Integrand size = 25, antiderivative size = 214 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}} \]
-1/12*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^(1/4)*2^(1/2)+1/18*arctan(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)+1/18*arctanh(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)+1/36*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^(3/4)*2^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.32 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\frac {x^2 \sqrt {1-x^3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},x^3,-\frac {x^3}{-10+6 \sqrt {3}}\right )}{4 \left (-5+3 \sqrt {3}\right ) \sqrt {-1+x^3}} \]
(x^2*Sqrt[1 - x^3]*AppellF1[2/3, 1/2, 1, 5/3, x^3, -(x^3/(-10 + 6*Sqrt[3]) )])/(4*(-5 + 3*Sqrt[3])*Sqrt[-1 + x^3])
Time = 0.22 (sec) , antiderivative size = 214, 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 |
| \(\displaystyle -\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (2 x-\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3-1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}\) |
-1/2*((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[- 1 + x^3])])/(Sqrt[2]*3^(3/4)) + ((2 + Sqrt[3])*ArcTan[((1 + Sqrt[3])*Sqrt[ -1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4)) + ((2 + Sqrt[3])*ArcTan h[(3^(1/4)*(1 + Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(6*Sqrt[2]*3^ (1/4)) + ((2 + Sqrt[3])*ArcTanh[(3^(1/4)*(1 - Sqrt[3] + 2*x))/(Sqrt[2]*Sqr t[-1 + x^3])])/(3*Sqrt[2]*3^(1/4))
3.1.89.3.1 Defintions of rubi rules used
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 = 41.40 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(\frac {2 \left (\sqrt {3}-1\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (6 \sqrt {3}-12\right ) \sqrt {x^{3}-1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (1-\sqrt {3}\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha \sqrt {3}-\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (1+2 \underline {\hspace {1.25 ex}}\alpha +\underline {\hspace {1.25 ex}}\alpha \sqrt {3}\right ) \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{2}+\underline {\hspace {1.25 ex}}\alpha +\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-\sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha +1\right ) \sqrt {x^{3}-1}}\right )}{18}\) | \(352\) |
| 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 {2 \left (1-\sqrt {3}\right )^{2} \sqrt {3}}{9}+\frac {\left (1-\sqrt {3}\right )^{2}}{3}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {3}}{9}+\frac {2}{3}-\frac {\sqrt {3}}{9}\right ) \Pi \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}{i \sqrt {3}+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 ) \Pi \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}\) | \(505\) |
| trager | \(\text {Expression too large to display}\) | \(4174\) |
2/3*(3^(1/2)-1)/(6*3^(1/2)-12)*(-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)/(-3^(1/2)+2*_alpha+1)*(-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),1/3*I*_alpha*3^(1/2)+1/2*I*_alpha+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. 1538 vs. \(2 (148) = 296\).
Time = 0.39 (sec) , antiderivative size = 1538, normalized size of antiderivative = 7.19 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\text {Too large to display} \]
-1/36*sqrt(14*sqrt(3) + 24)*arctan(-1/12*(3*x^2 - sqrt(3)*(x^2 + 10*x - 8) + 18*x - 12)*sqrt(14*sqrt(3) + 24)/sqrt(x^3 - 1)) - 1/72*sqrt(7*sqrt(3) + 3*sqrt(-56*sqrt(3) - 97) + 12)*log((x^8 + x^7 - 11*x^6 + 16*x^5 - 20*x^4 - 32*x^3 - 44*x^2 + (5*x^6 + 6*x^5 - 33*x^4 + 44*x^3 - 42*x^2 - sqrt(3)*(3 *x^6 + 4*x^5 - 17*x^4 + 28*x^3 - 22*x^2 + 8*x - 4) + (336*x^5 - 33*x^4 - 1 32*x^3 + 474*x^2 - sqrt(3)*(194*x^5 - 19*x^4 - 76*x^3 + 274*x^2 - 152*x + 76) - 264*x + 132)*sqrt(-56*sqrt(3) - 97) + 24*x - 4)*sqrt(x^3 - 1)*sqrt(7 *sqrt(3) + 3*sqrt(-56*sqrt(3) - 97) + 12) + 2*sqrt(3)*(x^7 + 8*x^6 + 7*x^4 - 16*x^3 - 8*x + 8) + 3*(26*x^7 - 12*x^6 - 48*x^5 + 98*x^4 - 96*x^3 + 48* x^2 - sqrt(3)*(15*x^7 - 7*x^6 - 28*x^5 + 56*x^4 - 56*x^3 + 28*x^2 - 8*x) - 16*x)*sqrt(-56*sqrt(3) - 97) - 8*x + 16)/(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*sqrt(3) + 3*sqrt(-56 *sqrt(3) - 97) + 12)*log((x^8 + x^7 - 11*x^6 + 16*x^5 - 20*x^4 - 32*x^3 - 44*x^2 - (5*x^6 + 6*x^5 - 33*x^4 + 44*x^3 - 42*x^2 - sqrt(3)*(3*x^6 + 4*x^ 5 - 17*x^4 + 28*x^3 - 22*x^2 + 8*x - 4) + (336*x^5 - 33*x^4 - 132*x^3 + 47 4*x^2 - sqrt(3)*(194*x^5 - 19*x^4 - 76*x^3 + 274*x^2 - 152*x + 76) - 264*x + 132)*sqrt(-56*sqrt(3) - 97) + 24*x - 4)*sqrt(x^3 - 1)*sqrt(7*sqrt(3) + 3*sqrt(-56*sqrt(3) - 97) + 12) + 2*sqrt(3)*(x^7 + 8*x^6 + 7*x^4 - 16*x^3 - 8*x + 8) + 3*(26*x^7 - 12*x^6 - 48*x^5 + 98*x^4 - 96*x^3 + 48*x^2 - sqrt( 3)*(15*x^7 - 7*x^6 - 28*x^5 + 56*x^4 - 56*x^3 + 28*x^2 - 8*x) - 16*x)*s...
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} - 10 + 6 \sqrt {3}\right )}\, dx \]
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}} \,d x } \]
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}} \,d x } \]
Timed out. \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {x^3-1}\,\left (x^3+6\,\sqrt {3}-10\right )} \,d x \]
\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=-6 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}-1}\, x}{x^{9}-21 x^{6}+12 x^{3}+8}d x \right )+\int \frac {\sqrt {x^{3}-1}\, x^{4}}{x^{9}-21 x^{6}+12 x^{3}+8}d x -10 \left (\int \frac {\sqrt {x^{3}-1}\, x}{x^{9}-21 x^{6}+12 x^{3}+8}d x \right ) \]