Integrand size = 24, antiderivative size = 150 \[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=\frac {2}{9} (e+2 f) \arctan \left (\frac {(1+x)^2}{3 \sqrt {-1-x^3}}\right )+\frac {2 \sqrt {2-\sqrt {3}} (e-f) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:
2/9*(e+2*f)*arctan(1/3*(1+x)^2/(-x^3-1)^(1/2))+2/9*(1/2*6^(1/2)-1/2*2^(1/2 ))*(e-f)*(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 = 20.38 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.83 \[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \sqrt {-\frac {i (1+x)}{-3 i+\sqrt {3}}} \left (-3 i f \sqrt {i+\sqrt {3}-2 i x} \left (-i-\sqrt {3}+\left (-i+\sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-3 i+\sqrt {3}}\right )+2 \sqrt {3} (e+2 f) \sqrt {-i+\sqrt {3}+2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\frac {\sqrt {-i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-3 i+\sqrt {3}}\right )\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {-i+\sqrt {3}+2 i x} \sqrt {-1-x^3}} \] Input:
Integrate[(e + f*x)/((2 - x)*Sqrt[-1 - x^3]),x]
Output:
(2*Sqrt[2/3]*Sqrt[((-I)*(1 + x))/(-3*I + Sqrt[3])]*((-3*I)*f*Sqrt[I + Sqrt [3] - (2*I)*x]*(-I - Sqrt[3] + (-I + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 2 *Sqrt[3]*(e + 2*f)*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x + x^2]*Elliptic Pi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[ 2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3*I + Sqrt[3])*Sqrt[-I + S qrt[3] + (2*I)*x]*Sqrt[-1 - x^3])
Time = 0.66 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2564, 27, 760, 2563, 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.
| \(\displaystyle \int \frac {e+f x}{(2-x) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2564 |
| \(\displaystyle \frac {1}{3} (e-f) \int \frac {1}{\sqrt {-x^3-1}}dx+\frac {1}{6} (e+2 f) \int \frac {2 (x+1)}{(2-x) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} (e-f) \int \frac {1}{\sqrt {-x^3-1}}dx+\frac {1}{3} (e+2 f) \int \frac {x+1}{(2-x) \sqrt {-x^3-1}}dx\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {1}{3} (e+2 f) \int \frac {x+1}{(2-x) \sqrt {-x^3-1}}dx+\frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} (e-f) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle \frac {2}{3} (e+2 f) \int \frac {1}{\frac {(x+1)^4}{-x^3-1}+9}d\frac {(x+1)^2}{\sqrt {-x^3-1}}+\frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} (e-f) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} (e-f) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {2}{9} \arctan \left (\frac {(x+1)^2}{3 \sqrt {-x^3-1}}\right ) (e+2 f)\) |
Input:
Int[(e + f*x)/((2 - x)*Sqrt[-1 - x^3]),x]
Output:
(2*(e + 2*f)*ArcTan[(1 + x)^2/(3*Sqrt[-1 - x^3])])/9 + (2*Sqrt[2 - Sqrt[3] ]*(e - f)*(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*3^(1/4)*Sqrt[-( (1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])
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 ] && NegQ[a]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*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]
Time = 0.50 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {2 i \left (e +2 f \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(246\) |
| elliptic | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (-e -2 f \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(248\) |
Input:
int((f*x+e)/(2-x)/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*I*f*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)*El lipticF(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*(e+2*f)*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+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+1/2*I*3^(1/2)),(I*3 ^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.38 \[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {1}{9} \, {\left (e + 2 \, f\right )} \arctan \left (\frac {{\left (x^{3} + 12 \, x^{2} - 6 \, x + 10\right )} \sqrt {-x^{3} - 1}}{6 \, {\left (x^{4} + x^{3} + x + 1\right )}}\right ) - \frac {2}{3} \, {\left (i \, e - i \, f\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:
integrate((f*x+e)/(2-x)/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
-1/9*(e + 2*f)*arctan(1/6*(x^3 + 12*x^2 - 6*x + 10)*sqrt(-x^3 - 1)/(x^4 + x^3 + x + 1)) - 2/3*(I*e - I*f)*weierstrassPInverse(0, -4, x)
\[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=- \int \frac {e}{x \sqrt {- x^{3} - 1} - 2 \sqrt {- x^{3} - 1}}\, dx - \int \frac {f x}{x \sqrt {- x^{3} - 1} - 2 \sqrt {- x^{3} - 1}}\, dx \] Input:
integrate((f*x+e)/(2-x)/(-x**3-1)**(1/2),x)
Output:
-Integral(e/(x*sqrt(-x**3 - 1) - 2*sqrt(-x**3 - 1)), x) - Integral(f*x/(x* sqrt(-x**3 - 1) - 2*sqrt(-x**3 - 1)), x)
\[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-x^{3} - 1} {\left (x - 2\right )}} \,d x } \] Input:
integrate((f*x+e)/(2-x)/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
-integrate((f*x + e)/(sqrt(-x^3 - 1)*(x - 2)), x)
\[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-x^{3} - 1} {\left (x - 2\right )}} \,d x } \] Input:
integrate((f*x+e)/(2-x)/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(-(f*x + e)/(sqrt(-x^3 - 1)*(x - 2)), x)
Time = 21.67 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.39 \[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {2\,f\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (e+2\,f\right )\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(-(e + f*x)/((- x^3 - 1)^(1/2)*(x - 2)),x)
Output:
(2*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(1/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^ (1/2)*1i)/2 - 3/2))^(1/2)*(e + 2*f)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2) *(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^( 1/2)*1i)/6 + 1/2, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2) *1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(- x^3 - 1)^(1/2)*(x^3 - x*(((3^ (1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*(( 3^(1/2)*1i)/2 + 1/2))^(1/2)) - (2*f*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(1/2) *((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1 /2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2) )^(1/2)*ellipticF(asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2) *1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/((- x^3 - 1)^(1/2)*(x^3 - x*(((3^(1 /2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^ (1/2)*1i)/2 + 1/2))^(1/2))
\[ \int \frac {e+f x}{(2-x) \sqrt {-1-x^3}} \, dx=i \left (\left (\int \frac {\sqrt {x^{3}+1}}{x^{4}-2 x^{3}+x -2}d x \right ) e +\left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{4}-2 x^{3}+x -2}d x \right ) f \right ) \] Input:
int((f*x+e)/(2-x)/(-x^3-1)^(1/2),x)
Output:
i*(int(sqrt(x**3 + 1)/(x**4 - 2*x**3 + x - 2),x)*e + int((sqrt(x**3 + 1)*x )/(x**4 - 2*x**3 + x - 2),x)*f)