Integrand size = 22, antiderivative size = 75 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\frac {2 \left (1+x^2\right ) \sqrt [4]{x^2+x^4}}{5 x^3}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}} \]
2/5*(x^2+1)*(x^4+x^2)^(1/4)/x^3+1/2*arctan(2^(1/4)*x/(x^4+x^2)^(1/4))*2^(1 /4)-1/2*arctanh(2^(1/4)*x/(x^4+x^2)^(1/4))*2^(1/4)
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\frac {\sqrt [4]{x^2+x^4} \left (4 \left (1+x^2\right )^{5/4}+5 \sqrt [4]{2} x^{5/2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-5 \sqrt [4]{2} x^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{10 x^3 \sqrt [4]{1+x^2}} \]
((x^2 + x^4)^(1/4)*(4*(1 + x^2)^(5/4) + 5*2^(1/4)*x^(5/2)*ArcTan[(2^(1/4)* Sqrt[x])/(1 + x^2)^(1/4)] - 5*2^(1/4)*x^(5/2)*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)]))/(10*x^3*(1 + x^2)^(1/4))
Time = 0.37 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2467, 25, 1388, 368, 996, 961, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4+x^2}}{x^4 \left (x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^2} \int -\frac {\sqrt [4]{x^2+1}}{x^{7/2} \left (1-x^4\right )}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^2} \int \frac {\sqrt [4]{x^2+1}}{x^{7/2} \left (1-x^4\right )}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^2} \int \frac {1}{x^{7/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \int \frac {1}{x^3 \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 996 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \int \frac {\left (1-x^2\right )^2}{x^3 \left (1-2 x^2\right )}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 961 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \int \left (\frac {1}{x^3}-\frac {x}{2 x^2-1}\right )d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^4+x^2} \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4}}-\frac {1}{5 x^{5/2}}\right )}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
(-2*(x^2 + x^4)^(1/4)*(-1/5*1/x^(5/2) - ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^2) ^(1/4)]/(2*2^(3/4)) + ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)]/(2*2^(3/4 ))))/(Sqrt[x]*(1 + x^2)^(1/4))
3.10.87.3.1 Defintions of rubi rules used
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[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^( n_)), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.) , x_Symbol] :> With[{k = Denominator[p]}, Simp[k*(a^(p + (m + 1)/n)/n) Su bst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p + q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && RationalQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]
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 = 11.70 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.53
| method | result | size |
| pseudoelliptic | \(\frac {-5 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x^{3}-10 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}} x^{3}+8 x^{2} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+8 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{20 x^{3}}\) | \(115\) |
| trager | \(\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x +4 \sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x -4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{\left (1+x \right ) x \left (x -1\right )}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{4}+x^{2}}\, x -4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{\left (1+x \right ) x \left (x -1\right )}\right )}{4}\) | \(259\) |
| risch | \(\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{4}+2 x^{2}+1\right )}{5 x^{3} \left (x^{2}+1\right )}+\frac {\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{6}+4 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )-4 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}\, x^{2}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}-5 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-4 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}}{\left (x^{2}+1\right )^{2} \left (x -1\right ) \left (1+x \right )}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{6}-4 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )-4 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}\, x^{2}-2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}+5 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-4 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}}{\left (x^{2}+1\right )^{2} \left (x -1\right ) \left (1+x \right )}\right )}{4}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{2} \left (x^{2}+1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{2}+1\right )}\) | \(606\) |
1/20*(-5*ln((-2^(1/4)*x-(x^2*(x^2+1))^(1/4))/(2^(1/4)*x-(x^2*(x^2+1))^(1/4 )))*2^(1/4)*x^3-10*arctan(1/2*2^(3/4)/x*(x^2*(x^2+1))^(1/4))*2^(1/4)*x^3+8 *x^2*(x^2*(x^2+1))^(1/4)+8*(x^2*(x^2+1))^(1/4))/x^3
Result contains complex when optimal does not.
Time = 1.27 (sec) , antiderivative size = 326, normalized size of antiderivative = 4.35 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {5 \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x + 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 5 i \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + i \cdot 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (3 i \, x^{3} + i \, x\right )} - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + 5 i \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - i \cdot 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (-3 i \, x^{3} - i \, x\right )} - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 5 \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 64 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{160 \, x^{3}} \]
-1/160*(5*8^(3/4)*x^3*log((4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 + 8^(3/4)*sqrt( x^4 + x^2)*x + 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 5*I *8^(3/4)*x^3*log(-(4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 + I*8^(3/4)*sqrt(x^4 + x^2)*x - 8^(1/4)*(3*I*x^3 + I*x) - 4*(x^4 + x^2)^(3/4))/(x^3 - x)) + 5*I*8 ^(3/4)*x^3*log(-(4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 - I*8^(3/4)*sqrt(x^4 + x^ 2)*x - 8^(1/4)*(-3*I*x^3 - I*x) - 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 5*8^(3 /4)*x^3*log((4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 - 8^(3/4)*sqrt(x^4 + x^2)*x - 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 64*(x^4 + x^2)^(1 /4)*(x^2 + 1))/x^3
\[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}} \,d x } \]
-2/585*(32*x^7 - 8*x^5 + 5*x^3 + 45*x)*(x^2 + 1)^(1/4)/(x^(15/2) - x^(7/2) ) - integrate(8/585*(32*x^6 - 8*x^4 + 5*x^2 + 45)*(x^2 + 1)^(1/4)/(x^(23/2 ) - 2*x^(15/2) + x^(7/2)), x)
Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\frac {2}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} - \frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) \]
2/5*(1/x^2 + 1)^(5/4) - 1/2*2^(1/4)*arctan(1/2*2^(3/4)*(1/x^2 + 1)^(1/4)) - 1/4*2^(1/4)*log(2^(1/4) + (1/x^2 + 1)^(1/4)) + 1/4*2^(1/4)*log(abs(-2^(1 /4) + (1/x^2 + 1)^(1/4)))
Timed out. \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\int \frac {{\left (x^4+x^2\right )}^{1/4}}{x^4-x^8} \,d x \]