Integrand size = 24, antiderivative size = 79 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {-x+x^3}}{1-x^2}-\frac {1}{4} \arctan \left (\frac {2 \sqrt {-x+x^3}}{-1-2 x+x^2}\right )-\frac {1}{4} \text {arctanh}\left (\frac {-\frac {1}{2}+x+\frac {x^2}{2}}{\sqrt {-x+x^3}}\right ) \]
(x^3-x)^(1/2)/(-x^2+1)-1/4*arctan(2*(x^3-x)^(1/2)/(x^2-2*x-1))-1/4*arctanh ((-1/2+x+1/2*x^2)/(x^3-x)^(1/2))
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {-4 x-(1-i) \sqrt {x} \sqrt {-1+x^2} \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )+(1+i) \sqrt {x} \sqrt {-1+x^2} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^2}}{\sqrt {x}}\right )}{4 \sqrt {x \left (-1+x^2\right )}} \]
(-4*x - (1 - I)*Sqrt[x]*Sqrt[-1 + x^2]*ArcTan[((1 + I)*Sqrt[x])/Sqrt[-1 + x^2]] + (1 + I)*Sqrt[x]*Sqrt[-1 + x^2]*ArcTan[((1/2 + I/2)*Sqrt[-1 + x^2]) /Sqrt[x]])/(4*Sqrt[x*(-1 + x^2)])
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2467, 25, 1388, 2035, 25, 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.
| \(\displaystyle \int \frac {x^4+1}{\sqrt {x^3-x} \left (x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-1} \int -\frac {x^4+1}{\sqrt {x} \sqrt {x^2-1} \left (1-x^4\right )}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-1} \int \frac {x^4+1}{\sqrt {x} \sqrt {x^2-1} \left (1-x^4\right )}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-1} \int \frac {x^4+1}{\sqrt {x} \left (-x^2-1\right ) \left (x^2-1\right )^{3/2}}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \int -\frac {x^4+1}{\left (x^2-1\right )^{3/2} \left (x^2+1\right )}d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \frac {x^4+1}{\left (x^2-1\right )^{3/2} \left (x^2+1\right )}d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \left (\frac {x^2}{\left (x^2-1\right )^{3/2}}+\frac {2}{\left (x^2-1\right )^{3/2} \left (x^2+1\right )}-\frac {1}{\left (x^2-1\right )^{3/2}}\right )d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )+\frac {\sqrt {x}}{2 \sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}\) |
(-2*Sqrt[x]*Sqrt[-1 + x^2]*(Sqrt[x]/(2*Sqrt[-1 + x^2]) + (1/8 - I/8)*ArcTa n[((1 + I)*Sqrt[x])/Sqrt[-1 + x^2]] + (1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[x] )/Sqrt[-1 + x^2]]))/Sqrt[-x + x^3]
3.11.47.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 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_)^(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]
Time = 7.87 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {x}{\sqrt {x \left (x^{2}-1\right )}}-\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )}{4}+\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )}{4}\) | \(104\) |
| default | \(\frac {\left (\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )-\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )\right ) \sqrt {x^{3}-x}-4 x}{4 \sqrt {x^{3}-x}}\) | \(113\) |
| pseudoelliptic | \(\frac {\left (\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )-\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{2}+\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )\right ) \sqrt {x^{3}-x}-4 x}{4 \sqrt {x^{3}-x}}\) | \(113\) |
| elliptic | \(-\frac {x}{\sqrt {x \left (x^{2}-1\right )}}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\) | \(223\) |
| trager | \(-\frac {\sqrt {x^{3}-x}}{x^{2}-1}+\frac {\ln \left (-\frac {300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-451008 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x -11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+14400 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+3528 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x +91 x^{2}-150 \sqrt {x^{3}-x}+11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+26 x -91}{{\left (72 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+x +3\right )}^{2}}\right )}{4}-9 \ln \left (-\frac {300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-451008 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x -11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+14400 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-300672 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+3528 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x +91 x^{2}-150 \sqrt {x^{3}-x}+11052 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+26 x -91}{{\left (72 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+x +3\right )}^{2}}\right ) \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+9 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \ln \left (\frac {-150336 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}+225504 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x +2826 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}+7200 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+150336 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}-10764 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -8 x^{2}-125 \sqrt {x^{3}-x}-2826 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+112 x +8}{{\left (72 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x -144 \operatorname {RootOf}\left (2592 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-3 x +1\right )}^{2}}\right )\) | \(559\) |
-x/(x*(x^2-1))^(1/2)-1/8*ln((x^2+2*(x^3-x)^(1/2)+2*x-1)/x)+1/4*arctan(((x^ 3-x)^(1/2)+x)/x)+1/8*ln((x^2-2*(x^3-x)^(1/2)+2*x-1)/x)+1/4*arctan(((x^3-x) ^(1/2)-x)/x)
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.33 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x}}\right ) + {\left (x^{2} - 1\right )} \log \left (\frac {x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} - 8 \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - 8 \, \sqrt {x^{3} - x}}{8 \, {\left (x^{2} - 1\right )}} \]
1/8*(2*(x^2 - 1)*arctan(1/2*(x^2 - 2*x - 1)/sqrt(x^3 - x)) + (x^2 - 1)*log ((x^4 + 8*x^3 + 2*x^2 - 4*sqrt(x^3 - x)*(x^2 + 2*x - 1) - 8*x + 1)/(x^4 + 2*x^2 + 1)) - 8*sqrt(x^3 - x))/(x^2 - 1)
\[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\int \frac {x^{4} + 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x}} \,d x } \]
\[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x}} \,d x } \]
Time = 0.05 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.94 \[ \int \frac {1+x^4}{\sqrt {-x+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {-x}\,\left (\frac {\sin \left (2\,\mathrm {asin}\left (\sqrt {-x}\right )\right )}{4\,\sqrt {1-x}}+\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2}\right )\,\sqrt {1-x}\,\sqrt {x+1}}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (1{}\mathrm {i};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )-\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2}+\frac {\sqrt {-x}\,\sqrt {1-x}}{2\,\sqrt {x+1}}\right )}{\sqrt {x^3-x}} \]
((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(-1i, asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) - (2*(-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticF (asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x + 1 )^(1/2)*ellipticPi(1i, asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) + ((-x)^(1/2 )*(sin(2*asin((-x)^(1/2)))/(4*(1 - x)^(1/2)) + ellipticE(asin((-x)^(1/2)), -1)/2)*(1 - x)^(1/2)*(x + 1)^(1/2))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(1 - x) ^(1/2)*(x + 1)^(1/2)*(ellipticF(asin((-x)^(1/2)), -1) - ellipticE(asin((-x )^(1/2)), -1)/2 + ((-x)^(1/2)*(1 - x)^(1/2))/(2*(x + 1)^(1/2))))/(x^3 - x) ^(1/2)