Integrand size = 18, antiderivative size = 102 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\frac {\sqrt [3]{1+x^4} \left (9-3 x^4-4 x^8\right )}{36 x^{12}}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2}{27} \log \left (-1+\sqrt [3]{1+x^4}\right )+\frac {1}{27} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
1/36*(x^4+1)^(1/3)*(-4*x^8-3*x^4+9)/x^12+2/27*3^(1/2)*arctan(1/3*3^(1/2)+2 /3*(x^4+1)^(1/3)*3^(1/2))-2/27*ln(-1+(x^4+1)^(1/3))+1/27*ln(1+(x^4+1)^(1/3 )+(x^4+1)^(2/3))
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\frac {1}{108} \left (-\frac {3 \sqrt [3]{1+x^4} \left (-9+3 x^4+4 x^8\right )}{x^{12}}+8 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )-8 \log \left (-1+\sqrt [3]{1+x^4}\right )+4 \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )\right ) \]
((-3*(1 + x^4)^(1/3)*(-9 + 3*x^4 + 4*x^8))/x^12 + 8*Sqrt[3]*ArcTan[(1 + 2* (1 + x^4)^(1/3))/Sqrt[3]] - 8*Log[-1 + (1 + x^4)^(1/3)] + 4*Log[1 + (1 + x ^4)^(1/3) + (1 + x^4)^(2/3)])/108
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {948, 25, 87, 51, 52, 69, 16, 1083, 217}
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.
| \(\displaystyle \int \frac {\left (x^4-3\right ) \sqrt [3]{x^4+1}}{x^{13}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{4} \int -\frac {\left (3-x^4\right ) \sqrt [3]{x^4+1}}{x^{16}}dx^4\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} \int \frac {\left (3-x^4\right ) \sqrt [3]{x^4+1}}{x^{16}}dx^4\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \int \frac {\sqrt [3]{x^4+1}}{x^{12}}dx^4+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \left (\frac {1}{6} \int \frac {1}{x^8 \left (x^4+1\right )^{2/3}}dx^4-\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \int \frac {1}{x^4 \left (x^4+1\right )^{2/3}}dx^4-\frac {\sqrt [3]{x^4+1}}{x^4}\right )-\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^4+1}}d\sqrt [3]{x^4+1}-\frac {3}{2} \int \frac {1}{x^8+\sqrt [3]{x^4+1}+1}d\sqrt [3]{x^4+1}-\frac {1}{2} \log \left (x^4\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )-\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{x^8+\sqrt [3]{x^4+1}+1}d\sqrt [3]{x^4+1}-\frac {1}{2} \log \left (x^4\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^4+1}\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )-\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (3 \int \frac {1}{-x^8-3}d\left (2 \sqrt [3]{x^4+1}+1\right )-\frac {1}{2} \log \left (x^4\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^4+1}\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )-\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (\frac {8}{3} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )-\frac {\log \left (x^4\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^4+1}\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )-\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )+\frac {\left (x^4+1\right )^{4/3}}{x^{12}}\right )\) |
((1 + x^4)^(4/3)/x^12 + (8*(-1/2*(1 + x^4)^(1/3)/x^8 + (-((1 + x^4)^(1/3)/ x^4) - (2*(-(Sqrt[3]*ArcTan[(1 + 2*(1 + x^4)^(1/3))/Sqrt[3]]) - Log[x^4]/2 + (3*Log[1 - (1 + x^4)^(1/3)])/2))/3)/6))/3)/4
3.15.42.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {4 x^{12}+7 x^{8}-6 x^{4}-9}{36 x^{12} \left (x^{4}+1\right )^{\frac {2}{3}}}-\frac {2 \left (-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right )}\) | \(76\) |
| pseudoelliptic | \(\frac {\left (-12 x^{8}-9 x^{4}+27\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+4 x^{12} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )+\ln \left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )-2 \ln \left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )\right )}{108 {\left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )}^{3} {\left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(110\) |
| meijerg | \(\frac {\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], -x^{4}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{12}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{8}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}\) | \(128\) |
| trager | \(-\frac {\left (4 x^{8}+3 x^{4}-9\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{36 x^{12}}-\frac {2 \ln \left (-\frac {333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-393 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+60 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+48 \left (x^{4}+1\right )^{\frac {2}{3}}-384 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-165 \left (x^{4}+1\right )^{\frac {1}{3}}+80}{x^{4}}\right )}{27}+\frac {2 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+162 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+40 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-495 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-165 \left (x^{4}+1\right )^{\frac {2}{3}}+195 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+48 \left (x^{4}+1\right )^{\frac {1}{3}}+100}{x^{4}}\right )}{9}\) | \(306\) |
-1/36*(4*x^12+7*x^8-6*x^4-9)/x^12/(x^4+1)^(2/3)-2/27/GAMMA(2/3)*(-2/3*GAMM A(2/3)*x^4*hypergeom([1,1,5/3],[2,2],-x^4)+(1/6*Pi*3^(1/2)-3/2*ln(3)+4*ln( x))*GAMMA(2/3))
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\frac {8 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 4 \, x^{12} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 8 \, x^{12} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (4 \, x^{8} + 3 \, x^{4} - 9\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{108 \, x^{12}} \]
1/108*(8*sqrt(3)*x^12*arctan(2/3*sqrt(3)*(x^4 + 1)^(1/3) + 1/3*sqrt(3)) + 4*x^12*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) - 8*x^12*log((x^4 + 1)^( 1/3) - 1) - 3*(4*x^8 + 3*x^4 - 9)*(x^4 + 1)^(1/3))/x^12
Timed out. \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{72 \, {\left (3 \, x^{4} + {\left (x^{4} + 1\right )}^{3} - 3 \, {\left (x^{4} + 1\right )}^{2} + 2\right )}} + \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{24 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} + \frac {1}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
2/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/72*(5*(x^4 + 1)^(7/3) - 13*(x^4 + 1)^(4/3) - 10*(x^4 + 1)^(1/3))/(3*x^4 + (x^4 + 1)^3 - 3*(x^4 + 1)^2 + 2) + 1/24*((x^4 + 1)^(4/3) + 2*(x^4 + 1)^(1/3))/(2*x^4 - (x^4 + 1)^2 + 1) + 1/27*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) - 2/27* log((x^4 + 1)^(1/3) - 1)
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {4 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 5 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 8 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{36 \, x^{12}} + \frac {1}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{27} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
2/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/36*(4*(x^4 + 1)^(7/3) - 5*(x^4 + 1)^(4/3) - 8*(x^4 + 1)^(1/3))/x^12 + 1/27*log((x^4 + 1 )^(2/3) + (x^4 + 1)^(1/3) + 1) - 2/27*log((x^4 + 1)^(1/3) - 1)
Time = 7.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.26 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^{13}} \, dx=\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {13\,{\left (x^4+1\right )}^{4/3}}{72}-\frac {5\,{\left (x^4+1\right )}^{7/3}}{72}}{{\left (x^4+1\right )}^3-3\,{\left (x^4+1\right )}^2+3\,x^4+2}-\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{1296}-\frac {25}{1296}\right )}{108}-\frac {\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{144}-\frac {1}{144}\right )}{36}+\frac {\frac {{\left (x^4+1\right )}^{1/3}}{12}+\frac {{\left (x^4+1\right )}^{4/3}}{24}}{2\,x^4-{\left (x^4+1\right )}^2+1}-\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {1}{8}-\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )+\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{12}+\frac {5}{24}-\frac {\sqrt {3}\,5{}\mathrm {i}}{24}\right )\,\left (-\frac {5}{216}+\frac {\sqrt {3}\,5{}\mathrm {i}}{216}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{12}+\frac {5}{24}+\frac {\sqrt {3}\,5{}\mathrm {i}}{24}\right )\,\left (\frac {5}{216}+\frac {\sqrt {3}\,5{}\mathrm {i}}{216}\right ) \]
((5*(x^4 + 1)^(1/3))/36 + (13*(x^4 + 1)^(4/3))/72 - (5*(x^4 + 1)^(7/3))/72 )/((x^4 + 1)^3 - 3*(x^4 + 1)^2 + 3*x^4 + 2) - (5*log((25*(x^4 + 1)^(1/3))/ 1296 - 25/1296))/108 - log((x^4 + 1)^(1/3)/144 - 1/144)/36 + ((x^4 + 1)^(1 /3)/12 + (x^4 + 1)^(4/3)/24)/(2*x^4 - (x^4 + 1)^2 + 1) - log((x^4 + 1)^(1/ 3)/4 - (3^(1/2)*1i)/8 + 1/8)*((3^(1/2)*1i)/72 - 1/72) + log((3^(1/2)*1i)/8 + (x^4 + 1)^(1/3)/4 + 1/8)*((3^(1/2)*1i)/72 + 1/72) - log((5*(x^4 + 1)^(1 /3))/12 - (3^(1/2)*5i)/24 + 5/24)*((3^(1/2)*5i)/216 - 5/216) + log((3^(1/2 )*5i)/24 + (5*(x^4 + 1)^(1/3))/12 + 5/24)*((3^(1/2)*5i)/216 + 5/216)