Integrand size = 29, antiderivative size = 81 \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \]
1/3*x/(x^4+1)^(1/4)-5/36*arctan(1/2*3^(1/4)*2^(3/4)*x/(x^4+1)^(1/4))*3^(3/ 4)*2^(1/4)-5/36*arctanh(1/2*3^(1/4)*2^(3/4)*x/(x^4+1)^(1/4))*3^(3/4)*2^(1/ 4)
Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \]
x/(3*(1 + x^4)^(1/4)) - (5*ArcTan[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(6*2^( 3/4)*3^(1/4)) - (5*ArcTanh[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(6*2^(3/4)*3^ (1/4))
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1387, 1024, 27, 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.
| \(\displaystyle \int \frac {2 x^4+1}{\sqrt [4]{x^4+1} \left (x^8-x^4-2\right )} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {2 x^4+1}{\left (x^4-2\right ) \left (x^4+1\right )^{5/4}}dx\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {1}{3} \int -\frac {5}{\left (2-x^4\right ) \sqrt [4]{x^4+1}}dx+\frac {x}{3 \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5}{3} \int \frac {1}{\left (2-x^4\right ) \sqrt [4]{x^4+1}}dx\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5}{3} \int \frac {1}{2-\frac {3 x^4}{x^4+1}}d\frac {x}{\sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5}{3} \left (\frac {\int \frac {1}{\sqrt {2}-\frac {\sqrt {3} x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}}{2 \sqrt {2}}+\frac {\int \frac {1}{\frac {\sqrt {3} x^2}{\sqrt {x^4+1}}+\sqrt {2}}d\frac {x}{\sqrt [4]{x^4+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5}{3} \left (\frac {\int \frac {1}{\sqrt {2}-\frac {\sqrt {3} x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}}{2 \sqrt {2}}+\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4} \sqrt [4]{3}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5}{3} \left (\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4} \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4} \sqrt [4]{3}}\right )\) |
x/(3*(1 + x^4)^(1/4)) - (5*(ArcTan[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^( 3/4)*3^(1/4)) + ArcTanh[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^(3/4)*3^(1/4 ))))/3
3.11.91.3.1 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[((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_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(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, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 6.91 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(\frac {10 \arctan \left (\frac {3^{\frac {3}{4}} 2^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{3 x}\right ) 2^{\frac {1}{4}} 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-5 \ln \left (\frac {-2^{\frac {3}{4}} 3^{\frac {1}{4}} x -2 \left (x^{4}+1\right )^{\frac {1}{4}}}{2^{\frac {3}{4}} 3^{\frac {1}{4}} x -2 \left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+24 x}{72 \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(104\) |
| trager | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
| risch | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
1/72*(10*arctan(1/3*3^(3/4)*2^(1/4)/x*(x^4+1)^(1/4))*2^(1/4)*3^(3/4)*(x^4+ 1)^(1/4)-5*ln((-2^(3/4)*3^(1/4)*x-2*(x^4+1)^(1/4))/(2^(3/4)*3^(1/4)*x-2*(x ^4+1)^(1/4)))*2^(1/4)*3^(3/4)*(x^4+1)^(1/4)+24*x)/(x^4+1)^(1/4)
Result contains complex when optimal does not.
Time = 3.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.86 \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=-\frac {5 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 24 \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 24^{\frac {3}{4}} {\left (5 \, x^{4} + 2\right )} + 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) + 5 \cdot 24^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (-\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 24 i \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (5 i \, x^{4} + 2 i\right )} - 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) + 5 \cdot 24^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (-\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 24 i \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (-5 i \, x^{4} - 2 i\right )} - 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) - 5 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 24 \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (5 \, x^{4} + 2\right )} + 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) - 192 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{576 \, {\left (x^{4} + 1\right )}} \]
-1/576*(5*24^(3/4)*(x^4 + 1)*log((24*sqrt(6)*(x^4 + 1)^(1/4)*x^3 + 24*24^( 1/4)*sqrt(x^4 + 1)*x^2 + 24^(3/4)*(5*x^4 + 2) + 48*(x^4 + 1)^(3/4)*x)/(x^4 - 2)) + 5*24^(3/4)*(I*x^4 + I)*log(-(24*sqrt(6)*(x^4 + 1)^(1/4)*x^3 + 24* I*24^(1/4)*sqrt(x^4 + 1)*x^2 - 24^(3/4)*(5*I*x^4 + 2*I) - 48*(x^4 + 1)^(3/ 4)*x)/(x^4 - 2)) + 5*24^(3/4)*(-I*x^4 - I)*log(-(24*sqrt(6)*(x^4 + 1)^(1/4 )*x^3 - 24*I*24^(1/4)*sqrt(x^4 + 1)*x^2 - 24^(3/4)*(-5*I*x^4 - 2*I) - 48*( x^4 + 1)^(3/4)*x)/(x^4 - 2)) - 5*24^(3/4)*(x^4 + 1)*log((24*sqrt(6)*(x^4 + 1)^(1/4)*x^3 - 24*24^(1/4)*sqrt(x^4 + 1)*x^2 - 24^(3/4)*(5*x^4 + 2) + 48* (x^4 + 1)^(3/4)*x)/(x^4 - 2)) - 192*(x^4 + 1)^(3/4)*x)/(x^4 + 1)
\[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int \frac {2 x^{4} + 1}{\left (x^{4} - 2\right ) \left (x^{4} + 1\right )^{\frac {5}{4}}}\, dx \]
\[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{4} + 1}{{\left (x^{8} - x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{4} + 1}{{\left (x^{8} - x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int -\frac {2\,x^4+1}{{\left (x^4+1\right )}^{1/4}\,\left (-x^8+x^4+2\right )} \,d x \]