Integrand size = 25, antiderivative size = 73 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{4} \left (-2+\sqrt {2}\right ) \text {arctanh}\left (\frac {-1+x}{\left (-1+\sqrt {2}\right ) \sqrt {-x+x^3}}\right )+\frac {1}{4} \left (2+\sqrt {2}\right ) \text {arctanh}\left (\frac {-1+x}{\left (1+\sqrt {2}\right ) \sqrt {-x+x^3}}\right ) \]
1/4*(-2+2^(1/2))*arctanh((-1+x)/(2^(1/2)-1)/(x^3-x)^(1/2))+1/4*(2+2^(1/2)) *arctanh((-1+x)/(1+2^(1/2))/(x^3-x)^(1/2))
Time = 10.97 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{4} \left (\left (2+\sqrt {2}\right ) \text {arctanh}\left (\frac {\left (-1+\sqrt {2}\right ) (-1+x)}{\sqrt {x \left (-1+x^2\right )}}\right )+\left (-2+\sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) (-1+x)}{\sqrt {x \left (-1+x^2\right )}}\right )\right ) \]
((2 + Sqrt[2])*ArcTanh[((-1 + Sqrt[2])*(-1 + x))/Sqrt[x*(-1 + x^2)]] + (-2 + Sqrt[2])*ArcTanh[((1 + Sqrt[2])*(-1 + x))/Sqrt[x*(-1 + x^2)]])/4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.74 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2467, 2035, 7279, 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.
| \(\displaystyle \int \frac {x-1}{\left (x^2-2 x-1\right ) \sqrt {x^3-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-1} \int \frac {1-x}{\sqrt {x} \left (-x^2+2 x+1\right ) \sqrt {x^2-1}}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \frac {1-x}{\left (-x^2+2 x+1\right ) \sqrt {x^2-1}}d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \left (-\frac {1}{\left (-2 x-2 \sqrt {2}+2\right ) \sqrt {x^2-1}}-\frac {1}{\left (-2 x+2 \sqrt {2}+2\right ) \sqrt {x^2-1}}\right )d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {\left (1+\sqrt {2}\right ) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (-1-\sqrt {2},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}+\frac {\left (1-\sqrt {2}\right ) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (-1+\sqrt {2},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}\) |
(2*Sqrt[x]*Sqrt[-1 + x^2]*(((1 + Sqrt[2])*Sqrt[1 - x]*Sqrt[1 + x]*Elliptic Pi[-1 - Sqrt[2], ArcSin[Sqrt[x]], -1])/(2*Sqrt[-1 + x^2]) + ((1 - Sqrt[2]) *Sqrt[1 - x]*Sqrt[1 + x]*EllipticPi[-1 + Sqrt[2], ArcSin[Sqrt[x]], -1])/(2 *Sqrt[-1 + x^2])))/Sqrt[-x + x^3]
3.10.56.3.1 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_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 2.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59
| method | result | size |
| elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\) | \(116\) |
| default | \(-\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{2}+\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -63721 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}-1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}+2862728 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -148735 x^{2}-2926449 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+886429 \sqrt {x^{3}-x}-1427856 x +1279121}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-3 x +7\right )}^{2}}\right )}{2}\) | \(543\) |
| trager | \(-\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{2}+\frac {\ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -493895 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-74648 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +66352 x^{2}-419247 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-152085 \sqrt {x^{3}-x}+40832 x +25520}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x +1\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \ln \left (\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x -63721 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{2}+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}-1038514 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}+2862728 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -148735 x^{2}-2926449 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+886429 \sqrt {x^{3}-x}-1427856 x +1279121}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-3 x +7\right )}^{2}}\right )}{2}\) | \(543\) |
1/2*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2-2^(1/2))*Ellipt icPi((1+x)^(1/2),-1/(-2-2^(1/2)),1/2*2^(1/2))+1/2*(1+x)^(1/2)*(2-2*x)^(1/2 )*(-x)^(1/2)/(x^3-x)^(1/2)/(-2+2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-2+2^(1 /2)),1/2*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (57) = 114\).
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.73 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 12 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} + 2 \, x^{2} - 12 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 4 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 1\right )} - 4 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) \]
1/8*sqrt(2)*log((x^4 + 12*x^3 + 4*sqrt(2)*sqrt(x^3 - x)*(x^2 + 2*x - 1) + 2*x^2 - 12*x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1)) + 1/4*log((x^4 + 4*x^3 + 2*x^2 - 4*sqrt(x^3 - x)*(x^2 + 1) - 4*x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1))
\[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int \frac {x - 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} - 2 x - 1\right )}\, dx \]
\[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}} \,d x } \]
\[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}} \,d x } \]
Time = 6.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {-1+x}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{\sqrt {2}+1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}\,\left (\sqrt {2}+1\right )}-\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\sqrt {2}-1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}\,\left (\sqrt {2}-1\right )} \]