Integrand size = 40, antiderivative size = 151 \[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {1-\sqrt {2} k+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{\sqrt {1-\sqrt {2} k+k^2}}-\frac {\arctan \left (\frac {\sqrt {1+\sqrt {2} k+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{\sqrt {1+\sqrt {2} k+k^2}} \]
-arctan((1-2^(1/2)*k+k^2)^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/(-1+x)/(k^2 *x-1))/(1-2^(1/2)*k+k^2)^(1/2)-arctan((1+2^(1/2)*k+k^2)^(1/2)*(x+(-k^2-1)* x^2+k^2*x^3)^(1/2)/(-1+x)/(k^2*x-1))/(1+2^(1/2)*k+k^2)^(1/2)
Time = 13.50 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.72 \[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {1-\sqrt {2} k+k^2} x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {1-\sqrt {2} k+k^2}}-\frac {\arctan \left (\frac {\sqrt {1+\sqrt {2} k+k^2} x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {1+\sqrt {2} k+k^2}} \]
-(ArcTan[(Sqrt[1 - Sqrt[2]*k + k^2]*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]]/Sqrt [1 - Sqrt[2]*k + k^2]) - ArcTan[(Sqrt[1 + Sqrt[2]*k + k^2]*x)/Sqrt[(-1 + x )*x*(-1 + k^2*x)]]/Sqrt[1 + Sqrt[2]*k + k^2]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.39 (sec) , antiderivative size = 1785, normalized size of antiderivative = 11.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
| \(\displaystyle \int \frac {k^4 x^4-1}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^4 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {1-k^4 x^4}{\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (k^4 x^4+1\right )}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {1-k^4 x^4}{\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (k^4 x^4+1\right )}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {1-k^4 x^4}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (k^4 x^4+1\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (k^4 x^4+1\right )}-\frac {1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\frac {(k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {\left (k+\sqrt {-\sqrt {-k^4}}\right )^2}{4 k \sqrt {-\sqrt {-k^4}}},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right ) \left (k-\sqrt {-\sqrt {-k^4}}\right )^2}{8 \sqrt {k} \left (k^2+\sqrt {-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\sqrt {k} (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right ) \left (k-\sqrt {-\sqrt {-k^4}}\right )}{4 \left (k^2+\sqrt {-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\sqrt [8]{-k^4} \arctan \left (\frac {\sqrt {\left (\sqrt [4]{-k^4}+1\right ) \left (k^2+\sqrt [4]{-k^4}\right )} \sqrt {x}}{\sqrt [8]{-k^4} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 \sqrt {\left (\sqrt [4]{-k^4}+1\right ) \left (k^2+\sqrt [4]{-k^4}\right )}}+\frac {k \arctan \left (\frac {\sqrt {\frac {k^6}{\left (-k^4\right )^{3/4}}+k^4+k^2+\left (-k^4\right )^{3/4}} \sqrt {x}}{k \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 \sqrt {\frac {k^6}{\left (-k^4\right )^{3/4}}+k^4+k^2+\left (-k^4\right )^{3/4}}}+\frac {\sqrt [4]{-\sqrt {-k^4}} \arctan \left (\frac {\sqrt {\left (\sqrt {-\sqrt {-k^4}}+1\right ) k^2-\sqrt {-k^4}+\sqrt {-\sqrt {-k^4}}} \sqrt {x}}{\sqrt [4]{-\sqrt {-k^4}} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 \sqrt {\left (\sqrt {-\sqrt {-k^4}}+1\right ) k^2-\sqrt {-k^4}+\sqrt {-\sqrt {-k^4}}}}+\frac {\sqrt [4]{-\sqrt {-k^4}} \text {arctanh}\left (\frac {\sqrt {\left (1-\sqrt {-\sqrt {-k^4}}\right ) k^2-\sqrt {-k^4}-\sqrt {-\sqrt {-k^4}}} \sqrt {x}}{\sqrt [4]{-\sqrt {-k^4}} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 \sqrt {\left (1-\sqrt {-\sqrt {-k^4}}\right ) k^2-\sqrt {-k^4}-\sqrt {-\sqrt {-k^4}}}}+\frac {\sqrt {k} \left (k+\sqrt {-\sqrt {-k^4}}\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 \left (k^2+\sqrt {-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {(k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\sqrt {k} (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 \left (k-\sqrt [4]{-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\sqrt {k} (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 \left (k+\sqrt [4]{-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (k+\sqrt [4]{-k^4}\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {k^3 \left (k-\sqrt [4]{-k^4}\right )^2}{4 \left (-k^4\right )^{5/4}},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 \sqrt {k} \left (k-\sqrt [4]{-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\sqrt [4]{-k^4} \left (k^2-\sqrt [4]{-k^4} k+\sqrt {-k^4}\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {\left (k+\sqrt [4]{-k^4}\right )^2}{4 k \sqrt [4]{-k^4}},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{7/2} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (k+\sqrt {-\sqrt {-k^4}}\right )^2 (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (-\frac {\left (k-\sqrt {-\sqrt {-k^4}}\right )^2}{4 k \sqrt {-\sqrt {-k^4}}},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 \sqrt {k} \left (k^2+\sqrt {-k^4}\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
(-2*Sqrt[x]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]*(((-k^4)^(1/8)*ArcTan[(Sqrt[(1 + (-k^4)^(1/4))*(k^2 + (-k^4)^(1/4))]*Sqrt[x])/((-k^4)^(1/8)*Sqrt[1 - (1 + k^2)*x + k^2*x^2])])/(4*Sqrt[(1 + (-k^4)^(1/4))*(k^2 + (-k^4)^(1/4))]) + (k*ArcTan[(Sqrt[k^2 + k^4 + k^6/(-k^4)^(3/4) + (-k^4)^(3/4)]*Sqrt[x])/(k* Sqrt[1 - (1 + k^2)*x + k^2*x^2])])/(4*Sqrt[k^2 + k^4 + k^6/(-k^4)^(3/4) + (-k^4)^(3/4)]) + ((-Sqrt[-k^4])^(1/4)*ArcTan[(Sqrt[-Sqrt[-k^4] + Sqrt[-Sqr t[-k^4]] + k^2*(1 + Sqrt[-Sqrt[-k^4]])]*Sqrt[x])/((-Sqrt[-k^4])^(1/4)*Sqrt [1 - (1 + k^2)*x + k^2*x^2])])/(4*Sqrt[-Sqrt[-k^4] + Sqrt[-Sqrt[-k^4]] + k ^2*(1 + Sqrt[-Sqrt[-k^4]])]) + ((-Sqrt[-k^4])^(1/4)*ArcTanh[(Sqrt[-Sqrt[-k ^4] - Sqrt[-Sqrt[-k^4]] + k^2*(1 - Sqrt[-Sqrt[-k^4]])]*Sqrt[x])/((-Sqrt[-k ^4])^(1/4)*Sqrt[1 - (1 + k^2)*x + k^2*x^2])])/(4*Sqrt[-Sqrt[-k^4] - Sqrt[- Sqrt[-k^4]] + k^2*(1 - Sqrt[-Sqrt[-k^4]])]) - ((1 + k*x)*Sqrt[(1 - (1 + k^ 2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^ 2/(4*k)])/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + (Sqrt[k]*(1 + k*x) *Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt[k]* Sqrt[x]], (1 + k)^2/(4*k)])/(4*(k - (-k^4)^(1/4))*Sqrt[1 - (1 + k^2)*x + k ^2*x^2]) + (Sqrt[k]*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2 ]*EllipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(4*(k + (-k^4)^(1 /4))*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + (Sqrt[k]*(k - Sqrt[-Sqrt[-k^4]])*( 1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcT...
3.21.88.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.96 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(\frac {\left (\arctan \left (\frac {2^{\frac {1}{4}} \sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {\sqrt {2}\, k^{2}-2 \,\operatorname {csgn}\left (k \right ) k +\sqrt {2}}}\right ) \sqrt {\sqrt {2}\, k^{2}+2 \,\operatorname {csgn}\left (k \right ) k +\sqrt {2}}+\arctan \left (\frac {2^{\frac {1}{4}} \sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {\sqrt {2}\, k^{2}+2 \,\operatorname {csgn}\left (k \right ) k +\sqrt {2}}}\right ) \sqrt {\sqrt {2}\, k^{2}-2 \,\operatorname {csgn}\left (k \right ) k +\sqrt {2}}\right ) 2^{\frac {1}{4}}}{\sqrt {\sqrt {2}\, k^{2}+2 \,\operatorname {csgn}\left (k \right ) k +\sqrt {2}}\, \sqrt {\sqrt {2}\, k^{2}-2 \,\operatorname {csgn}\left (k \right ) k +\sqrt {2}}}\) | \(160\) |
| default | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (k^{4} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{4}+1}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (k^{4}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{k^{4}}\) | \(255\) |
| elliptic | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (k^{4} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{4}+1}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (k^{4}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{k^{4}}\) | \(255\) |
(arctan(2^(1/4)*((-1+x)*x*(k^2*x-1))^(1/2)/x/(2^(1/2)*k^2-2*csgn(k)*k+2^(1 /2))^(1/2))*(2^(1/2)*k^2+2*csgn(k)*k+2^(1/2))^(1/2)+arctan(2^(1/4)*((-1+x) *x*(k^2*x-1))^(1/2)/x/(2^(1/2)*k^2+2*csgn(k)*k+2^(1/2))^(1/2))*(2^(1/2)*k^ 2-2*csgn(k)*k+2^(1/2))^(1/2))*2^(1/4)/(2^(1/2)*k^2+2*csgn(k)*k+2^(1/2))^(1 /2)/(2^(1/2)*k^2-2*csgn(k)*k+2^(1/2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1197 vs. \(2 (129) = 258\).
Time = 0.32 (sec) , antiderivative size = 1197, normalized size of antiderivative = 7.93 \[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=\text {Too large to display} \]
-1/4*sqrt(-(k^2 + 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/( k^4 + 1))*log((k^4*x^4 + 4*k^2*x^2 - 2*(k^4 + k^2)*x^3 + 4*sqrt(1/2)*((k^6 + k^2)*x^3 - (k^6 + k^4 + k^2 + 1)*x^2 + (k^4 + 1)*x)*sqrt(k^2/(k^8 + 2*k ^4 + 1)) - 2*(k^2 + 1)*x + 2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(2*k^2*x - (k^4 + k^2)*x^2 - k^2 + 2*sqrt(1/2)*(k^4 + (k^6 + k^2)*x^2 - (k^6 + k^4 + k^2 + 1)*x + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt(-(k^2 + 2*sqrt(1/2)* (k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1)) + 1)/(k^4*x^4 + 1)) + 1/4*sqrt(-(k^2 + 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/ (k^4 + 1))*log((k^4*x^4 + 4*k^2*x^2 - 2*(k^4 + k^2)*x^3 + 4*sqrt(1/2)*((k^ 6 + k^2)*x^3 - (k^6 + k^4 + k^2 + 1)*x^2 + (k^4 + 1)*x)*sqrt(k^2/(k^8 + 2* k^4 + 1)) - 2*(k^2 + 1)*x - 2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(2*k^2*x - (k^4 + k^2)*x^2 - k^2 + 2*sqrt(1/2)*(k^4 + (k^6 + k^2)*x^2 - (k^6 + k^4 + k^2 + 1)*x + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt(-(k^2 + 2*sqrt(1/2) *(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1)) + 1)/(k^4*x^4 + 1)) - 1/4*sqrt(-(k^2 - 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1) /(k^4 + 1))*log((k^4*x^4 + 4*k^2*x^2 - 2*(k^4 + k^2)*x^3 - 4*sqrt(1/2)*((k ^6 + k^2)*x^3 - (k^6 + k^4 + k^2 + 1)*x^2 + (k^4 + 1)*x)*sqrt(k^2/(k^8 + 2 *k^4 + 1)) - 2*(k^2 + 1)*x + 2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(2*k^2*x - (k^4 + k^2)*x^2 - k^2 - 2*sqrt(1/2)*(k^4 + (k^6 + k^2)*x^2 - (k^6 + k^4 + k^2 + 1)*x + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt(-(k^2 - 2*sqrt(...
Timed out. \[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=\int { \frac {k^{4} x^{4} - 1}{{\left (k^{4} x^{4} + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
\[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=\int { \frac {k^{4} x^{4} - 1}{{\left (k^{4} x^{4} + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
Timed out. \[ \int \frac {-1+k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^4 x^4\right )} \, dx=\int \frac {k^4\,x^4-1}{\left (k^4\,x^4+1\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}} \,d x \]