Integrand size = 20, antiderivative size = 110 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \] Output:
2/3*(1/2*6^(1/2)+1/2*2^(1/2))*(1+x)^(1/2)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2 )*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*3^(3/4)/((1+x)/(1+x +3^(1/2))^2)^(1/2)/(x^2-x+1)^(1/2)
Result contains complex when optimal does not.
Time = 20.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {i (1+x) \sqrt {1+\frac {6 i}{\left (-3 i+\sqrt {3}\right ) (1+x)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (3 i+\sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \] Input:
Integrate[1/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]
Output:
(I*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[2/3 - (4*I)/((3 *I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/S qrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/(Sqrt[(-I)/(3*I + Sqrt[3])] *Sqrt[1 - x + x^2])
Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1151, 759}
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.
\(\displaystyle \int \frac {1}{\sqrt {x+1} \sqrt {x^2-x+1}} \, dx\) |
\(\Big \downarrow \) 1151 |
\(\displaystyle \frac {\sqrt {x^3+1} \int \frac {1}{\sqrt {x^3+1}}dx}{\sqrt {x+1} \sqrt {x^2-x+1}}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}\) |
Input:
Int[1/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]
Output:
(2*Sqrt[2 + Sqrt[3]]*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*E llipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^ (1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])
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[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + c*x^2)^FracPart[p]/(a*d + c *e*x^3)^FracPart[p]) Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x] /; F reeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] && !IntegerQ[p]
Time = 1.91 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\left (-i \sqrt {3}+3\right ) \sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )}{x^{3}+1}\) | \(137\) |
elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}}{\sqrt {x^{3}+1}\, \sqrt {x +1}\, \sqrt {x^{2}-x +1}}\) | \(145\) |
Input:
int(1/(x+1)^(1/2)/(x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-I*3^(1/2)+3)*(x+1)^(1/2)*(x^2-x+1)^(1/2)*(-2*(x+1)/(I*3^(1/2)-3))^(1/2)* ((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(I*3^(1/2)-3))^ (1/2)*EllipticF((-2*(x+1)/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+ 3))^(1/2))/(x^3+1)
Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.05 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=2 \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:
integrate(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="fricas")
Output:
2*weierstrassPInverse(0, -4, x)
\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{\sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \] Input:
integrate(1/(1+x)**(1/2)/(x**2-x+1)**(1/2),x)
Output:
Integral(1/(sqrt(x + 1)*sqrt(x**2 - x + 1)), x)
\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1}} \,d x } \] Input:
integrate(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(x^2 - x + 1)*sqrt(x + 1)), x)
\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1}} \,d x } \] Input:
integrate(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(x^2 - x + 1)*sqrt(x + 1)), x)
Timed out. \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{\sqrt {x+1}\,\sqrt {x^2-x+1}} \,d x \] Input:
int(1/((x + 1)^(1/2)*(x^2 - x + 1)^(1/2)),x)
Output:
int(1/((x + 1)^(1/2)*(x^2 - x + 1)^(1/2)), x)
\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}}{x^{3}+1}d x \] Input:
int(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x)
Output:
int((sqrt(x + 1)*sqrt(x**2 - x + 1))/(x**3 + 1),x)