Integrand size = 31, antiderivative size = 93 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=-\frac {1}{10 x^5}+\frac {17}{24 x^3}-\frac {93}{16 x}-\frac {x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac {29 \arctan (x)}{8}-\frac {2207 \arctan \left (\frac {x}{\sqrt {2}}\right )}{128 \sqrt {2}} \]
-1/10/x^5+17/24/x^3-93/16/x-1/32*x*(-5*x^2+3)/(x^4+3*x^2+2)^2-1/128*x*(999 *x^2+1771)/(x^4+3*x^2+2)+29/8*arctan(x)-2207/256*arctan(1/2*x*2^(1/2))*2^( 1/2)
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=\frac {-\frac {2 \left (768-3136 x^2+30816 x^4+170702 x^6+246477 x^8+137120 x^{10}+26145 x^{12}\right )}{x^5 \left (2+3 x^2+x^4\right )^2}+13920 \arctan (x)-33105 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )}{3840} \]
((-2*(768 - 3136*x^2 + 30816*x^4 + 170702*x^6 + 246477*x^8 + 137120*x^10 + 26145*x^12))/(x^5*(2 + 3*x^2 + x^4)^2) + 13920*ArcTan[x] - 33105*Sqrt[2]* ArcTan[x/Sqrt[2]])/3840
Time = 0.44 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2198, 27, 2198, 25, 2195, 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.
\(\displaystyle \int \frac {5 x^6+3 x^4+x^2+4}{x^6 \left (x^4+3 x^2+2\right )^3} \, dx\) |
\(\Big \downarrow \) 2198 |
\(\displaystyle -\frac {1}{8} \int -\frac {25 x^8-81 x^6+136 x^4-80 x^2+64}{4 x^6 \left (x^4+3 x^2+2\right )^2}dx-\frac {x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{32} \int \frac {25 x^8-81 x^6+136 x^4-80 x^2+64}{x^6 \left (x^4+3 x^2+2\right )^2}dx-\frac {x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 2198 |
\(\displaystyle \frac {1}{32} \left (-\frac {1}{4} \int -\frac {-999 x^8+681 x^6+736 x^4-352 x^2+128}{x^6 \left (x^4+3 x^2+2\right )}dx-\frac {x \left (999 x^2+1771\right )}{4 \left (x^4+3 x^2+2\right )}\right )-\frac {x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \int \frac {-999 x^8+681 x^6+736 x^4-352 x^2+128}{x^6 \left (x^4+3 x^2+2\right )}dx-\frac {x \left (999 x^2+1771\right )}{4 \left (x^4+3 x^2+2\right )}\right )-\frac {x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 2195 |
\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \int \left (-\frac {2207}{x^2+2}+\frac {744}{x^2}-\frac {272}{x^4}+\frac {64}{x^6}+\frac {464}{x^2+1}\right )dx-\frac {x \left (999 x^2+1771\right )}{4 \left (x^4+3 x^2+2\right )}\right )-\frac {x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (464 \arctan (x)-\frac {2207 \arctan \left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {64}{5 x^5}+\frac {272}{3 x^3}-\frac {744}{x}\right )-\frac {x \left (999 x^2+1771\right )}{4 \left (x^4+3 x^2+2\right )}\right )-\frac {x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}\) |
-1/32*(x*(3 - 5*x^2))/(2 + 3*x^2 + x^4)^2 + (-1/4*(x*(1771 + 999*x^2))/(2 + 3*x^2 + x^4) + (-64/(5*x^5) + 272/(3*x^3) - 744/x + 464*ArcTan[x] - (220 7*ArcTan[x/Sqrt[2]])/Sqrt[2])/4)/32
3.1.99.3.1 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_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) ^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x ^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {-\frac {1743}{128} x^{12}-\frac {857}{12} x^{10}-\frac {82159}{640} x^{8}-\frac {85351}{960} x^{6}-\frac {321}{20} x^{4}+\frac {49}{30} x^{2}-\frac {2}{5}}{x^{5} \left (x^{4}+3 x^{2}+2\right )^{2}}+\frac {29 \arctan \left (x \right )}{8}-\frac {2207 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{256}\) | \(66\) |
default | \(-\frac {\frac {311}{8} x^{3}+\frac {337}{4} x}{16 \left (x^{2}+2\right )^{2}}-\frac {2207 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{256}-\frac {1}{10 x^{5}}+\frac {17}{24 x^{3}}-\frac {93}{16 x}+\frac {-\frac {43}{8} x^{3}-\frac {45}{8} x}{\left (x^{2}+1\right )^{2}}+\frac {29 \arctan \left (x \right )}{8}\) | \(68\) |
(-1743/128*x^12-857/12*x^10-82159/640*x^8-85351/960*x^6-321/20*x^4+49/30*x ^2-2/5)/x^5/(x^4+3*x^2+2)^2+29/8*arctan(x)-2207/256*arctan(1/2*x*2^(1/2))* 2^(1/2)
Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.33 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=-\frac {52290 \, x^{12} + 274240 \, x^{10} + 492954 \, x^{8} + 341404 \, x^{6} + 61632 \, x^{4} + 33105 \, \sqrt {2} {\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 6272 \, x^{2} - 13920 \, {\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (x\right ) + 1536}{3840 \, {\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} \]
-1/3840*(52290*x^12 + 274240*x^10 + 492954*x^8 + 341404*x^6 + 61632*x^4 + 33105*sqrt(2)*(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)*arctan(1/2*sqrt(2) *x) - 6272*x^2 - 13920*(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)*arctan(x) + 1536)/(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=\frac {29 \operatorname {atan}{\left (x \right )}}{8} - \frac {2207 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{256} + \frac {- 26145 x^{12} - 137120 x^{10} - 246477 x^{8} - 170702 x^{6} - 30816 x^{4} + 3136 x^{2} - 768}{1920 x^{13} + 11520 x^{11} + 24960 x^{9} + 23040 x^{7} + 7680 x^{5}} \]
29*atan(x)/8 - 2207*sqrt(2)*atan(sqrt(2)*x/2)/256 + (-26145*x**12 - 137120 *x**10 - 246477*x**8 - 170702*x**6 - 30816*x**4 + 3136*x**2 - 768)/(1920*x **13 + 11520*x**11 + 24960*x**9 + 23040*x**7 + 7680*x**5)
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=-\frac {2207}{256} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {26145 \, x^{12} + 137120 \, x^{10} + 246477 \, x^{8} + 170702 \, x^{6} + 30816 \, x^{4} - 3136 \, x^{2} + 768}{1920 \, {\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} + \frac {29}{8} \, \arctan \left (x\right ) \]
-2207/256*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/1920*(26145*x^12 + 137120*x^10 + 246477*x^8 + 170702*x^6 + 30816*x^4 - 3136*x^2 + 768)/(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5) + 29/8*arctan(x)
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.72 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=-\frac {2207}{256} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {999 \, x^{7} + 4768 \, x^{5} + 7291 \, x^{3} + 3554 \, x}{128 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac {1395 \, x^{4} - 170 \, x^{2} + 24}{240 \, x^{5}} + \frac {29}{8} \, \arctan \left (x\right ) \]
-2207/256*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/128*(999*x^7 + 4768*x^5 + 7291 *x^3 + 3554*x)/(x^4 + 3*x^2 + 2)^2 - 1/240*(1395*x^4 - 170*x^2 + 24)/x^5 + 29/8*arctan(x)
Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx=\frac {29\,\mathrm {atan}\left (x\right )}{8}-\frac {2207\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{256}-\frac {\frac {1743\,x^{12}}{128}+\frac {857\,x^{10}}{12}+\frac {82159\,x^8}{640}+\frac {85351\,x^6}{960}+\frac {321\,x^4}{20}-\frac {49\,x^2}{30}+\frac {2}{5}}{x^{13}+6\,x^{11}+13\,x^9+12\,x^7+4\,x^5} \]