Integrand size = 28, antiderivative size = 34 \[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x+x^2+x^3}}{1+x+x^2}\right ) \]
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\frac {\sqrt {2} \sqrt {x} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {1+x+x^2}}\right )}{\sqrt {x \left (1+x+x^2\right )}} \]
-((Sqrt[2]*Sqrt[x]*Sqrt[1 + x + x^2]*ArcTanh[(Sqrt[2]*Sqrt[x])/Sqrt[1 + x + x^2]])/Sqrt[x*(1 + x + x^2)])
Time = 0.40 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2467, 25, 2035, 2537, 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.
| \(\displaystyle \int \frac {x^2-1}{\left (x^2-x+1\right ) \sqrt {x^3+x^2+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2+x+1} \int -\frac {1-x^2}{\sqrt {x} \left (x^2-x+1\right ) \sqrt {x^2+x+1}}dx}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2+x+1} \int \frac {1-x^2}{\sqrt {x} \left (x^2-x+1\right ) \sqrt {x^2+x+1}}dx}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+x+1} \int \frac {1-x^2}{\left (x^2-x+1\right ) \sqrt {x^2+x+1}}d\sqrt {x}}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 2537 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+x+1} \int \frac {1}{1-2 x}d\frac {\sqrt {x}}{\sqrt {x^2+x+1}}}{\sqrt {x^3+x^2+x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt {2} \sqrt {x} \sqrt {x^2+x+1} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{\sqrt {x^3+x^2+x}}\) |
-((Sqrt[2]*Sqrt[x]*Sqrt[1 + x + x^2]*ArcTanh[(Sqrt[2]*Sqrt[x])/Sqrt[1 + x + x^2]])/Sqrt[x + x^2 + x^3])
3.5.17.3.1 Defintions of rubi rules used
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[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Co eff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Simp[A Subst[Int[1/(d - (b*d - a*e )*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]
Time = 3.44 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right )\) | \(25\) |
| pseudoelliptic | \(-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right )\) | \(25\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}+x^{2}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}-x +1}\right )}{2}\) | \(62\) |
| elliptic | \(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \left (-\frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{4}-\frac {5}{8}+\frac {3 i \sqrt {3}}{8}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \left (-\frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{4}-\frac {5}{8}+\frac {i \sqrt {3}}{8}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}\) | \(425\) |
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 14 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 3 \, x + 1\right )} + 19 \, x^{2} + 14 \, x + 1}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) \]
1/4*sqrt(2)*log((x^4 + 14*x^3 - 4*sqrt(2)*sqrt(x^3 + x^2 + x)*(x^2 + 3*x + 1) + 19*x^2 + 14*x + 1)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1))
\[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {x \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{3} + x^{2} + x} {\left (x^{2} - x + 1\right )}} \,d x } \]
\[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{3} + x^{2} + x} {\left (x^{2} - x + 1\right )}} \,d x } \]
Time = 0.09 (sec) , antiderivative size = 227, normalized size of antiderivative = 6.68 \[ \int \frac {-1+x^2}{\left (1-x+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\frac {\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-1;\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,1{}\mathrm {i}}{\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}} \]
-((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)* 1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1 /2)*(3^(1/2) + 1i)*(ellipticPi(((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/ 2), asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1 /2)*1i)/2 + 1/2)) - ellipticF(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3 ^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) + ellipticPi(-1, asin((x/((3^( 1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) )*1i)/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)