\(\int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx\) [506]

Optimal result

Integrand size = 20, antiderivative size = 110 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \]

[Out]

2/3*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*(1+x)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*((x^2-x+1)/(1+x
+3^(1/2))^2)^(1/2)*3^(3/4)/(x^2-x+1)^(1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (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 = {727, 224} \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}} \]

[In]

Int[1/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]

[Out]

(2*Sqrt[2 + Sqrt[3]]*Sqrt[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)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

Rule 224

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]*Sq
rt[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]

Rule 727

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(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]
/; FreeQ[{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] &&  !Intege
rQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^3} \int \frac {1}{\sqrt {1+x^3}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2 \sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {i (1+x) \sqrt {1+\frac {6 i}{\left (-3 i+\sqrt {3}\right ) (1+x)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (3 i+\sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \]

[In]

Integrate[1/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]

[Out]

(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])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/(Sqrt[(-I)/(3*I + Sqrt[3]
)]*Sqrt[1 - x + x^2])

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25

method	result	size
default	\(\frac {\left (3-i \sqrt {3}\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )}{x^{3}+1}\)	\(137\)
elliptic	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{\sqrt {x^{3}+1}\, \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\)	\(145\)

[In]

int(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(3-I*3^(1/2))*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1
/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(
1/2)+3))^(1/2))/(x^3+1)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.05 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=2 \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \]

[In]

integrate(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="fricas")

[Out]

2*weierstrassPInverse(0, -4, x)

Sympy [F]

\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{\sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \]

[In]

integrate(1/(1+x)**(1/2)/(x**2-x+1)**(1/2),x)

[Out]

Integral(1/(sqrt(x + 1)*sqrt(x**2 - x + 1)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1}} \,d x } \]

[In]

integrate(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - x + 1)*sqrt(x + 1)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1}} \,d x } \]

[In]

integrate(1/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - x + 1)*sqrt(x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{\sqrt {x+1}\,\sqrt {x^2-x+1}} \,d x \]

[In]

int(1/((x + 1)^(1/2)*(x^2 - x + 1)^(1/2)),x)

[Out]

int(1/((x + 1)^(1/2)*(x^2 - x + 1)^(1/2)), x)