Integrand size = 18, antiderivative size = 107 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {\sqrt [3]{1+x^4} \left (-81-45 x^4+6 x^8-10 x^{12}\right )}{432 x^{16}}+\frac {5 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}}-\frac {5}{324} \log \left (-1+\sqrt [3]{1+x^4}\right )+\frac {5}{648} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
1/432*(x^4+1)^(1/3)*(-10*x^12+6*x^8-45*x^4-81)/x^16+5/324*3^(1/2)*arctan(1 /3*3^(1/2)+2/3*(x^4+1)^(1/3)*3^(1/2))-5/324*ln(-1+(x^4+1)^(1/3))+5/648*ln( 1+(x^4+1)^(1/3)+(x^4+1)^(2/3))
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {-\frac {3 \sqrt [3]{1+x^4} \left (81+45 x^4-6 x^8+10 x^{12}\right )}{x^{16}}+20 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )-20 \log \left (-1+\sqrt [3]{1+x^4}\right )+10 \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )}{1296} \]
((-3*(1 + x^4)^(1/3)*(81 + 45*x^4 - 6*x^8 + 10*x^12))/x^16 + 20*Sqrt[3]*Ar cTan[(1 + 2*(1 + x^4)^(1/3))/Sqrt[3]] - 20*Log[-1 + (1 + x^4)^(1/3)] + 10* Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)])/1296
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {948, 87, 51, 52, 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.
\(\displaystyle \int \frac {\sqrt [3]{x^4+1} \left (x^4+3\right )}{x^{17}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt [3]{x^4+1} \left (x^4+3\right )}{x^{20}}dx^4\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{4} \left (-\int \frac {\sqrt [3]{x^4+1}}{x^{16}}dx^4-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{9} \int \frac {1}{x^{12} \left (x^4+1\right )^{2/3}}dx^4-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {5}{6} \int \frac {1}{x^8 \left (x^4+1\right )^{2/3}}dx^4+\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {5}{6} \left (-\frac {2}{3} \int \frac {1}{x^4 \left (x^4+1\right )^{2/3}}dx^4-\frac {\sqrt [3]{x^4+1}}{x^4}\right )+\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {5}{6} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^4+1}}d\sqrt [3]{x^4+1}-\frac {3}{2} \int \frac {1}{x^8+\sqrt [3]{x^4+1}+1}d\sqrt [3]{x^4+1}-\frac {1}{2} \log \left (x^4\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )+\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {5}{6} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{x^8+\sqrt [3]{x^4+1}+1}d\sqrt [3]{x^4+1}-\frac {1}{2} \log \left (x^4\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^4+1}\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )+\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {5}{6} \left (-\frac {2}{3} \left (3 \int \frac {1}{-x^8-3}d\left (2 \sqrt [3]{x^4+1}+1\right )-\frac {1}{2} \log \left (x^4\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^4+1}\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )+\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {5}{6} \left (-\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )-\frac {\log \left (x^4\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^4+1}\right )\right )-\frac {\sqrt [3]{x^4+1}}{x^4}\right )+\frac {\sqrt [3]{x^4+1}}{2 x^8}\right )-\frac {3 \left (x^4+1\right )^{4/3}}{4 x^{16}}+\frac {\sqrt [3]{x^4+1}}{3 x^{12}}\right )\) |
((1 + x^4)^(1/3)/(3*x^12) - (3*(1 + x^4)^(4/3))/(4*x^16) + ((1 + x^4)^(1/3 )/(2*x^8) + (5*(-((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)/9)/4
3.16.52.3.1 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 = 2.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {10 x^{16}+4 x^{12}+39 x^{8}+126 x^{4}+81}{432 x^{16} \left (x^{4}+1\right )^{\frac {2}{3}}}-\frac {5 \left (-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )\right )}{324 \Gamma \left (\frac {2}{3}\right )}\) | \(81\) |
pseudoelliptic | \(\frac {\left (-30 x^{12}+18 x^{8}-135 x^{4}-243\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+10 x^{16} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )+\ln \left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )-2 \ln \left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )\right )}{1296 {\left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )}^{4} {\left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )}^{4}}\) | \(115\) |
meijerg | \(-\frac {\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], -x^{4}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{27}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{12}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{8}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], -x^{4}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{16}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{8}}+\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}\) | \(144\) |
trager | \(-\frac {\left (10 x^{12}-6 x^{8}+45 x^{4}+81\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{432 x^{16}}-\frac {5 \ln \left (-\frac {333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-393 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+60 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}+48 \left (x^{4}+1\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-165 \left (x^{4}+1\right )^{\frac {1}{3}}-384 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+80}{x^{4}}\right )}{324}+\frac {5 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+162 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+40 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-165 \left (x^{4}+1\right )^{\frac {2}{3}}-495 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+48 \left (x^{4}+1\right )^{\frac {1}{3}}+195 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+100}{x^{4}}\right )}{108}\) | \(311\) |
-1/432*(10*x^16+4*x^12+39*x^8+126*x^4+81)/x^16/(x^4+1)^(2/3)-5/324/GAMMA(2 /3)*(-2/3*GAMMA(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 = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {20 \, \sqrt {3} x^{16} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 10 \, x^{16} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 20 \, x^{16} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (10 \, x^{12} - 6 \, x^{8} + 45 \, x^{4} + 81\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{1296 \, x^{16}} \]
1/1296*(20*sqrt(3)*x^16*arctan(2/3*sqrt(3)*(x^4 + 1)^(1/3) + 1/3*sqrt(3)) + 10*x^16*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) - 20*x^16*log((x^4 + 1)^(1/3) - 1) - 3*(10*x^12 - 6*x^8 + 45*x^4 + 81)*(x^4 + 1)^(1/3))/x^16
Timed out. \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (85) = 170\).
Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5}{324} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {20 \, {\left (x^{4} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{432 \, {\left ({\left (x^{4} + 1\right )}^{4} - 4 \, x^{4} - 4 \, {\left (x^{4} + 1\right )}^{3} + 6 \, {\left (x^{4} + 1\right )}^{2} - 3\right )}} + \frac {5 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{216 \, {\left (3 \, x^{4} + {\left (x^{4} + 1\right )}^{3} - 3 \, {\left (x^{4} + 1\right )}^{2} + 2\right )}} + \frac {5}{648} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{324} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
5/324*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/432*(20*(x^4 + 1)^(10/3) - 72*(x^4 + 1)^(7/3) + 93*(x^4 + 1)^(4/3) + 40*(x^4 + 1)^(1/3 ))/((x^4 + 1)^4 - 4*x^4 - 4*(x^4 + 1)^3 + 6*(x^4 + 1)^2 - 3) + 1/216*(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) + 5/648*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1 ) - 5/324*log((x^4 + 1)^(1/3) - 1)
Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5}{324} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {10 \, {\left (x^{4} + 1\right )}^{\frac {10}{3}} - 36 \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} + 87 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} + 20 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{432 \, x^{16}} + \frac {5}{648} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{324} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
5/324*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/432*(10*(x^4 + 1)^(10/3) - 36*(x^4 + 1)^(7/3) + 87*(x^4 + 1)^(4/3) + 20*(x^4 + 1)^(1/3 ))/x^16 + 5/648*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) - 5/324*log((x^ 4 + 1)^(1/3) - 1)
Time = 6.67 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.50 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^{17}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{11664}-\frac {25}{11664}\right )}{324}-\frac {5\,\ln \left (\frac {25\,{\left (x^4+1\right )}^{1/3}}{2916}-\frac {25}{2916}\right )}{162}-\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{108}+\frac {13\,{\left (x^4+1\right )}^{4/3}}{216}-\frac {5\,{\left (x^4+1\right )}^{7/3}}{216}}{{\left (x^4+1\right )}^3-3\,{\left (x^4+1\right )}^2+3\,x^4+2}+\frac {\frac {5\,{\left (x^4+1\right )}^{1/3}}{54}+\frac {31\,{\left (x^4+1\right )}^{4/3}}{144}-\frac {{\left (x^4+1\right )}^{7/3}}{6}+\frac {5\,{\left (x^4+1\right )}^{10/3}}{108}}{4\,{\left (x^4+1\right )}^3-6\,{\left (x^4+1\right )}^2-{\left (x^4+1\right )}^4+4\,x^4+3}-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{18}+\frac {5}{36}-\frac {\sqrt {3}\,5{}\mathrm {i}}{36}\right )\,\left (-\frac {5}{324}+\frac {\sqrt {3}\,5{}\mathrm {i}}{324}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{18}+\frac {5}{36}+\frac {\sqrt {3}\,5{}\mathrm {i}}{36}\right )\,\left (\frac {5}{324}+\frac {\sqrt {3}\,5{}\mathrm {i}}{324}\right )+\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {5}{72}-\frac {\sqrt {3}\,5{}\mathrm {i}}{72}\right )\,\left (-\frac {5}{648}+\frac {\sqrt {3}\,5{}\mathrm {i}}{648}\right )-\ln \left (\frac {5\,{\left (x^4+1\right )}^{1/3}}{36}+\frac {5}{72}+\frac {\sqrt {3}\,5{}\mathrm {i}}{72}\right )\,\left (\frac {5}{648}+\frac {\sqrt {3}\,5{}\mathrm {i}}{648}\right ) \]
(5*log((25*(x^4 + 1)^(1/3))/11664 - 25/11664))/324 - (5*log((25*(x^4 + 1)^ (1/3))/2916 - 25/2916))/162 - ((5*(x^4 + 1)^(1/3))/108 + (13*(x^4 + 1)^(4/ 3))/216 - (5*(x^4 + 1)^(7/3))/216)/((x^4 + 1)^3 - 3*(x^4 + 1)^2 + 3*x^4 + 2) + ((5*(x^4 + 1)^(1/3))/54 + (31*(x^4 + 1)^(4/3))/144 - (x^4 + 1)^(7/3)/ 6 + (5*(x^4 + 1)^(10/3))/108)/(4*(x^4 + 1)^3 - 6*(x^4 + 1)^2 - (x^4 + 1)^4 + 4*x^4 + 3) - log((5*(x^4 + 1)^(1/3))/18 - (3^(1/2)*5i)/36 + 5/36)*((3^( 1/2)*5i)/324 - 5/324) + log((3^(1/2)*5i)/36 + (5*(x^4 + 1)^(1/3))/18 + 5/3 6)*((3^(1/2)*5i)/324 + 5/324) + log((5*(x^4 + 1)^(1/3))/36 - (3^(1/2)*5i)/ 72 + 5/72)*((3^(1/2)*5i)/648 - 5/648) - log((3^(1/2)*5i)/72 + (5*(x^4 + 1) ^(1/3))/36 + 5/72)*((3^(1/2)*5i)/648 + 5/648)