Integrand size = 17, antiderivative size = 67 \[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \]
-1/2*x/(x^4+1)^(1/4)-1/8*arctan(2^(1/4)*x/(x^4+1)^(1/4))*2^(3/4)-1/8*arcta nh(2^(1/4)*x/(x^4+1)^(1/4))*2^(3/4)
Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \]
-1/2*x/(1 + x^4)^(1/4) - ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(4*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(4*2^(1/4))
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {1388, 907, 25, 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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{x^4+1} \left (x^8-1\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {1}{\left (x^4-1\right ) \left (x^4+1\right )^{5/4}}dx\) |
| \(\Big \downarrow \) 907 |
| \(\displaystyle \frac {1}{2} \int -\frac {1}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}dx-\frac {x}{2 \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {1}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}dx-\frac {x}{2 \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle -\frac {1}{2} \int \frac {1}{1-\frac {2 x^4}{x^4+1}}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {x}{2 \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {2} x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} x^2}{\sqrt {x^4+1}}+1}d\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {x}{2 \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {2} x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\right )-\frac {x}{2 \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}\right )-\frac {x}{2 \sqrt [4]{x^4+1}}\) |
-1/2*x/(1 + x^4)^(1/4) + (-1/2*ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/2^(1/4) - ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4)))/2
3.9.88.3.1 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q} , x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || ! LtQ[q, -1]) && NeQ[p, -1]
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]))
Time = 6.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.25
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+\ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+8 x}{16 \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(84\) |
| risch | \(-\frac {x}{2 \left (x^{4}+1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+3 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-4 \left (x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\) | \(237\) |
| trager | \(-\frac {x}{2 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+3 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\) | \(238\) |
-1/16*(-2*arctan(1/2*2^(3/4)/x*(x^4+1)^(1/4))*2^(3/4)*(x^4+1)^(1/4)+ln((2^ (1/4)*x+(x^4+1)^(1/4))/(-2^(1/4)*x+(x^4+1)^(1/4)))*2^(3/4)*(x^4+1)^(1/4)+8 *x)/(x^4+1)^(1/4)
Result contains complex when optimal does not.
Time = 3.04 (sec) , antiderivative size = 312, normalized size of antiderivative = 4.66 \[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - 2^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{4} + i\right )} - 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - 2^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{4} - i\right )} - 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - 2^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{4} + 1\right )}} \]
-1/32*(2^(3/4)*(x^4 + 1)*log((4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 4*2^(1/4)*sq rt(x^4 + 1)*x^2 + 2^(3/4)*(3*x^4 + 1) + 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 2^(3/4)*(-I*x^4 - I)*log(-(4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 4*I*2^(1/4)*sqr t(x^4 + 1)*x^2 - 2^(3/4)*(3*I*x^4 + I) - 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 2^(3/4)*(I*x^4 + I)*log(-(4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 - 4*I*2^(1/4)*sqr t(x^4 + 1)*x^2 - 2^(3/4)*(-3*I*x^4 - I) - 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 2^(3/4)*(x^4 + 1)*log((4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 - 4*2^(1/4)*sqrt(x^ 4 + 1)*x^2 - 2^(3/4)*(3*x^4 + 1) + 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) + 16*(x ^4 + 1)^(3/4)*x)/(x^4 + 1)
\[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int \frac {1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )^{\frac {5}{4}}}\, dx \]
\[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int \frac {1}{{\left (x^4+1\right )}^{1/4}\,\left (x^8-1\right )} \,d x \]