Integrand size = 25, antiderivative size = 152 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{3^{3/4}}+\frac {\sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]
-1/3*arctan((1-x)*(3+2*3^(1/2))^(1/2)/(-x^3+1)^(1/2))*2^(1/2)*3^(1/4)+1/3* (1-x)*EllipticF((1-x-3^(1/2))/(1-x+3^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*((x^2+x +1)/(1-x+3^(1/2))^2)^(1/2)*3^(1/4)/(-x^3+1)^(1/2)/((1-x)/(1-x+3^(1/2))^2)^ (1/2)
Result contains complex when optimal does not.
Time = 10.54 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.53 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 i \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {i \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \left (3 i+(1+2 i) \sqrt {3}+\left (3+(2+i) \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+2 \left (1+\sqrt {3}\right ) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {1-x^3}} \]
((2*I)*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((I*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x) /(1 + (-1)^(1/3))]*(3*I + (1 + 2*I)*Sqrt[3] + (3 + (2 + I)*Sqrt[3])*x)*Ell ipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqr t[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + 2*(1 + Sqrt[3])*Sqrt[1 + x + x^2] *EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^( 2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((3 + (2 + I)*Sqrt[3])*Sqrt[1 - x^3])
Time = 0.49 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2566, 27, 759, 2565, 216}
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.
| \(\displaystyle \int \frac {x}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}} \, dx\) |
| \(\Big \downarrow \) 2566 |
| \(\displaystyle \frac {\int -\frac {6 \left (-x-\sqrt {3}+1\right )}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx}{6 \left (3-\sqrt {3}\right )}-\frac {\left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^3}}dx}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^3}}dx}{3-\sqrt {3}}-\frac {\int \frac {-x-\sqrt {3}+1}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {2 \left (2-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {\int \frac {-x-\sqrt {3}+1}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle \frac {2 \left (2-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \int \frac {1}{\frac {\left (3+2 \sqrt {3}\right ) (1-x)^2}{1-x^3}+1}d\frac {1-x}{\sqrt {1-x^3}}}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \left (2-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\left (3-\sqrt {3}\right ) \sqrt {3+2 \sqrt {3}}}\) |
(-2*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt[1 - x^3]])/((3 - Sqrt[3])*Sq rt[3 + 2*Sqrt[3]]) + (2*(2 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt [3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(3 - Sqrt[3])*Sqrt[(1 - x)/(1 + Sqrt[ 3] - x)^2]*Sqrt[1 - x^3])
3.2.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d) S ubst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*(x/c))/Sqrt[a + b*x ^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c ^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^ 3), 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d ^3)) Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/(c*d*(b*c^3 - 28*a* d^3)) Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3) , 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (124 ) = 248\).
Time = 2.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-1-\sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(257\) |
| elliptic | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-1-\sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(257\) |
2/3*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I*3 ^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*Ell ipticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/ 2+1/2*I*3^(1/2)))^(1/2))-2/3*I*(-1-3^(1/2))*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2 ))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1 /2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)/(-3/2-3^(1/2)+1/2*I*3^(1/2))*EllipticPi (1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-3/2-3^(1/ 2)+1/2*I*3^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {1}{3} \, {\left (i \, \sqrt {3} - 3 i\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{6} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (-\frac {3^{\frac {1}{4}} \sqrt {2} \sqrt {-x^{3} + 1} {\left (3 \, x^{2} - \sqrt {3} {\left (x^{2} - 2 \, x + 4\right )} + 6 \, x\right )}}{12 \, {\left (x^{3} - 1\right )}}\right ) \]
-1/3*(I*sqrt(3) - 3*I)*weierstrassPInverse(0, 4, x) + 1/6*3^(1/4)*sqrt(2)* arctan(-1/12*3^(1/4)*sqrt(2)*sqrt(-x^3 + 1)*(3*x^2 - sqrt(3)*(x^2 - 2*x + 4) + 6*x)/(x^3 - 1))
\[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=- \int \frac {x}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx \]
\[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {x}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}} \,d x } \]
\[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {x}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Hanged} \]