Integrand size = 31, antiderivative size = 87 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {2}{27 x^6}+\frac {13}{108 x^4}-\frac {13}{54 x^2}+\frac {25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {1237 \arctan \left (\frac {1+x^2}{\sqrt {2}}\right )}{1944 \sqrt {2}}+\frac {61 \log (x)}{243}-\frac {61}{972} \log \left (3+2 x^2+x^4\right ) \]
-2/27/x^6+13/108/x^4-13/54/x^2+25/648*(-7*x^2+1)/(x^4+2*x^2+3)+61/243*ln(x )-61/972*ln(x^4+2*x^2+3)-1237/3888*arctan(1/2*(x^2+1)*2^(1/2))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {-\frac {576}{x^6}+\frac {936}{x^4}-\frac {1872}{x^2}-\frac {300 \left (-1+7 x^2\right )}{3+2 x^2+x^4}+1952 \log (x)-\sqrt {2} \left (1237 i+244 \sqrt {2}\right ) \log \left (-i+\sqrt {2}-i x^2\right )+\sqrt {2} \left (1237 i-244 \sqrt {2}\right ) \log \left (i+\sqrt {2}+i x^2\right )}{7776} \]
(-576/x^6 + 936/x^4 - 1872/x^2 - (300*(-1 + 7*x^2))/(3 + 2*x^2 + x^4) + 19 52*Log[x] - Sqrt[2]*(1237*I + 244*Sqrt[2])*Log[-I + Sqrt[2] - I*x^2] + Sqr t[2]*(1237*I - 244*Sqrt[2])*Log[I + Sqrt[2] + I*x^2])/7776
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2194, 2177, 27, 2159, 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.
\(\displaystyle \int \frac {5 x^6+3 x^4+x^2+4}{x^7 \left (x^4+2 x^2+3\right )^2} \, dx\) |
\(\Big \downarrow \) 2194 |
\(\displaystyle \frac {1}{2} \int \frac {5 x^6+3 x^4+x^2+4}{x^8 \left (x^4+2 x^2+3\right )^2}dx^2\) |
\(\Big \downarrow \) 2177 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{8} \int \frac {2 \left (-175 x^8+400 x^6+300 x^4-180 x^2+432\right )}{81 x^8 \left (x^4+2 x^2+3\right )}dx^2+\frac {25 \left (1-7 x^2\right )}{324 \left (x^4+2 x^2+3\right )}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{324} \int \frac {-175 x^8+400 x^6+300 x^4-180 x^2+432}{x^8 \left (x^4+2 x^2+3\right )}dx^2+\frac {25 \left (1-7 x^2\right )}{324 \left (x^4+2 x^2+3\right )}\right )\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{324} \int \left (\frac {-244 x^2-1481}{3 \left (x^4+2 x^2+3\right )}+\frac {244}{3 x^2}+\frac {156}{x^4}-\frac {156}{x^6}+\frac {144}{x^8}\right )dx^2+\frac {25 \left (1-7 x^2\right )}{324 \left (x^4+2 x^2+3\right )}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{324} \left (-\frac {1237 \arctan \left (\frac {x^2+1}{\sqrt {2}}\right )}{3 \sqrt {2}}-\frac {48}{x^6}+\frac {78}{x^4}-\frac {156}{x^2}+\frac {244 \log \left (x^2\right )}{3}-\frac {122}{3} \log \left (x^4+2 x^2+3\right )\right )+\frac {25 \left (1-7 x^2\right )}{324 \left (x^4+2 x^2+3\right )}\right )\) |
((25*(1 - 7*x^2))/(324*(3 + 2*x^2 + x^4)) + (-48/x^6 + 78/x^4 - 156/x^2 - (1237*ArcTan[(1 + x^2)/Sqrt[2]])/(3*Sqrt[2]) + (244*Log[x^2])/3 - (122*Log [3 + 2*x^2 + x^4])/3)/324)/2
3.2.8.3.1 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[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {2}{27 x^{6}}+\frac {13}{108 x^{4}}-\frac {13}{54 x^{2}}+\frac {61 \ln \left (x \right )}{243}-\frac {\frac {525 x^{2}}{4}-\frac {75}{4}}{486 \left (x^{4}+2 x^{2}+3\right )}-\frac {61 \ln \left (x^{4}+2 x^{2}+3\right )}{972}-\frac {1237 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{3888}\) | \(73\) |
risch | \(\frac {-\frac {331}{648} x^{8}-\frac {209}{648} x^{6}-\frac {5}{9} x^{4}+\frac {23}{108} x^{2}-\frac {2}{9}}{x^{6} \left (x^{4}+2 x^{2}+3\right )}+\frac {61 \ln \left (x \right )}{243}-\frac {61 \ln \left (1530169 x^{4}+3060338 x^{2}+4590507\right )}{972}-\frac {1237 \sqrt {2}\, \arctan \left (\frac {\left (1237 x^{2}+1237\right ) \sqrt {2}}{2474}\right )}{3888}\) | \(77\) |
-2/27/x^6+13/108/x^4-13/54/x^2+61/243*ln(x)-1/486*(525/4*x^2-75/4)/(x^4+2* x^2+3)-61/972*ln(x^4+2*x^2+3)-1237/3888*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/ 2))
Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.32 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {1986 \, x^{8} + 1254 \, x^{6} + 2160 \, x^{4} + 1237 \, \sqrt {2} {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 828 \, x^{2} + 244 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 976 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x\right ) + 864}{3888 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} \]
-1/3888*(1986*x^8 + 1254*x^6 + 2160*x^4 + 1237*sqrt(2)*(x^10 + 2*x^8 + 3*x ^6)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 828*x^2 + 244*(x^10 + 2*x^8 + 3*x^6)*l og(x^4 + 2*x^2 + 3) - 976*(x^10 + 2*x^8 + 3*x^6)*log(x) + 864)/(x^10 + 2*x ^8 + 3*x^6)
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {61 \log {\left (x \right )}}{243} - \frac {61 \log {\left (x^{4} + 2 x^{2} + 3 \right )}}{972} - \frac {1237 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{3888} + \frac {- 331 x^{8} - 209 x^{6} - 360 x^{4} + 138 x^{2} - 144}{648 x^{10} + 1296 x^{8} + 1944 x^{6}} \]
61*log(x)/243 - 61*log(x**4 + 2*x**2 + 3)/972 - 1237*sqrt(2)*atan(sqrt(2)* x**2/2 + sqrt(2)/2)/3888 + (-331*x**8 - 209*x**6 - 360*x**4 + 138*x**2 - 1 44)/(648*x**10 + 1296*x**8 + 1944*x**6)
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {1237}{3888} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {331 \, x^{8} + 209 \, x^{6} + 360 \, x^{4} - 138 \, x^{2} + 144}{648 \, {\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} - \frac {61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {61}{486} \, \log \left (x^{2}\right ) \]
-1237/3888*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/648*(331*x^8 + 209*x^ 6 + 360*x^4 - 138*x^2 + 144)/(x^10 + 2*x^8 + 3*x^6) - 61/972*log(x^4 + 2*x ^2 + 3) + 61/486*log(x^2)
Time = 0.44 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {1237}{3888} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {122 \, x^{4} - 281 \, x^{2} + 441}{1944 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {671 \, x^{6} + 702 \, x^{4} - 351 \, x^{2} + 216}{2916 \, x^{6}} - \frac {61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {61}{486} \, \log \left (x^{2}\right ) \]
-1237/3888*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/1944*(122*x^4 - 281*x ^2 + 441)/(x^4 + 2*x^2 + 3) - 1/2916*(671*x^6 + 702*x^4 - 351*x^2 + 216)/x ^6 - 61/972*log(x^4 + 2*x^2 + 3) + 61/486*log(x^2)
Time = 8.64 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {61\,\ln \left (x\right )}{243}-\frac {61\,\ln \left (x^4+2\,x^2+3\right )}{972}-\frac {\frac {331\,x^8}{648}+\frac {209\,x^6}{648}+\frac {5\,x^4}{9}-\frac {23\,x^2}{108}+\frac {2}{9}}{x^{10}+2\,x^8+3\,x^6}-\frac {1237\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{3888} \]