqrt(2*x^2 + 1) + 3*x), x) + 2/3*integrate((8*x^5 + 14*x^3 - 3*x^2 + 8*x)/(8*x^6 + 12*x^4 + 9*x^2 + 1), x) + 1/6*log(2*x^2 + 1) - 2*log(x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.02 (sec) , antiderivative size = 929, normalized size of antiderivative = 2.76 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
integrate((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x , algorithm="giac")
-x^2 + 1/20*(sqrt(5*sqrt(2) - 1) + 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x + s qrt(2)*(2*sqrt(2) + 3)^(1/4) - 2*sqrt(2*x^2 + 1))/(2*sqrt(2) + 3)^(1/4)) + 1/20*(sqrt(5*sqrt(2) - 1) - 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x - sqrt(2) *(2*sqrt(2) + 3)^(1/4) - 2*sqrt(2*x^2 + 1))/(2*sqrt(2) + 3)^(1/4)) + 1/20* (sqrt(5*sqrt(2) - 1) + 8)*arctan(8*(1/8)^(3/4)*(2*x + (1/8)^(1/4)*sqrt(-sq rt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/20*(sqrt(5*sqrt(2) - 1) - 8)*arctan(8*( 1/8)^(3/4)*(2*x - (1/8)^(1/4)*sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/2 0*(sqrt(5*sqrt(2) - 1) - 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x + sqrt(2)*(-2 *sqrt(2) + 3)^(1/4) - 2*sqrt(2*x^2 + 1))/(-2*sqrt(2) + 3)^(1/4)) - 1/20*(s qrt(5*sqrt(2) - 1) + 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x - sqrt(2)*(-2*sqr t(2) + 3)^(1/4) - 2*sqrt(2*x^2 + 1))/(-2*sqrt(2) + 3)^(1/4)) + 1/40*sqrt(5 *sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 + sqrt(2)*(sqrt(2)*x - s qrt(2*x^2 + 1))*(2*sqrt(2) + 3)^(1/4) + sqrt(2*sqrt(2) + 3)) - 1/40*sqrt(5 *sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 - sqrt(2)*(sqrt(2)*x - s qrt(2*x^2 + 1))*(2*sqrt(2) + 3)^(1/4) + sqrt(2*sqrt(2) + 3)) + 1/40*sqrt(5 *sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 + sqrt(2)*(sqrt(2)*x - s qrt(2*x^2 + 1))*(-2*sqrt(2) + 3)^(1/4) + sqrt(-2*sqrt(2) + 3)) - 1/40*sqrt (5*sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 - sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + 1))*(-2*sqrt(2) + 3)^(1/4) + sqrt(-2*sqrt(2) + 3)) - 1/40*sq rt(5*sqrt(2) + 1)*log(x^2 + (1/8)^(1/4)*x*sqrt(-sqrt(2) + 2) + 1/2*sqrt...
Time = 7.76 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.60 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
log(x + 1i)*(2/5 - 1i/5) - log(x - (2^(1/2)*(x^2 + 1/2)^(1/2))/2 + 1i/2)*( 1/5 - 1i/10) + log(x + (2^(1/2)*(x^2 + 1/2)^(1/2))/2 - 1i/2)*(1/5 + 1i/10) - 2*log(x) + log(x + (- 1/4 - 1i/4)^(1/2))*((- 1/4 - 1i/4)^(3/2)*(1/5 + 2 i/5) + (3/20 - 1i/5)) - (2^(1/2)*asinh(2^(1/2)*x))/2 - log(x - (- 1 + 1i)^ (1/2)/2)*((- 1/4 + 1i/4)^(3/2)*(1/5 - 2i/5) - (3/20 + 1i/5)) + log(x + (- 1 + 1i)^(1/2)/2)*((- 1/4 + 1i/4)^(3/2)*(1/5 - 2i/5) + (3/20 + 1i/5)) - log (x - (- 1/4 - 1i/4)^(1/2))*((- 1/4 - 1i/4)^(3/2)*(1/5 + 2i/5) - (3/20 - 1i /5)) - x^2 - (2^(1/2)*(log((1/4 - 1i/4)^(1/2)*(x^2 + 1/2)^(1/2) - (- 1/4 - 1i/4)^(1/2)*x + 1/2) - log(x + (- 1/4 - 1i/4)^(1/2)))*((- 1/4 - 1i/4)^(1/ 2) + 4*(- 1/4 - 1i/4)^(3/2) + 4*(- 1/4 - 1i/4)^(5/2) + (1/4 - 3i/4)))/(2*( 1/4 - 1i/4)^(1/2)*(10*(- 1/4 - 1i/4)^(1/2) + 48*(- 1/4 - 1i/4)^(3/2) + 48* (- 1/4 - 1i/4)^(5/2))) + (2^(1/2)*(log(x - (- 1 + 1i)^(1/2)/2) - log((- 1/ 4 + 1i/4)^(1/2)*x + (1/4 + 1i/4)^(1/2)*(x^2 + 1/2)^(1/2) + 1/2))*((- 1/4 + 1i/4)^(1/2) + 4*(- 1/4 + 1i/4)^(3/2) + 4*(- 1/4 + 1i/4)^(5/2) - (1/4 + 3i /4)))/(2*(1/4 + 1i/4)^(1/2)*(10*(- 1/4 + 1i/4)^(1/2) + 48*(- 1/4 + 1i/4)^( 3/2) + 48*(- 1/4 + 1i/4)^(5/2))) + (2^(1/2)*(log(x + (- 1 + 1i)^(1/2)/2) - log((1/4 + 1i/4)^(1/2)*(x^2 + 1/2)^(1/2) - (- 1/4 + 1i/4)^(1/2)*x + 1/2)) *((- 1/4 + 1i/4)^(1/2) + 4*(- 1/4 + 1i/4)^(3/2) + 4*(- 1/4 + 1i/4)^(5/2) + (1/4 + 3i/4)))/(2*(1/4 + 1i/4)^(1/2)*(10*(- 1/4 + 1i/4)^(1/2) + 48*(- 1/4 + 1i/4)^(3/2) + 48*(- 1/4 + 1i/4)^(5/2))) + (2^(1/2)*(log(x - (- 1/4 -...