Integrand size = 27, antiderivative size = 64 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {x}{\sqrt [4]{1+x^4}}+\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}} \]
x/(x^4+1)^(1/4)+1/6*arctan(3^(1/4)*x/(x^4+1)^(1/4))*3^(3/4)+1/6*arctanh(3^ (1/4)*x/(x^4+1)^(1/4))*3^(3/4)
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {x}{\sqrt [4]{1+x^4}}+\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}} \]
x/(1 + x^4)^(1/4) + ArcTan[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4)) + ArcT anh[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4))
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, 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.
| \(\displaystyle \int \frac {x^4-2}{\sqrt [4]{x^4+1} \left (2 x^8+x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {x^4-2}{\left (x^4+1\right )^{5/4} \left (2 x^4-1\right )}dx\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {1}{3} \int \frac {3}{\left (1-2 x^4\right ) \sqrt [4]{x^4+1}}dx+\frac {x}{\sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\left (1-2 x^4\right ) \sqrt [4]{x^4+1}}dx+\frac {x}{\sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \int \frac {1}{1-\frac {3 x^4}{x^4+1}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {x}{\sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{1-\frac {\sqrt {3} x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} x^2}{\sqrt {x^4+1}}+1}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {x}{\sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{1-\frac {\sqrt {3} x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {x}{\sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {x}{\sqrt [4]{x^4+1}}\) |
x/(1 + x^4)^(1/4) + ArcTan[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4)) + ArcT anh[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4))
3.9.50.3.1 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 = 5.87 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.31
| method | result | size |
| pseudoelliptic | \(\frac {-2 \arctan \left (\frac {3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{3 x}\right ) 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+\ln \left (\frac {3^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}{-3^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+12 x}{12 \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(84\) |
| trager | \(\frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{3} x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) x^{4}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )}{2 x^{4}-1}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right )}{2 x^{4}-1}\right )}{12}\) | \(223\) |
| risch | \(\frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) x^{4}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )}{2 x^{4}-1}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right )}{2 x^{4}-1}\right )}{12}\) | \(224\) |
1/12*(-2*arctan(1/3*3^(3/4)/x*(x^4+1)^(1/4))*3^(3/4)*(x^4+1)^(1/4)+ln((3^( 1/4)*x+(x^4+1)^(1/4))/(-3^(1/4)*x+(x^4+1)^(1/4)))*3^(3/4)*(x^4+1)^(1/4)+12 *x)/(x^4+1)^(1/4)
Result contains complex when optimal does not.
Time = 2.99 (sec) , antiderivative size = 318, normalized size of antiderivative = 4.97 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 3^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 3^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (-\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 i \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (4 i \, x^{4} + i\right )} - 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 3^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (-\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 i \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (-4 i \, x^{4} - i\right )} - 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) - 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 24 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{24 \, {\left (x^{4} + 1\right )}} \]
1/24*(3^(3/4)*(x^4 + 1)*log((6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 + 6*3^(1/4)*sqr t(x^4 + 1)*x^2 + 3^(3/4)*(4*x^4 + 1) + 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) + 3^(3/4)*(I*x^4 + I)*log(-(6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 + 6*I*3^(1/4)*sqr t(x^4 + 1)*x^2 - 3^(3/4)*(4*I*x^4 + I) - 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) + 3^(3/4)*(-I*x^4 - I)*log(-(6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 - 6*I*3^(1/4)* sqrt(x^4 + 1)*x^2 - 3^(3/4)*(-4*I*x^4 - I) - 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) - 3^(3/4)*(x^4 + 1)*log((6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 - 6*3^(1/4)*sq rt(x^4 + 1)*x^2 - 3^(3/4)*(4*x^4 + 1) + 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) + 24*(x^4 + 1)^(3/4)*x)/(x^4 + 1)
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int \frac {x^{4} - 2}{\left (x^{4} + 1\right )^{\frac {5}{4}} \cdot \left (2 x^{4} - 1\right )}\, dx \]
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (2 \, x^{8} + x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (2 \, x^{8} + x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int \frac {x^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (2\,x^8+x^4-1\right )} \,d x \]