Integrand size = 29, antiderivative size = 97 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=-\sqrt {\frac {1}{5}+\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\sqrt {\frac {1}{5}-\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]
-1/5*(5+10*I)^(1/2)*arctan((1-2*I)^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))-1/5* (5-10*I)^(1/2)*arctan((1+2*I)^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=-\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (\sqrt {1+2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1-2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{\sqrt {5} \sqrt {x \left (-1-x+x^2\right )}} \]
-((Sqrt[x]*Sqrt[-1 - x + x^2]*(Sqrt[1 + 2*I]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[x] )/Sqrt[-1 - x + x^2]] + Sqrt[1 - 2*I]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[x])/Sqrt[ -1 - x + x^2]]))/(Sqrt[5]*Sqrt[x*(-1 - x + x^2)]))
Time = 0.84 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2467, 25, 2035, 7276, 2009}
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.
| \(\displaystyle \int \frac {x^2-1}{\left (x^2+1\right ) \sqrt {x^3-x^2-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-x-1} \int -\frac {1-x^2}{\sqrt {x} \left (x^2+1\right ) \sqrt {x^2-x-1}}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-x-1} \int \frac {1-x^2}{\sqrt {x} \left (x^2+1\right ) \sqrt {x^2-x-1}}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^2-x-1}}d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \left (\frac {2}{\left (x^2+1\right ) \sqrt {x^2-x-1}}-\frac {1}{\sqrt {x^2-x-1}}\right )d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (\frac {\arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )}{2 \sqrt {1-2 i}}+\frac {\arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )}{2 \sqrt {1+2 i}}\right )}{\sqrt {x^3-x^2-x}}\) |
(-2*Sqrt[x]*Sqrt[-1 - x + x^2]*(ArcTan[(Sqrt[1 - 2*I]*Sqrt[x])/Sqrt[-1 - x + x^2]]/(2*Sqrt[1 - 2*I]) + ArcTan[(Sqrt[1 + 2*I]*Sqrt[x])/Sqrt[-1 - x + x^2]]/(2*Sqrt[1 + 2*I])))/Sqrt[-x - x^2 + x^3]
3.14.46.3.1 Defintions of rubi rules used
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_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(191\) vs. \(2(77)=154\).
Time = 18.80 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.98
| method | result | size |
| default | \(-\frac {-\sqrt {5}\, \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\sqrt {5}\, \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (5+\sqrt {5}\right ) \left (\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )\right )}{5 \sqrt {2+2 \sqrt {5}}}\) | \(192\) |
| pseudoelliptic | \(-\frac {-\sqrt {5}\, \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\sqrt {5}\, \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (5+\sqrt {5}\right ) \left (\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )\right )}{5 \sqrt {2+2 \sqrt {5}}}\) | \(192\) |
| trager | \(\frac {\operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {325 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{5} x^{2}-325 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{5}+90 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{3} x^{2}+520 x \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{3}+80 \sqrt {x^{3}-x^{2}-x}\, \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}-90 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{3}-31 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right ) x^{2}+248 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right ) x +192 \sqrt {x^{3}-x^{2}-x}+31 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{{\left (5 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} x -5 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}-x -3\right )}^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \ln \left (\frac {65 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{4} x^{2}-65 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{4}+34 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}-104 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x +80 \sqrt {x^{3}-x^{2}-x}\, \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}-34 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right )-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x -160 \sqrt {x^{3}-x^{2}-x}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right )}{{\left (5 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} x -5 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+3 x +1\right )}^{2}}\right )}{10}\) | \(654\) |
| elliptic | \(\text {Expression too large to display}\) | \(873\) |
-1/5/(2+2*5^(1/2))^(1/2)*(-5^(1/2)*ln((-(x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2)) ^(1/2)+x*5^(1/2)+x^2-x-1)/x)+5^(1/2)*ln(((x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2) )^(1/2)+x*5^(1/2)+x^2-x-1)/x)+(5+5^(1/2))*(arctan(((-2+2*5^(1/2))^(1/2)*x- 2*(x*(x^2-x-1))^(1/2))/x/(2+2*5^(1/2))^(1/2))-arctan(((-2+2*5^(1/2))^(1/2) *x+2*(x*(x^2-x-1))^(1/2))/x/(2+2*5^(1/2))^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (69) = 138\).
Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 3.02 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\frac {1}{40} \, \sqrt {5} \sqrt {-8 i - 4} \log \left (\frac {5 \, x^{4} + \left (20 i - 10\right ) \, x^{3} + \sqrt {5} \sqrt {-8 i - 4} \sqrt {x^{3} - x^{2} - x} {\left (-\left (2 i - 1\right ) \, x^{2} + \left (2 i + 4\right ) \, x + 2 i - 1\right )} - \left (20 i + 30\right ) \, x^{2} - \left (20 i - 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{40} \, \sqrt {5} \sqrt {-8 i - 4} \log \left (\frac {5 \, x^{4} + \left (20 i - 10\right ) \, x^{3} + \sqrt {5} \sqrt {-8 i - 4} \sqrt {x^{3} - x^{2} - x} {\left (\left (2 i - 1\right ) \, x^{2} - \left (2 i + 4\right ) \, x - 2 i + 1\right )} - \left (20 i + 30\right ) \, x^{2} - \left (20 i - 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {5} \sqrt {8 i - 4} \log \left (\frac {5 \, x^{4} - \left (20 i + 10\right ) \, x^{3} + \sqrt {5} \sqrt {8 i - 4} \sqrt {x^{3} - x^{2} - x} {\left (\left (2 i + 1\right ) \, x^{2} - \left (2 i - 4\right ) \, x - 2 i - 1\right )} + \left (20 i - 30\right ) \, x^{2} + \left (20 i + 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{40} \, \sqrt {5} \sqrt {8 i - 4} \log \left (\frac {5 \, x^{4} - \left (20 i + 10\right ) \, x^{3} + \sqrt {5} \sqrt {8 i - 4} \sqrt {x^{3} - x^{2} - x} {\left (-\left (2 i + 1\right ) \, x^{2} + \left (2 i - 4\right ) \, x + 2 i + 1\right )} + \left (20 i - 30\right ) \, x^{2} + \left (20 i + 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) \]
1/40*sqrt(5)*sqrt(-8*I - 4)*log((5*x^4 + (20*I - 10)*x^3 + sqrt(5)*sqrt(-8 *I - 4)*sqrt(x^3 - x^2 - x)*(-(2*I - 1)*x^2 + (2*I + 4)*x + 2*I - 1) - (20 *I + 30)*x^2 - (20*I - 10)*x + 5)/(x^4 + 2*x^2 + 1)) - 1/40*sqrt(5)*sqrt(- 8*I - 4)*log((5*x^4 + (20*I - 10)*x^3 + sqrt(5)*sqrt(-8*I - 4)*sqrt(x^3 - x^2 - x)*((2*I - 1)*x^2 - (2*I + 4)*x - 2*I + 1) - (20*I + 30)*x^2 - (20*I - 10)*x + 5)/(x^4 + 2*x^2 + 1)) + 1/40*sqrt(5)*sqrt(8*I - 4)*log((5*x^4 - (20*I + 10)*x^3 + sqrt(5)*sqrt(8*I - 4)*sqrt(x^3 - x^2 - x)*((2*I + 1)*x^ 2 - (2*I - 4)*x - 2*I - 1) + (20*I - 30)*x^2 + (20*I + 10)*x + 5)/(x^4 + 2 *x^2 + 1)) - 1/40*sqrt(5)*sqrt(8*I - 4)*log((5*x^4 - (20*I + 10)*x^3 + sqr t(5)*sqrt(8*I - 4)*sqrt(x^3 - x^2 - x)*(-(2*I + 1)*x^2 + (2*I - 4)*x + 2*I + 1) + (20*I - 30)*x^2 + (20*I + 10)*x + 5)/(x^4 + 2*x^2 + 1))
\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 1\right )}} \,d x } \]
\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 1\right )}} \,d x } \]
Time = 0.15 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.16 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\left (\sqrt {5}+1\right )\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )+\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )+\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]
-((x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1 /2)*(5^(1/2) + 1)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*(ellipti cPi(- (5^(1/2)*1i)/2 - 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/ 2 + 1/2)/(5^(1/2)/2 - 1/2)) - ellipticF(asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2)) + ellipticPi((5^(1/2)*1i)/2 + 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2))) )/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2)