Integrand size = 29, antiderivative size = 111 \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}} \]
-1/4*arctan(1/2*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*2^(3/4)/x/(x^4+1) )*2^(3/4)+1/4*arctanh(1/2*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*2^(3/4) /x/(x^4+1))*2^(3/4)
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \left (\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )\right )}{2 \sqrt [4]{2} \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \]
((-1 + x^4)^(1/4)*(1 + x^4)*(ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)] + ArcTan h[(2^(1/4)*x)/(-1 + x^4)^(1/4)]))/(2*2^(1/4)*((-1 + x^4)*(1 + x^4)^4)^(1/4 ))
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {7239, 2058, 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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{x^{20}+3 x^{16}+2 x^{12}-2 x^8-3 x^4-1}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\sqrt [4]{\left (x^4-1\right ) \left (x^4+1\right )^4}}dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \int \frac {1}{\sqrt [4]{x^4-1} \left (x^4+1\right )}dx}{\sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \int \frac {1}{1-\frac {2 x^4}{x^4-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{\sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {2} x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} x^2}{\sqrt {x^4-1}}+1}d\frac {x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {2} x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2}}\right )}{\sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2}}\right )}{\sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}\) |
((-1 + x^4)^(1/4)*(1 + x^4)*(ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/(2*2^(1/ 4)) + ArcTanh[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/(2*2^(1/4))))/(-((1 - x^4)*(1 + x^4)^4))^(1/4)
3.17.48.3.1 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[((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[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.05 (sec) , antiderivative size = 639, normalized size of antiderivative = 5.76
| method | result | size |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-3 x^{16} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{11}-8 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{12}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{6}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{7}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{8}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {3 x^{16} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{11}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{12}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{6}-4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{7}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{8}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}\) | \(639\) |
-1/8*RootOf(_Z^4-8)*ln((-3*x^16*RootOf(_Z^4-8)+2*(x^20+3*x^16+2*x^12-2*x^8 -3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^11-8*RootOf(_Z^4-8)*x^12-(x^20+3*x^16+2 *x^12-2*x^8-3*x^4-1)^(1/2)*RootOf(_Z^4-8)^3*x^6+4*(x^20+3*x^16+2*x^12-2*x^ 8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^7-6*RootOf(_Z^4-8)*x^8-(x^20+3*x^16+2* x^12-2*x^8-3*x^4-1)^(1/2)*RootOf(_Z^4-8)^3*x^2+2*(x^20+3*x^16+2*x^12-2*x^8 -3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^3+4*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^ (3/4)*x+RootOf(_Z^4-8))/(x^4+1)^4)+1/8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*ln((3 *x^16*RootOf(_Z^2+RootOf(_Z^4-8)^2)-2*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^( 1/4)*RootOf(_Z^4-8)^2*x^11+8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^12-(x^20+3*x^ 16+2*x^12-2*x^8-3*x^4-1)^(1/2)*RootOf(_Z^4-8)^2*RootOf(_Z^2+RootOf(_Z^4-8) ^2)*x^6-4*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^7+6* RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^8-(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/2) *RootOf(_Z^4-8)^2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^2-2*(x^20+3*x^16+2*x^12- 2*x^8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^3+4*(x^20+3*x^16+2*x^12-2*x^8-3*x^ 4-1)^(3/4)*x-RootOf(_Z^2+RootOf(_Z^4-8)^2))/(x^4+1)^4)
Result contains complex when optimal does not.
Time = 2.89 (sec) , antiderivative size = 666, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} + 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} - 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) + \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (3 i \, x^{16} + 8 i \, x^{12} + 6 i \, x^{8} - i\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (i \, x^{6} + i \, x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) - \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (-3 i \, x^{16} - 8 i \, x^{12} - 6 i \, x^{8} + i\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (-i \, x^{6} - i \, x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) \]
1/16*2^(3/4)*log((2^(3/4)*(3*x^16 + 8*x^12 + 6*x^8 - 1) + 4*sqrt(2)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(1/4)*(x^11 + 2*x^7 + x^3) + 4*2^(1 /4)*sqrt(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)*(x^6 + x^2) + 4*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(3/4)*x)/(x^16 + 4*x^12 + 6*x^8 + 4*x^4 + 1)) - 1/16*2^(3/4)*log(-(2^(3/4)*(3*x^16 + 8*x^12 + 6*x^8 - 1) - 4 *sqrt(2)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(1/4)*(x^11 + 2*x^7 + x^3) + 4*2^(1/4)*sqrt(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)*(x^6 + x^2) - 4*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(3/4)*x)/(x^16 + 4* x^12 + 6*x^8 + 4*x^4 + 1)) + 1/16*I*2^(3/4)*log((2^(3/4)*(3*I*x^16 + 8*I*x ^12 + 6*I*x^8 - I) - 4*sqrt(2)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1 )^(1/4)*(x^11 + 2*x^7 + x^3) - 4*2^(1/4)*sqrt(x^20 + 3*x^16 + 2*x^12 - 2*x ^8 - 3*x^4 - 1)*(I*x^6 + I*x^2) + 4*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^ 4 - 1)^(3/4)*x)/(x^16 + 4*x^12 + 6*x^8 + 4*x^4 + 1)) - 1/16*I*2^(3/4)*log( (2^(3/4)*(-3*I*x^16 - 8*I*x^12 - 6*I*x^8 + I) - 4*sqrt(2)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(1/4)*(x^11 + 2*x^7 + x^3) - 4*2^(1/4)*sqrt(x ^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)*(-I*x^6 - I*x^2) + 4*(x^20 + 3* x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(3/4)*x)/(x^16 + 4*x^12 + 6*x^8 + 4*x^4 + 1))
\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int \frac {1}{\sqrt [4]{x^{20} + 3 x^{16} + 2 x^{12} - 2 x^{8} - 3 x^{4} - 1}}\, dx \]
\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int { \frac {1}{{\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int { \frac {1}{{\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int \frac {1}{{\left (x^{20}+3\,x^{16}+2\,x^{12}-2\,x^8-3\,x^4-1\right )}^{1/4}} \,d x \]