Integrand size = 29, antiderivative size = 76 \[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\frac {\left (1-14 x^4\right ) \sqrt [4]{-1-x^4}}{5 x^5}+\frac {3}{2} \arctan \left (\frac {x \left (-1-x^4\right )^{3/4}}{1+x^4}\right )-\frac {3}{2} \text {arctanh}\left (\frac {x \left (-1-x^4\right )^{3/4}}{1+x^4}\right ) \]
1/5*(-14*x^4+1)*(-x^4-1)^(1/4)/x^5+3/2*arctan(x*(-x^4-1)^(3/4)/(x^4+1))-3/ 2*arctanh(x*(-x^4-1)^(3/4)/(x^4+1))
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\frac {\left (1-14 x^4\right ) \sqrt [4]{-1-x^4}}{5 x^5}-\frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{-1-x^4}}\right )+\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1-x^4}}\right ) \]
((1 - 14*x^4)*(-1 - x^4)^(1/4))/(5*x^5) - (3*ArcTan[x/(-1 - x^4)^(1/4)])/2 + (3*ArcTanh[x/(-1 - x^4)^(1/4)])/2
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1050, 25, 1053, 27, 996, 827, 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.
| \(\displaystyle \int \frac {\sqrt [4]{-x^4-1} \left (x^4-1\right )}{x^6 \left (2 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{5} \int -\frac {13 x^4+14}{x^2 \left (-x^4-1\right )^{3/4} \left (2 x^4+1\right )}dx+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [4]{-x^4-1}}{5 x^5}-\frac {1}{5} \int \frac {13 x^4+14}{x^2 \left (-x^4-1\right )^{3/4} \left (2 x^4+1\right )}dx\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{5} \left (-\int -\frac {15 x^2}{\left (-x^4-1\right )^{3/4} \left (2 x^4+1\right )}dx-\frac {14 \sqrt [4]{-x^4-1}}{x}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (15 \int \frac {x^2}{\left (-x^4-1\right )^{3/4} \left (2 x^4+1\right )}dx-\frac {14 \sqrt [4]{-x^4-1}}{x}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
| \(\Big \downarrow \) 996 |
| \(\displaystyle \frac {1}{5} \left (15 \int \frac {x^2}{\sqrt {-x^4-1} \left (1-\frac {x^4}{-x^4-1}\right )}d\frac {x}{\sqrt [4]{-x^4-1}}-\frac {14 \sqrt [4]{-x^4-1}}{x}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {1}{5} \left (15 \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {-x^4-1}}}d\frac {x}{\sqrt [4]{-x^4-1}}-\frac {1}{2} \int \frac {1}{\frac {x^2}{\sqrt {-x^4-1}}+1}d\frac {x}{\sqrt [4]{-x^4-1}}\right )-\frac {14 \sqrt [4]{-x^4-1}}{x}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{5} \left (15 \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {-x^4-1}}}d\frac {x}{\sqrt [4]{-x^4-1}}-\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-x^4-1}}\right )\right )-\frac {14 \sqrt [4]{-x^4-1}}{x}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} \left (15 \left (\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^4-1}}\right )-\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-x^4-1}}\right )\right )-\frac {14 \sqrt [4]{-x^4-1}}{x}\right )+\frac {\sqrt [4]{-x^4-1}}{5 x^5}\) |
(-1 - x^4)^(1/4)/(5*x^5) + ((-14*(-1 - x^4)^(1/4))/x + 15*(-1/2*ArcTan[x/( -1 - x^4)^(1/4)] + ArcTanh[x/(-1 - x^4)^(1/4)]/2))/5
3.11.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^n*(m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1 ))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && G tQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 4.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25
| method | result | size |
| pseudoelliptic | \(\frac {-15 \ln \left (\frac {\left (-x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}+15 \ln \left (\frac {\left (-x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right ) x^{5}+30 \arctan \left (\frac {\left (-x^{4}-1\right )^{\frac {1}{4}}}{x}\right ) x^{5}-56 \left (-x^{4}-1\right )^{\frac {1}{4}} x^{4}+4 \left (-x^{4}-1\right )^{\frac {1}{4}}}{20 x^{5}}\) | \(95\) |
| trager | \(-\frac {\left (14 x^{4}-1\right ) \left (-x^{4}-1\right )^{\frac {1}{4}}}{5 x^{5}}+\frac {3 \ln \left (\frac {2 \left (-x^{4}-1\right )^{\frac {3}{4}} x +2 \sqrt {-x^{4}-1}\, x^{2}+2 \left (-x^{4}-1\right )^{\frac {1}{4}} x^{3}-1}{2 x^{4}+1}\right )}{4}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-1}\, x^{2}+2 \left (-x^{4}-1\right )^{\frac {3}{4}} x -2 \left (-x^{4}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{4}+1}\right )}{4}\) | \(150\) |
| risch | \(\frac {14 x^{8}+13 x^{4}-1}{5 x^{5} \left (-x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {3 \ln \left (\frac {2 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{9}+2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{6}+x^{8}+2 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x^{3}+4 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{2}+2 x^{4}+2 \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x +1}{\left (x^{4}+1\right )^{2} \left (2 x^{4}+1\right )}\right )}{4}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{9}-2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{6}+x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {-x^{12}-3 x^{8}-3 x^{4}-1}\, x^{2}+2 x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (-x^{12}-3 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} x +1}{\left (x^{4}+1\right )^{2} \left (2 x^{4}+1\right )}\right )}{4}\right ) {\left (-\left (x^{4}+1\right )^{3}\right )}^{\frac {1}{4}}}{\left (-x^{4}-1\right )^{\frac {3}{4}}}\) | \(425\) |
1/20*(-15*ln(((-x^4-1)^(1/4)-x)/x)*x^5+15*ln(((-x^4-1)^(1/4)+x)/x)*x^5+30* arctan((-x^4-1)^(1/4)/x)*x^5-56*(-x^4-1)^(1/4)*x^4+4*(-x^4-1)^(1/4))/x^5
Result contains complex when optimal does not.
Time = 2.02 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.64 \[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\frac {30 \, x^{5} \log \left (-\frac {2 \, {\left (2 \, {\left (-x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {-x^{4} - 1} x^{2} + 2 \, {\left (-x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{2 \, x^{4} + 1}\right ) - 15 i \, x^{5} \log \left (-\frac {2 \, {\left (2 i \, {\left (-x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {-x^{4} - 1} x^{2} - 2 i \, {\left (-x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{2 \, x^{4} + 1}\right ) + 15 i \, x^{5} \log \left (-\frac {2 \, {\left (-2 i \, {\left (-x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {-x^{4} - 1} x^{2} + 2 i \, {\left (-x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{2 \, x^{4} + 1}\right ) - 8 \, {\left (14 \, x^{4} - 1\right )} {\left (-x^{4} - 1\right )}^{\frac {1}{4}}}{40 \, x^{5}} \]
1/40*(30*x^5*log(-2*(2*(-x^4 - 1)^(1/4)*x^3 + 2*sqrt(-x^4 - 1)*x^2 + 2*(-x ^4 - 1)^(3/4)*x - 1)/(2*x^4 + 1)) - 15*I*x^5*log(-2*(2*I*(-x^4 - 1)^(1/4)* x^3 - 2*sqrt(-x^4 - 1)*x^2 - 2*I*(-x^4 - 1)^(3/4)*x - 1)/(2*x^4 + 1)) + 15 *I*x^5*log(-2*(-2*I*(-x^4 - 1)^(1/4)*x^3 - 2*sqrt(-x^4 - 1)*x^2 + 2*I*(-x^ 4 - 1)^(3/4)*x - 1)/(2*x^4 + 1)) - 8*(14*x^4 - 1)*(-x^4 - 1)^(1/4))/x^5
\[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{- x^{4} - 1}}{x^{6} \cdot \left (2 x^{4} + 1\right )}\, dx \]
\[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 1\right )} {\left (-x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 1\right )} {\left (-x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{-1-x^4} \left (-1+x^4\right )}{x^6 \left (1+2 x^4\right )} \, dx=\int \frac {\left (x^4-1\right )\,{\left (-x^4-1\right )}^{1/4}}{x^6\,\left (2\,x^4+1\right )} \,d x \]