Integrand size = 21, antiderivative size = 86 \[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=-\frac {\left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}} \]
-(x^4-x^2)^(3/4)/x/(x^2-1)-1/4*arctan(2^(1/4)*x/(x^4-x^2)^(1/4))*2^(3/4)-1 /4*arctanh(2^(1/4)*x/(x^4-x^2)^(1/4))*2^(3/4)
Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x}+2^{3/4} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
-1/4*(Sqrt[x]*(4*Sqrt[x] + 2^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x ])/(-1 + x^2)^(1/4)] + 2^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/ (-1 + x^2)^(1/4)]))/(x^2*(-1 + x^2))^(1/4)
Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2467, 25, 1388, 368, 25, 907, 902, 756, 216, 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.
| \(\displaystyle \int \frac {1}{\left (x^4-1\right ) \sqrt [4]{x^4-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int -\frac {1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^4\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^4\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\sqrt {x} \left (-x^2-1\right ) \left (x^2-1\right )^{5/4}}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int -\frac {1}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 907 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{x^2-1} \left (x^2+1\right )}d\sqrt {x}+\frac {\sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{2} \int \frac {1}{1-2 x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}+\frac {\sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {2} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}+\frac {1}{2} \int \frac {1}{\sqrt {2} x+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {\sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {2} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2}}\right )+\frac {\sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2}}\right )+\frac {\sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
(-2*Sqrt[x]*(-1 + x^2)^(1/4)*(Sqrt[x]/(2*(-1 + x^2)^(1/4)) + (ArcTan[(2^(1 /4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(2*2^(1/4)) + ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(2*2^(1/4)))/2))/(-x^2 + x^4)^(1/4)
3.12.63.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q} , x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || ! LtQ[q, -1]) && NeQ[p, -1]
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_.)*(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]
Time = 11.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2 x}\right )\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+8 x}{8 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(99\) |
| risch | \(-\frac {x}{\left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (x^{2}+1\right ) x}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{\left (x^{2}+1\right ) x}\right )}{8}\) | \(247\) |
| trager | \(-\frac {\left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}-1\right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (x^{2}+1\right ) x}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\sqrt {x^{4}-x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{\left (x^{2}+1\right ) x}\right )}{8}\) | \(257\) |
-1/8/(x^4-x^2)^(1/4)*(2^(3/4)*(ln((-2^(1/4)*x-(x^4-x^2)^(1/4))/(2^(1/4)*x- (x^4-x^2)^(1/4)))-2*arctan(1/2*2^(3/4)/x*(x^4-x^2)^(1/4)))*(x^4-x^2)^(1/4) +8*x)
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.29 \[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 2^{\frac {3}{4}} {\left (x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 2^{\frac {3}{4}} {\left (-i \, x^{3} + i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{3} - i \, x\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 2^{\frac {3}{4}} {\left (i \, x^{3} - i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{3} + i \, x\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 16 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{16 \, {\left (x^{3} - x\right )}} \]
-1/16*(2^(3/4)*(x^3 - x)*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 + 2^(3/4)*(3 *x^3 - x) + 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 2^(3/4)*(x^3 - x)*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(3*x^3 - x) - 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 2^( 3/4)*(-I*x^3 + I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(3*I*x ^3 - I*x) + 4*I*2^(1/4)*sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^(3/4))/(x^3 + x) ) - 2^(3/4)*(I*x^3 - I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)* (-3*I*x^3 + I*x) - 4*I*2^(1/4)*sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^(3/4))/(x ^3 + x)) + 16*(x^4 - x^2)^(3/4))/(x^3 - x)
\[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \]
1/4*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) - 1/8*2^(3/4)*log(2^(1/ 4) + (-1/x^2 + 1)^(1/4)) + 1/8*2^(3/4)*log(2^(1/4) - (-1/x^2 + 1)^(1/4)) - 1/(-1/x^2 + 1)^(1/4)
Timed out. \[ \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{\left (x^4-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]