Integrand size = 22, antiderivative size = 79 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {\left (x^2+x^4\right )^{3/4}}{x \left (1+x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2 \sqrt [4]{2}} \]
(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 = 1.29 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {\sqrt {x} \left (4 \sqrt {x}-2^{3/4} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-2^{3/4} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{4 \sqrt [4]{x^2+x^4}} \]
(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)]))/(4*(x^2 + x^4)^(1/4))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.57, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2467, 25, 1388, 368, 1012}
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.
| \(\displaystyle \int \frac {x^2}{\left (x^4-1\right ) \sqrt [4]{x^4+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int -\frac {x^{3/2}}{\sqrt [4]{x^2+1} \left (1-x^4\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^{3/2}}{\sqrt [4]{x^2+1} \left (1-x^4\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^{3/2}}{\left (1-x^2\right ) \left (x^2+1\right )^{5/4}}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^2}{\left (1-x^2\right ) \left (x^2+1\right )^{5/4}}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle -\frac {2 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {9}{4},\frac {2 x^2}{x^2+1}\right )}{5 \left (x^2+1\right ) \sqrt [4]{x^4+x^2}}\) |
3.11.49.3.1 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_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 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 = 8.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.43
| method | result | size |
| pseudoelliptic | \(\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+8 x}{8 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(113\) |
| risch | \(\frac {x}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{\left (1+x \right ) x \left (x -1\right )}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (1+x \right ) x \left (x -1\right )}\right )}{8}\) | \(242\) |
| trager | \(\frac {\left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (1+x \right ) x \left (x -1\right )}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{\left (1+x \right ) x \left (x -1\right )}\right )}{8}\) | \(248\) |
1/8*(2*arctan(1/2*2^(3/4)/x*(x^2*(x^2+1))^(1/4))*2^(3/4)*(x^2*(x^2+1))^(1/ 4)-ln((-2^(1/4)*x-(x^2*(x^2+1))^(1/4))/(2^(1/4)*x-(x^2*(x^2+1))^(1/4)))*2^ (3/4)*(x^2*(x^2+1))^(1/4)+8*x)/(x^2*(x^2+1))^(1/4)
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 341, normalized size of antiderivative = 4.32 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{3} + x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 2^{\frac {3}{4}} {\left (x^{3} + x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 2^{\frac {3}{4}} {\left (-i \, x^{3} - i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{3} + i \, x\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 2^{\frac {3}{4}} {\left (i \, x^{3} + i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{3} - i \, x\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 16 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{16 \, {\left (x^{3} + x\right )}} \]
-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)
\[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{2}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2}}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
-2/3*(x^3 + x)*x^(3/2)/((x^4 - 1)*(x^2 + 1)^(1/4)) - integrate(8/3*(x^2 + 1)^(3/4)*x^(3/2)/(x^8 - 2*x^4 + 1), x)
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {1}{{\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \]
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(abs(-2^(1/4) + (1/x^2 + 1)^(1/4)) ) + 1/(1/x^2 + 1)^(1/4)
Timed out. \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^2}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-1\right )} \,d x \]