Integrand size = 40, antiderivative size = 4030 \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx =\text {Too large to display} \]
1/2*x^2*(-1+2*2^(1/2))-2^(1/2)*(-1/3*(x^4+2*x^2+4*x+1)^(1/2)+1/3*(1+x)*(x^ 4+2*x^2+4*x+1)^(1/2)+4*I*(-13+3*33^(1/2))^(1/3)*(x^4+2*x^2+4*x+1)^(1/2)/(4 *2^(2/3)*(-I+3^(1/2))-2*I*(-13+3*33^(1/2))^(1/3)+6*I*x*(-13+3*33^(1/2))^(1 /3)+2^(1/3)*(3^(1/2)+I)*(-13+3*33^(1/2))^(2/3))-8*2^(2/3)*EllipticE((26-6* 33^(1/2)+6*x*(-13+3*33^(1/2))+(-13-13*I*3^(1/2)+9*I*11^(1/2)+3*33^(1/2))*( -26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-26+6*33^(1/2))^(2/3))^(1/2)/((39+1 3*I*3^(1/2)-9*I*11^(1/2)-9*33^(1/2)+4*(3-I*3^(1/2))*(-26+6*33^(1/2))^(1/3) )/(39-13*I*3^(1/2)+9*I*11^(1/2)-9*33^(1/2)+4*(3+I*3^(1/2))*(-26+6*33^(1/2) )^(1/3)))^(1/2)/(26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-13+13*I*3^(1/2)-9*I* 11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*33^(1/ 2))^(2/3))^(1/2),((84+28*I*3^(1/2)-12*I*11^(1/2)-12*33^(1/2)+(3-I*3^(1/2)- 3*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3))/(84-28*I*3^(1/2)+12*I*11^ (1/2)-12*33^(1/2)+(3+I*3^(1/2)+3*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^( 1/3)))^(1/2))*(x^4+2*x^2+4*x+1)^(1/2)*3^(1/2)/(-13+3*33^(1/2)+4*(-26+6*33^ (1/2))^(1/3))^(1/2)*(I*(-19899+x*(59697-10335*33^(1/2))+3445*33^(1/2)+(-26 +6*33^(1/2))^(2/3)*(-2574+466*33^(1/2))+(-26+6*33^(1/2))^(1/3)*(-19899+344 5*33^(1/2)))/(-39-13*I*3^(1/2)+9*I*11^(1/2)+9*33^(1/2)+4*I*(3*I+3^(1/2))*( -26+6*33^(1/2))^(1/3))/(26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-13+13*I*3^(1/ 2)-9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6 *33^(1/2))^(2/3)))^(1/2)/(4*2^(2/3)-(-13+3*33^(1/2))^(1/3)+3*x*(-13+3*3...
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 13.55 (sec) , antiderivative size = 3168, normalized size of antiderivative = 0.79 \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\text {Result too large to show} \]
(Sqrt[9 - 4*Sqrt[2]]*x^2)/2 - (Sqrt[2]*x*Sqrt[1 + 4*x + 2*x^2 + x^4])/3 - (2*Sqrt[2]*((6*(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])^2*(-(EllipticF[ ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3 *#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]) *(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*# 1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*( Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]) + EllipticPi[(1 + Root[1 + 3 *#1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Roo t[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - R oot[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^ 2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Roo t[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*(1 + Root[1 + 3*#1 - #1^2 + #1 ^3 & , 1, 0]))*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])/((x - Root [1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]))]*(-1 - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])*Sqrt[(x - Root[1 + ...
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.
\(\displaystyle \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {x^4+2 x^2+4 x+1}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \sqrt {9-4 \sqrt {2}} x^2-\sqrt {2} \int \sqrt {x^4+2 x^2+4 x+1}dx\) |
3.3.81.3.1 Defintions of rubi rules used
Time = 2.31 (sec) , antiderivative size = 4640, normalized size of antiderivative = 1.15
method | result | size |
default | \(\text {Expression too large to display}\) | \(4640\) |
parts | \(\text {Expression too large to display}\) | \(4640\) |
elliptic | \(\text {Expression too large to display}\) | \(4646\) |
1/2*x^2*(-1+2*2^(1/2))-2^(1/2)*(1/3*x*(x^4+2*x^2+4*x+1)^(1/2)+4/3*(-1/6*(2 6+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-4/3+1/2*I*3^(1/2)*(-1/3*(26+ 6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*((1/2*(26+6*33^(1/2))^(1/3)- 4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+ 6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^( 1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/ 3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*(x +1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)^2*((-1/3*(26+6*3 3^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3) +4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8 /3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^ (1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1 /3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*( (-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33 ^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^( 1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26 +6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6 *33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)- 1/3))^(1/2)/(1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/ 2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/3*(26+6*...
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int { x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} \,d x } \]
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int \left (x \left (-1 + 2 \sqrt {2}\right ) - \sqrt {2} \sqrt {x^{4} + 2 x^{2} + 4 x + 1}\right )\, dx \]
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int { x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} \,d x } \]
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int { x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} \,d x } \]
Timed out. \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int x\,\left (2\,\sqrt {2}-1\right )-\sqrt {2}\,\sqrt {x^4+2\,x^2+4\,x+1} \,d x \]