Integrand size = 25, antiderivative size = 222 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}} \] Output:
1/36*arctan(1/2*3^(1/4)*(1-x)*(1-3^(1/2))*2^(1/2)/(x^3-1)^(1/2))*(2-3^(1/2 ))*3^(3/4)*2^(1/2)+1/18*arctan(1/2*3^(1/4)*(1+2*x+3^(1/2))*2^(1/2)/(x^3-1) ^(1/2))*(2-3^(1/2))*3^(3/4)*2^(1/2)+1/12*arctanh(1/2*3^(1/4)*(1-x)*(1+3^(1 /2))*2^(1/2)/(x^3-1)^(1/2))*(2-3^(1/2))*3^(1/4)*2^(1/2)-1/18*arctanh(1/6*( 1-3^(1/2))*(x^3-1)^(1/2)*3^(1/4)*2^(1/2))*(2-3^(1/2))*3^(1/4)*2^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.29 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=-\frac {x^2 \sqrt {1-x^3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},x^3,\frac {x^3}{10+6 \sqrt {3}}\right )}{\left (20+12 \sqrt {3}\right ) \sqrt {-1+x^3}} \] Input:
Integrate[x/(Sqrt[-1 + x^3]*(-10 - 6*Sqrt[3] + x^3)),x]
Output:
-((x^2*Sqrt[1 - x^3]*AppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/(10 + 6*Sqrt[3])] )/((20 + 12*Sqrt[3])*Sqrt[-1 + x^3]))
Time = 0.22 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {990}
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.
| \(\displaystyle \int \frac {x}{\sqrt {x^3-1} \left (x^3-6 \sqrt {3}-10\right )} \, dx\) |
| \(\Big \downarrow \) 990 |
| \(\displaystyle \frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (2 x+\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3-1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}\) |
Input:
Int[x/(Sqrt[-1 + x^3]*(-10 - 6*Sqrt[3] + x^3)),x]
Output:
((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x ^3])])/(6*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3] + 2*x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(3*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*Arc Tanh[(3^(1/4)*(1 + Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(2*Sqrt[2] *3^(3/4)) - ((2 - Sqrt[3])*ArcTanh[((1 - Sqrt[3])*Sqrt[-1 + x^3])/(Sqrt[2] *3^(3/4))])/(3*Sqrt[2]*3^(3/4))
Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wi th[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, Simp[q*(2 - r)*( ArcTanh[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[-a, 2]*r^(3/2)))]/(3*Sqrt[2]*R t[-a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTanh[Rt[-a, 2]*Sqrt[r]*(1 + r)*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(2*Sqrt[2]*Rt[-a, 2]*d*r^(3/2))) , x] - Simp[q*(2 - r)*(ArcTan[Rt[-a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sqrt[2]*S qrt[a + b*x^3]))]/(3*Sqrt[2]*Rt[-a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(Ar cTan[Rt[-a, 2]*(1 - r)*Sqrt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sq rt[2]*Rt[-a, 2]*d*Sqrt[r])), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a* d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 61.05 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(-\frac {\left (1+\sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \left (2+\sqrt {3}\right ) \sqrt {x^{3}-1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha \sqrt {3}+\underline {\hspace {1.25 ex}}\alpha +2\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \left (1+2 \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha \sqrt {3}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{2}+\underline {\hspace {1.25 ex}}\alpha +\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-1-2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}-1}}\right )}{18}\) | \(347\) |
| elliptic | \(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {\left (1+\sqrt {3}\right )^{2}}{3}-\frac {2 \left (1+\sqrt {3}\right )^{2} \sqrt {3}}{9}+\frac {2}{3}+\frac {\sqrt {3}}{9}-\frac {2 \left (1+\sqrt {3}\right ) \sqrt {3}}{9}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1+\sqrt {3}\right )^{2} \sqrt {3}}{6}-\frac {i \left (1+\sqrt {3}\right )^{2}}{3}-\frac {\left (1+\sqrt {3}\right )^{2} \sqrt {3}}{3}+\frac {\left (1+\sqrt {3}\right )^{2}}{2}+\frac {i \left (1+\sqrt {3}\right ) \sqrt {3}}{6}-\frac {i \left (1+\sqrt {3}\right )}{3}-\frac {\left (1+\sqrt {3}\right ) \sqrt {3}}{3}+1-\frac {i}{3}+\frac {\sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (1+\sqrt {3}\right ) \sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (1+\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{2}+3 \underline {\hspace {1.25 ex}}\alpha +3+2 \sqrt {3}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right )\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}-\frac {i \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \sqrt {3}}{6}+\frac {1}{2}-\frac {i}{3}-\frac {\sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \sqrt {x^{3}-1}}\right )}{27}\) | \(485\) |
| trager | \(\text {Expression too large to display}\) | \(4181\) |
Input:
int(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/9*(1+3^(1/2))/(2+3^(1/2))*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1 /2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2* I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*3^(1/2)*EllipticPi(((- 1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),-1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/2+1/ 2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-1/18*2^(1/2)*sum((-_alpha*3^(1/2) +_alpha+2)/(-1-2*_alpha-3^(1/2))*(-3-I*3^(1/2))*((-1+x)/(-3-I*3^(1/2)))^(1 /2)*((-I*3^(1/2)+2*x+1)/(3-I*3^(1/2)))^(1/2)*((I*3^(1/2)+2*x+1)/(I*3^(1/2) +3))^(1/2)/(x^3-1)^(1/2)*(1+2*_alpha-_alpha*3^(1/2))*EllipticPi(((-1+x)/(- 3/2-1/2*I*3^(1/2)))^(1/2),-1/2*I*_alpha+1/3*I*_alpha*3^(1/2)-1/2*_alpha*3^ (1/2)+_alpha+1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^( 1/2)),_alpha=RootOf(_Z^2+(1+3^(1/2))*_Z+2*3^(1/2)+4))
Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (146) = 292\).
Time = 0.21 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.97 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx =\text {Too large to display} \] Input:
integrate(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="fricas")
Output:
-1/6*sqrt(7/6*sqrt(3) - 2)*arctan(1/3*sqrt(x^3 - 1)*(sqrt(3)*(x - 10) + 3* x - 18)*sqrt(7/6*sqrt(3) - 2)/(x^2 - x)) - 1/72*sqrt(7/2*sqrt(3) - 6)*log( (x^8 + 10*x^7 + 22*x^6 - 20*x^5 - 20*x^4 - 80*x^3 - 8*x^2 + 2*(5*x^6 + 30* x^5 + 6*x^4 - 28*x^3 + 60*x^2 + 3*sqrt(3)*(x^6 + 6*x^5 + 2*x^4 - 4*x^3 + 1 2*x^2 - 8*x) - 24*x + 32)*sqrt(x^3 - 1)*sqrt(7/2*sqrt(3) - 6) + 4*sqrt(3)* (x^7 + 2*x^6 + 7*x^4 + 14*x^3 - 8*x - 16) + 64*x + 112)/(x^8 + 4*x^7 + 16* x^6 + 16*x^5 + 28*x^4 - 32*x^3 + 64*x^2 - 32*x + 16)) + 1/72*sqrt(7/2*sqrt (3) - 6)*log((x^8 + 10*x^7 + 22*x^6 - 20*x^5 - 20*x^4 - 80*x^3 - 8*x^2 - 2 *(5*x^6 + 30*x^5 + 6*x^4 - 28*x^3 + 60*x^2 + 3*sqrt(3)*(x^6 + 6*x^5 + 2*x^ 4 - 4*x^3 + 12*x^2 - 8*x) - 24*x + 32)*sqrt(x^3 - 1)*sqrt(7/2*sqrt(3) - 6) + 4*sqrt(3)*(x^7 + 2*x^6 + 7*x^4 + 14*x^3 - 8*x - 16) + 64*x + 112)/(x^8 + 4*x^7 + 16*x^6 + 16*x^5 + 28*x^4 - 32*x^3 + 64*x^2 - 32*x + 16)) + 1/72* sqrt(14*sqrt(3) - 24)*log((x^8 + 16*x^7 + 112*x^6 + 16*x^5 + 112*x^4 - 224 *x^3 + 64*x^2 - 2*(5*x^6 + 54*x^5 + 96*x^4 + 56*x^3 - 36*x^2 + 3*sqrt(3)*( x^6 + 10*x^5 + 20*x^4 + 8*x^3 - 4*x^2 - 8*x) - 24*x - 16)*sqrt(x^3 - 1)*sq rt(14*sqrt(3) - 24) + 16*sqrt(3)*(x^7 + 2*x^6 + 6*x^5 - 5*x^4 + 2*x^3 - 6* x^2 + 4*x - 4) - 128*x + 112)/(x^8 - 8*x^7 + 16*x^6 + 16*x^5 - 56*x^4 - 32 *x^3 + 64*x^2 + 64*x + 16))
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} - 6 \sqrt {3} - 10\right )}\, dx \] Input:
integrate(x/(-10+x**3-6*3**(1/2))/(x**3-1)**(1/2),x)
Output:
Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x**3 - 6*sqrt(3) - 10)), x)
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} - 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}} \,d x } \] Input:
integrate(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} - 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}} \,d x } \] Input:
integrate(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)
Timed out. \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=\int -\frac {x}{\sqrt {x^3-1}\,\left (-x^3+6\,\sqrt {3}+10\right )} \,d x \] Input:
int(-x/((x^3 - 1)^(1/2)*(6*3^(1/2) - x^3 + 10)),x)
Output:
int(-x/((x^3 - 1)^(1/2)*(6*3^(1/2) - x^3 + 10)), x)
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10-6 \sqrt {3}+x^3\right )} \, dx=6 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}-1}\, x}{x^{9}-21 x^{6}+12 x^{3}+8}d x \right )+\int \frac {\sqrt {x^{3}-1}\, x^{4}}{x^{9}-21 x^{6}+12 x^{3}+8}d x -10 \left (\int \frac {\sqrt {x^{3}-1}\, x}{x^{9}-21 x^{6}+12 x^{3}+8}d x \right ) \] Input:
int(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x)
Output:
6*sqrt(3)*int((sqrt(x**3 - 1)*x)/(x**9 - 21*x**6 + 12*x**3 + 8),x) + int(( sqrt(x**3 - 1)*x**4)/(x**9 - 21*x**6 + 12*x**3 + 8),x) - 10*int((sqrt(x**3 - 1)*x)/(x**9 - 21*x**6 + 12*x**3 + 8),x)