Integrand size = 23, antiderivative size = 156 \[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {-1-x^3}}\right )}{3^{3/4}}+\frac {2 \sqrt {\frac {7}{6}-\frac {2}{\sqrt {3}}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
-1/3*arctanh((1+x)*(3+2*3^(1/2))^(1/2)/(-x^3-1)^(1/2))*2^(1/2)*3^(1/4)+2/3 *(1+x)*EllipticF((1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))*(1/3*6^(1/2)-1 /2*2^(1/2))*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*3^(3/4)/(-x^3-1)^(1/2)/((-1- x)/(1+x-3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.38 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.35 \[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {\left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {2 i \left (1+\sqrt {3}\right ) \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{3+(2+i) \sqrt {3}}\right )}{\sqrt {-1-x^3}} \]
(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/( 1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((2*I)*(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 = 0.45 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2566, 27, 760, 2565, 219}
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.
| \(\displaystyle \int \frac {x}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2566 |
| \(\displaystyle \frac {\left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {-x^3-1}}dx}{3-\sqrt {3}}-\frac {\int -\frac {6 \left (x-\sqrt {3}+1\right )}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}}dx}{6 \left (3-\sqrt {3}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {-x^3-1}}dx}{3-\sqrt {3}}+\frac {\int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}}dx}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {\int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}}dx}{3-\sqrt {3}}+\frac {2 \left (2-\sqrt {3}\right )^{3/2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle \frac {2 \left (2-\sqrt {3}\right )^{3/2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2 \int \frac {1}{1-\frac {\left (3+2 \sqrt {3}\right ) (x+1)^2}{-x^3-1}}d\frac {x+1}{\sqrt {-x^3-1}}}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (2-\sqrt {3}\right )^{3/2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {-x^3-1}}\right )}{\left (3-\sqrt {3}\right ) \sqrt {3+2 \sqrt {3}}}\) |
(-2*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[-1 - x^3]])/((3 - Sqrt[3])* Sqrt[3 + 2*Sqrt[3]]) + (2*(2 - Sqrt[3])^(3/2)*(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])
3.2.38.3.1 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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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 ] && NegQ[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]
Time = 2.37 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.62
| method | result | size |
| default | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (-1-\sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(253\) |
| elliptic | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (-1-\sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(253\) |
-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 = 217, normalized size of antiderivative = 1.39 \[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=-\frac {1}{3} \, {\left (-i \, \sqrt {3} + 3 i\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) + \frac {1}{12} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (\frac {x^{8} - 16 \, x^{7} + 112 \, x^{6} - 16 \, x^{5} + 112 \, x^{4} + 224 \, x^{3} + 2 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (x^{6} - 18 \, x^{5} + 12 \, x^{4} - 40 \, x^{3} - 36 \, x^{2} + \sqrt {3} {\left (x^{6} - 6 \, x^{5} + 24 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + 24 \, x + 16\right )} - 24 \, x - 32\right )} \sqrt {-x^{3} - 1} + 64 \, x^{2} - 16 \, \sqrt {3} {\left (x^{7} - 2 \, x^{6} + 6 \, x^{5} + 5 \, x^{4} + 2 \, x^{3} + 6 \, x^{2} + 4 \, x + 4\right )} + 128 \, x + 112}{x^{8} + 8 \, x^{7} + 16 \, x^{6} - 16 \, x^{5} - 56 \, x^{4} + 32 \, x^{3} + 64 \, x^{2} - 64 \, x + 16}\right ) \]
-1/3*(-I*sqrt(3) + 3*I)*weierstrassPInverse(0, -4, x) + 1/12*3^(1/4)*sqrt( 2)*log((x^8 - 16*x^7 + 112*x^6 - 16*x^5 + 112*x^4 + 224*x^3 + 2*3^(1/4)*sq rt(2)*(x^6 - 18*x^5 + 12*x^4 - 40*x^3 - 36*x^2 + sqrt(3)*(x^6 - 6*x^5 + 24 *x^4 + 8*x^3 + 12*x^2 + 24*x + 16) - 24*x - 32)*sqrt(-x^3 - 1) + 64*x^2 - 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))
\[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {x}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt {3}\right )}\, dx \]
\[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {x}{\sqrt {-x^{3} - 1} {\left (x + \sqrt {3} + 1\right )}} \,d x } \]
Exception generated. \[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[2]%%%} / %%%{%%{[2,4]:[1,0,-3]%%},[2]%%%} Error: Bad Ar gument Va
Timed out. \[ \int \frac {x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\text {Hanged} \]