Integrand size = 26, antiderivative size = 383 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\left ((-2)^{2/3}-3\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{27 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3} \sqrt [3]{3}} \] Output:
1/27*(-1)^(2/3)*(3*(-3)^(2/3)-2^(2/3))*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/( 24-18*(-3)^(2/3)*2^(1/3))^(1/2))*3^(5/6)/(1+(-1)^(1/3))^2/(8-6*(-3)^(2/3)* 2^(1/3))^(1/2)-1/81*((-2)^(2/3)-3*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2* x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*3^(5/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1 /3)*3^(2/3))^(1/2)+1/81*(2^(2/3)-3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^( 1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*3^(5/6)/(-8+6*2^(1/3)*3^(2/3))^(1/2 )+1/36*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(1/3)*3^(2/3)/(1+(-1)^(1/3))^2+1 /108*(-1)^(1/3)*3^(2/3)*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*2^(1/3)-1/108*ln( 6+3*2^(2/3)*3^(1/3)*x+x^2)*2^(1/3)*3^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.16 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^3}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \] Input:
Integrate[x^4/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (Log[x - #1]*#1^3)/ (36 + 162*#1 + 12*#1^2 + #1^4) & ]/6
Time = 1.78 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2466, 2009}
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.
| \(\displaystyle \int \frac {x^4}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {(-1)^{2/3} \left (2-\sqrt [3]{-3} 2^{2/3} x\right )}{7558272 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {(-1)^{2/3} \left ((-2)^{2/3} \sqrt [3]{3} x+2\right )}{22674816 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {\sqrt [3]{3} x+\sqrt [3]{2}}{11337408\ 6^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (-\frac {\left (2 (-1)^{2/3} \sqrt [6]{2} 3^{5/6}-9 \sqrt {6}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{204073344 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{11337408 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{34012224 \sqrt {6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}+\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{7558272\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\sqrt [3]{-\frac {1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{22674816\ 2^{2/3}}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{22674816\ 2^{2/3} \sqrt [3]{3}}\right )\) |
Input:
Int[x^4/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
1259712*(-1/204073344*((2*(-1)^(2/3)*2^(1/6)*3^(5/6) - 9*Sqrt[6])*ArcTan[( 3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)] + ((-1)^(2/3)*(3*(-3)^(2/3) - 2^(2/3))*ArcTan[(2^(1/ 6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(11337 408*3^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[2*(4 - 3*(-3)^(2/3)*2^(1/3))]) - ((9 - 2^(2/3)*3^(1/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3 *2^(1/3)*3^(2/3))]])/(34012224*Sqrt[6*(-4 + 3*2^(1/3)*3^(2/3))]) + Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(7558272*2^(2/3)*3^(1/3)*(1 + (-1)^(1/3))^2 ) + ((-1/3)^(1/3)*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(22674816*2^(2/3) ) - Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(22674816*2^(2/3)*3^(1/3)))
Int[(u_.)*(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Sim p[1/(3^(3*p)*a^(2*p)) Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]* x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3* (-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.15
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(56\) |
Input:
int(x^4/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/6*sum(_R^4/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4 +324*_Z^3+108*_Z^2+216))
Timed out. \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \] Input:
integrate(x^4/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (15695178850368 t^{6} - 2066242608 t^{4} + 1845163152 t^{3} - 1180980 t^{2} - 1944 t - 1, \left ( t \mapsto t \log {\left (\frac {614714526178551746208 t^{5}}{57121295165} - \frac {1270857362386176 t^{4}}{57121295165} - \frac {80483053187684376 t^{3}}{57121295165} + \frac {72431318325103884 t^{2}}{57121295165} - \frac {45358602689088 t}{57121295165} + x - \frac {44532180783}{57121295165} \right )} \right )\right )} \] Input:
integrate(x**4/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(15695178850368*_t**6 - 2066242608*_t**4 + 1845163152*_t**3 - 11809 80*_t**2 - 1944*_t - 1, Lambda(_t, _t*log(614714526178551746208*_t**5/5712 1295165 - 1270857362386176*_t**4/57121295165 - 80483053187684376*_t**3/571 21295165 + 72431318325103884*_t**2/57121295165 - 45358602689088*_t/5712129 5165 + x - 44532180783/57121295165)))
\[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(x^4/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
integrate(x^4/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(x^4/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate(x^4/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
Time = 22.52 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.02 \[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (-\frac {5038848\,\left (1377495072\,x+17006112\,x\,\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )-104976\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^2+158112\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^3+1946\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^4+3\,x\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^5-4251528\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^2+3927852\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^3-1188\,{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^4-{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^5+7558272\,\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )+33519046752\right )}{{\mathrm {root}\left (z^6+1944\,z^5+1180980\,z^4-1845163152\,z^3+2066242608\,z^2-15695178850368,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{7596}+\frac {217\,z^3}{1845828}-\frac {5\,z^2}{66449808}-\frac {z}{8073651672}-\frac {1}{15695178850368},z,k\right ) \] Input:
int(x^4/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)
Output:
symsum(log(-(5038848*(1377495072*x + 17006112*x*root(z^6 + 1944*z^5 + 1180 980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 15695178850368, z, k) - 104976 *x*root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 1 5695178850368, z, k)^2 + 158112*x*root(z^6 + 1944*z^5 + 1180980*z^4 - 1845 163152*z^3 + 2066242608*z^2 - 15695178850368, z, k)^3 + 1946*x*root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 15695178850368, z, k)^4 + 3*x*root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152*z^3 + 206624 2608*z^2 - 15695178850368, z, k)^5 - 4251528*root(z^6 + 1944*z^5 + 1180980 *z^4 - 1845163152*z^3 + 2066242608*z^2 - 15695178850368, z, k)^2 + 3927852 *root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 156 95178850368, z, k)^3 - 1188*root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152 *z^3 + 2066242608*z^2 - 15695178850368, z, k)^4 - root(z^6 + 1944*z^5 + 11 80980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 15695178850368, z, k)^5 + 75 58272*root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 15695178850368, z, k) + 33519046752))/root(z^6 + 1944*z^5 + 1180980*z^4 - 1845163152*z^3 + 2066242608*z^2 - 15695178850368, z, k)^5)*root(z^6 - z^ 4/7596 + (217*z^3)/1845828 - (5*z^2)/66449808 - z/8073651672 - 1/156951788 50368, z, k), k, 1, 6)
\[ \int \frac {x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int \frac {x^{4}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \] Input:
int(x^4/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
int(x**4/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)