Integrand size = 22, antiderivative size = 156 \[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=-\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:
-2/9*2^(2/3)*arctanh(3^(1/2)*(1+2^(1/3)*x)/(-x^3-1)^(1/2))*3^(1/2)+2/9*(1/ 2*6^(1/2)-1/2*2^(1/2))*(1+x)*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*EllipticF(( 1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))*3^(3/4)/(-(1+x)/(1+x-3^(1/2))^2) ^(1/2)/(-x^3-1)^(1/2)
Result contains complex when optimal does not.
Time = 10.48 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.34 \[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {\left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {i 2^{2/3} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3}}{\sqrt [3]{-1}+2^{2/3}},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt {-1-x^3}} \] Input:
Integrate[x/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
Output:
(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))]) + (I*2^(2/3)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/ 3)), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1) ^(1/3) + 2^(2/3))))/Sqrt[-1 - x^3]
Time = 0.75 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2564, 760, 2562, 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.
| \(\displaystyle \int \frac {x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\sqrt {-x^3-1}}dx-\frac {1}{3} \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {1}{3} \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 2562 |
| \(\displaystyle \frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2}{3} 2^{2/3} \int \frac {1}{1-\frac {3 \left (\sqrt [3]{2} x+1\right )^2}{-x^3-1}}d\frac {\sqrt [3]{2} x+1}{\sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {-x^3-1}}\right )}{3 \sqrt {3}}\) |
Input:
Int[x/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
Output:
(-2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3]])/(3*Sqrt[3]) + (2*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*El lipticF[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(3*3 ^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])
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] :> Simp[2*(e/d) Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*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*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[(2*d*e + c*f)/(3*c*d) Int[1/Sqrt[a + b*x^3], x], x] + Si mp[(d*e - c*f)/(3*c*d) Int[(c - 2*d*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*c^3 - 4*a* d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (123 ) = 246\).
Time = 2.37 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}\right )}\) | \(249\) |
| elliptic | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}\right )}\) | \(249\) |
Input:
int(x/(2^(2/3)+x)/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-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*2^(2/3)*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)/(1/2+1/2*I*3^(1/2)+2^(2/3))*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)/(1/2+1/2*I*3^(1/2)+ 2^(2/3)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (119) = 238\).
Time = 0.15 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.59 \[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {1}{6} \cdot 2^{\frac {1}{6}} \sqrt {\frac {2}{3}} \log \left (\frac {x^{18} - 1440 \, x^{15} + 17400 \, x^{12} + 21056 \, x^{9} - 10368 \, x^{6} - 15360 \, x^{3} + 6 \cdot 2^{\frac {1}{6}} \sqrt {\frac {2}{3}} {\left (126 \, x^{14} - 2664 \, x^{11} + 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac {2}{3}} {\left (x^{16} - 310 \, x^{13} + 2332 \, x^{10} + 2656 \, x^{7} - 256 \, x^{4} - 512 \, x\right )} - 2^{\frac {1}{3}} {\left (17 \, x^{15} - 1058 \, x^{12} + 2528 \, x^{9} + 5408 \, x^{6} + 2560 \, x^{3} + 512\right )}\right )} \sqrt {-x^{3} - 1} - 24 \cdot 2^{\frac {2}{3}} {\left (x^{17} - 121 \, x^{14} + 478 \, x^{11} + 1144 \, x^{8} + 608 \, x^{5} + 64 \, x^{2}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} - 176 \, x^{13} + 83 \, x^{10} + 680 \, x^{7} + 544 \, x^{4} + 128 \, x\right )} - 2048}{x^{18} + 24 \, x^{15} + 240 \, x^{12} + 1280 \, x^{9} + 3840 \, x^{6} + 6144 \, x^{3} + 4096}\right ) - \frac {2}{3} i \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:
integrate(x/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
1/6*2^(1/6)*sqrt(2/3)*log((x^18 - 1440*x^15 + 17400*x^12 + 21056*x^9 - 103 68*x^6 - 15360*x^3 + 6*2^(1/6)*sqrt(2/3)*(126*x^14 - 2664*x^11 + 4608*x^5 + 2304*x^2 + 2^(2/3)*(x^16 - 310*x^13 + 2332*x^10 + 2656*x^7 - 256*x^4 - 5 12*x) - 2^(1/3)*(17*x^15 - 1058*x^12 + 2528*x^9 + 5408*x^6 + 2560*x^3 + 51 2))*sqrt(-x^3 - 1) - 24*2^(2/3)*(x^17 - 121*x^14 + 478*x^11 + 1144*x^8 + 6 08*x^5 + 64*x^2) + 48*2^(1/3)*(5*x^16 - 176*x^13 + 83*x^10 + 680*x^7 + 544 *x^4 + 128*x) - 2048)/(x^18 + 24*x^15 + 240*x^12 + 1280*x^9 + 3840*x^6 + 6 144*x^3 + 4096)) - 2/3*I*weierstrassPInverse(0, -4, x)
\[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {x}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac {2}{3}}\right )}\, dx \] Input:
integrate(x/(2**(2/3)+x)/(-x**3-1)**(1/2),x)
Output:
Integral(x/(sqrt(-(x + 1)*(x**2 - x + 1))*(x + 2**(2/3))), x)
\[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {x}{\sqrt {-x^{3} - 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate(x/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate(x/(sqrt(-x^3 - 1)*(x + 2^(2/3))), x)
\[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {x}{\sqrt {-x^{3} - 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate(x/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(x/(sqrt(-x^3 - 1)*(x + 2^(2/3))), x)
Timed out. \[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {x}{\sqrt {-x^3-1}\,\left (x+2^{2/3}\right )} \,d x \] Input:
int(x/((- x^3 - 1)^(1/2)*(x + 2^(2/3))),x)
Output:
int(x/((- x^3 - 1)^(1/2)*(x + 2^(2/3))), x)
\[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=-\left (\int \frac {x}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \right ) i \] Input:
int(x/(2^(2/3)+x)/(-x^3-1)^(1/2),x)
Output:
- int(x/(sqrt(x**3 + 1)*2**(2/3) + sqrt(x**3 + 1)*x),x)*i