Integrand size = 31, antiderivative size = 346 \[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=-\frac {\left (c+d-\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \arctan \left (\frac {\sqrt {c^2-c d+d^2} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c+d-\sqrt {3} d\right )^2}{\left (c+d+\sqrt {3} d\right )^2},\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (c+d+\sqrt {3} d\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \] Output:
-(c+d-3^(1/2)*d)*(1-x)*((x^2+x+1)/(1-3^(1/2)-x)^2)^(1/2)*arctan((c^2-c*d+d ^2)^(1/2)*(-(1-x)/(1-3^(1/2)-x)^2)^(1/2)/d^(1/2)/(c+d)^(1/2)/((x^2+x+1)/(1 -3^(1/2)-x)^2)^(1/2))/d^(1/2)/(c+d)^(1/2)/(c^2-c*d+d^2)^(1/2)/(-(1-x)/(1-3 ^(1/2)-x)^2)^(1/2)/(-x^3+1)^(1/2)+4*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*(1-x )*((x^2+x+1)/(1-3^(1/2)-x)^2)^(1/2)*EllipticPi((1+3^(1/2)-x)/(1-3^(1/2)-x) ,(c+d-3^(1/2)*d)^2/(c+d+3^(1/2)*d)^2,2*I-I*3^(1/2))/(c+d+3^(1/2)*d)/(-(1-x )/(1-3^(1/2)-x)^2)^(1/2)/(-x^3+1)^(1/2)
Result contains complex when optimal does not.
Time = 10.83 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.68 \[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \left (\sqrt [3]{-1}+x\right ) \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3} c+\left (-3+\sqrt {3}\right ) d\right ) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{-c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt {1-x^3}} \] Input:
Integrate[(1 - Sqrt[3] - x)/((c + d*x)*Sqrt[1 - x^3]),x]
Output:
(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-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))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c + (-3 + Sqrt[3])*d)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1) ^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c - (-1)^(1/3)*d)))/(3*d*Sqrt[ 1 - x^3])
Time = 1.66 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2568, 2538, 412, 435, 104, 218}
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.
| \(\displaystyle \int \frac {-x-\sqrt {3}+1}{\sqrt {1-x^3} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2568 |
| \(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \int \frac {1}{\left (c+\sqrt {3} d+d-\frac {\left (c-\sqrt {3} d+d\right ) \left (-x+\sqrt {3}+1\right )}{-x-\sqrt {3}+1}\right ) \sqrt {1-\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}d\left (-\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2538 |
| \(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (\left (c+\sqrt {3} d+d\right ) \int \frac {1}{\sqrt {1-\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \left (\left (c+\sqrt {3} d+d\right )^2-\frac {\left (c-\sqrt {3} d+d\right )^2 \left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}\right )}d\left (-\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )-\left (c-\sqrt {3} d+d\right ) \int -\frac {-x+\sqrt {3}+1}{\sqrt {1-\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \left (\left (c+\sqrt {3} d+d\right )^2-\frac {\left (c-\sqrt {3} d+d\right )^2 \left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}\right ) \left (-x-\sqrt {3}+1\right )}d\left (-\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (-\left (c-\sqrt {3} d+d\right ) \int -\frac {-x+\sqrt {3}+1}{\sqrt {1-\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \left (\left (c+\sqrt {3} d+d\right )^2-\frac {\left (c-\sqrt {3} d+d\right )^2 \left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}\right ) \left (-x-\sqrt {3}+1\right )}d\left (-\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )-\frac {\operatorname {EllipticPi}\left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c+\sqrt {3} d+d\right )}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (-\frac {1}{2} \left (c-\sqrt {3} d+d\right ) \int \frac {1}{\sqrt {\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}+1} \left (\frac {\left (-x+\sqrt {3}+1\right ) \left (c-\sqrt {3} d+d\right )^2}{-x-\sqrt {3}+1}+\left (c+\sqrt {3} d+d\right )^2\right ) \sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}d\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c+\sqrt {3} d+d\right )}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (-\left (c-\sqrt {3} d+d\right ) \int \frac {1}{-4 \sqrt {3} d (c+d)-\frac {4 \left (2+\sqrt {3}\right ) \left (c^2-d c+d^2\right ) \sqrt {\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}}d\frac {\sqrt {\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}+1}}{\sqrt {\frac {\left (-x+\sqrt {3}+1\right )^2}{\left (-x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7}}-\frac {\operatorname {EllipticPi}\left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c+\sqrt {3} d+d\right )}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (-\frac {\operatorname {EllipticPi}\left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {7+4 \sqrt {3}} \left (c+\sqrt {3} d+d\right )}-\frac {\left (c-\sqrt {3} d+d\right ) \arctan \left (\frac {\sqrt {2+\sqrt {3}} \left (-x+\sqrt {3}+1\right ) \sqrt {c^2-c d+d^2}}{\sqrt [4]{3} \sqrt {d} \left (-x-\sqrt {3}+1\right ) \sqrt {c+d}}\right )}{4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2}}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}\) |
Input:
Int[(1 - Sqrt[3] - x)/((c + d*x)*Sqrt[1 - x^3]),x]
Output:
(-4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x) ^2]*(-1/4*((c + d - Sqrt[3]*d)*ArcTan[(Sqrt[2 + Sqrt[3]]*Sqrt[c^2 - c*d + d^2]*(1 + Sqrt[3] - x))/(3^(1/4)*Sqrt[d]*Sqrt[c + d]*(1 - Sqrt[3] - x))])/ (3^(1/4)*Sqrt[2 + Sqrt[3]]*Sqrt[d]*Sqrt[c + d]*Sqrt[c^2 - c*d + d^2]) - El lipticPi[(c + d - Sqrt[3]*d)^2/(c + d + Sqrt[3]*d)^2, ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]]/(Sqrt[7 + 4*Sqrt[3]]*(c + d + Sqr t[3]*d))))/(Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[1 - x^3])
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[a Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[b Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{q = Simplify[(-1 + Sqrt[3])*(f/e)]}, Simp[4*3^(1/4)*Sqrt[2 + Sqrt[3]]*f*(1 - q*x)*(Sqrt[(1 + q*x + q^2*x^2)/(1 - Sqrt[3] - q*x)^2]/(q *Sqrt[a + b*x^3]*Sqrt[-(1 - q*x)/(1 - Sqrt[3] - q*x)^2])) Subst[Int[1/((( 1 + Sqrt[3])*d + c*q + ((1 - Sqrt[3])*d + c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 + 4* Sqrt[3] + x^2]), x], x, (1 + Sqrt[3] - q*x)/(-1 + Sqrt[3] + q*x)], x]] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 - 3*Sq rt[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Time = 0.43 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}+1}}+\frac {2 i \left (\sqrt {3}\, d -c -d \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(268\) |
| elliptic | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}+1}}+\frac {2 i \left (\sqrt {3}\, d -c -d \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(268\) |
Input:
int((1-3^(1/2)-x)/(d*x+c)/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*I/d*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)*E llipticF(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*(3^(1/2)*d-c-d)/d^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)/(-1/2+1/2*I*3^(1/2)+c/d)*Ellip ticPi(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)+c/d),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))
Timed out. \[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=\text {Timed out} \] Input:
integrate((1-3^(1/2)-x)/(d*x+c)/(-x^3+1)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=- \int \frac {\sqrt {3}}{c \sqrt {1 - x^{3}} + d x \sqrt {1 - x^{3}}}\, dx - \int \frac {x}{c \sqrt {1 - x^{3}} + d x \sqrt {1 - x^{3}}}\, dx - \int \left (- \frac {1}{c \sqrt {1 - x^{3}} + d x \sqrt {1 - x^{3}}}\right )\, dx \] Input:
integrate((1-3**(1/2)-x)/(d*x+c)/(-x**3+1)**(1/2),x)
Output:
-Integral(sqrt(3)/(c*sqrt(1 - x**3) + d*x*sqrt(1 - x**3)), x) - Integral(x /(c*sqrt(1 - x**3) + d*x*sqrt(1 - x**3)), x) - Integral(-1/(c*sqrt(1 - x** 3) + d*x*sqrt(1 - x**3)), x)
\[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=\int { -\frac {x + \sqrt {3} - 1}{\sqrt {-x^{3} + 1} {\left (d x + c\right )}} \,d x } \] Input:
integrate((1-3^(1/2)-x)/(d*x+c)/(-x^3+1)^(1/2),x, algorithm="maxima")
Output:
-integrate((x + sqrt(3) - 1)/(sqrt(-x^3 + 1)*(d*x + c)), x)
\[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=\int { -\frac {x + \sqrt {3} - 1}{\sqrt {-x^{3} + 1} {\left (d x + c\right )}} \,d x } \] Input:
integrate((1-3^(1/2)-x)/(d*x+c)/(-x^3+1)^(1/2),x, algorithm="giac")
Output:
integrate(-(x + sqrt(3) - 1)/(sqrt(-x^3 + 1)*(d*x + c)), x)
Timed out. \[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=\text {Hanged} \] Input:
int(-(x + 3^(1/2) - 1)/((1 - x^3)^(1/2)*(c + d*x)),x)
Output:
\text{Hanged}
\[ \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {1-x^3}} \, dx=\sqrt {3}\, \left (\int \frac {\sqrt {-x^{3}+1}}{d \,x^{4}+c \,x^{3}-d x -c}d x \right )-\left (\int \frac {\sqrt {-x^{3}+1}}{d \,x^{4}+c \,x^{3}-d x -c}d x \right )+\int \frac {\sqrt {-x^{3}+1}\, x}{d \,x^{4}+c \,x^{3}-d x -c}d x \] Input:
int((1-3^(1/2)-x)/(d*x+c)/(-x^3+1)^(1/2),x)
Output:
sqrt(3)*int(sqrt( - x**3 + 1)/(c*x**3 - c + d*x**4 - d*x),x) - int(sqrt( - x**3 + 1)/(c*x**3 - c + d*x**4 - d*x),x) + int((sqrt( - x**3 + 1)*x)/(c*x **3 - c + d*x**4 - d*x),x)