Integrand size = 40, antiderivative size = 4030 \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx =\text {Too large to display} \] Output:
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 = 10.04 (sec) , antiderivative size = 3168, normalized size of antiderivative = 0.79 \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Sqrt[9 - 4*Sqrt[2]]*x - Sqrt[2]*Sqrt[1 + 4*x + 2*x^2 + x^4],x]
Output:
(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 |
| \(\displaystyle \frac {1}{2} \sqrt {9-4 \sqrt {2}} x^2-\sqrt {2} \int \sqrt {x^4+2 x^2+4 x+1}dx\) |
Input:
Int[Sqrt[9 - 4*Sqrt[2]]*x - Sqrt[2]*Sqrt[1 + 4*x + 2*x^2 + x^4],x]
Output:
$Aborted
Time = 1.82 (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 | \(\text {Expression too large to display}\) | \(4646\) |
Input:
int(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x,method=_RETURNVERB OSE)
Output:
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*...
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int { x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} \,d x } \] Input:
integrate(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x, algorithm=" fricas")
Output:
integral(2*sqrt(2)*x - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1) - x, x)
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int \left (x \left (-1 + 2 \sqrt {2}\right ) - \sqrt {2} \sqrt {x^{4} + 2 x^{2} + 4 x + 1}\right )\, dx \] Input:
integrate(-2**(1/2)*(x**4+2*x**2+4*x+1)**(1/2)+x*(-1+2*2**(1/2)),x)
Output:
Integral(x*(-1 + 2*sqrt(2)) - sqrt(2)*sqrt(x**4 + 2*x**2 + 4*x + 1), x)
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int { x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} \,d x } \] Input:
integrate(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x, algorithm=" maxima")
Output:
1/2*x^2*(2*sqrt(2) - 1) - sqrt(2)*integrate(sqrt(x^3 - x^2 + 3*x + 1)*sqrt (x + 1), x)
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int { x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} \,d x } \] Input:
integrate(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x, algorithm=" giac")
Output:
integrate(x*(2*sqrt(2) - 1) - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1), x)
Timed out. \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx=\int x\,\left (2\,\sqrt {2}-1\right )-\sqrt {2}\,\sqrt {x^4+2\,x^2+4\,x+1} \,d x \] Input:
int(x*(2*2^(1/2) - 1) - 2^(1/2)*(4*x + 2*x^2 + x^4 + 1)^(1/2),x)
Output:
int(x*(2*2^(1/2) - 1) - 2^(1/2)*(4*x + 2*x^2 + x^4 + 1)^(1/2), x)
\[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx =\text {Too large to display} \] Input:
int(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x)
Output:
( - 30*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2)*x - 5*sqrt(x**4 + 2*x**2 + 4* x + 1)*sqrt(2) - 14*sqrt(2)*int(sqrt(x**4 + 2*x**2 + 4*x + 1)/(x**4 + 2*x* *2 + 4*x + 1),x) + 10*sqrt(2)*int((sqrt(x**4 + 2*x**2 + 4*x + 1)*x**3)/(x* *4 + 2*x**2 + 4*x + 1),x) - 60*sqrt(2)*int((sqrt(x**4 + 2*x**2 + 4*x + 1)* x**2)/(x**4 + 2*x**2 + 4*x + 1),x) + 10*sqrt(2)*int((sqrt(x**4 + 2*x**2 + 4*x + 1)*x)/(x**4 + 2*x**2 + 4*x + 1),x) - 48*sqrt(2)*log(x**2 - 2) + 105* sqrt(2)*log( - 46*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2) - 65*sqrt(x**4 + 2 *x**2 + 4*x + 1) - 62*sqrt(2)*x**2 - 32*sqrt(2)*x - 8*sqrt(2) - 87*x**2 - 44*x - 11) + 21*sqrt(2)*log( - 41*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2)*x - 58*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2) + 58*sqrt(x**4 + 2*x**2 + 4*x + 1)*x + 82*sqrt(x**4 + 2*x**2 + 4*x + 1) - 55*sqrt(2)*x**3 - 106*sqrt(2)*x **2 - 47*sqrt(2)*x - 10*sqrt(2) + 78*x**3 + 150*x**2 + 66*x + 14) + 6*sqrt (2)*log( - 4*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2) + 5*sqrt(x**4 + 2*x**2 + 4*x + 1) + 2*sqrt(2)*x**2 - 8*sqrt(2)*x - 8*sqrt(2) - 3*x**2 + 10*x + 11 ) + 3*sqrt(2)*log( - 3*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2) + 4*sqrt(x**4 + 2*x**2 + 4*x + 1) + 4*sqrt(2)*x**2 + 2*sqrt(2)*x - 6*x**2 - 3*x) - 90*s qrt(2)*log(sqrt(x**4 + 2*x**2 + 4*x + 1) + x**2) + 90*sqrt(2)*log(sqrt(x** 4 + 2*x**2 + 4*x + 1) - x**2) - 24*sqrt(2)*log(sqrt(2) + 2*x + 2) + 15*sqr t(2)*log(46*sqrt(x**4 + 2*x**2 + 4*x + 1)*sqrt(2) + 65*sqrt(x**4 + 2*x**2 + 4*x + 1) - 62*sqrt(2)*x**2 - 32*sqrt(2)*x - 8*sqrt(2) - 87*x**2 - 44*...