Integrand size = 31, antiderivative size = 72 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=-\frac {x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {369 \arctan (x)}{8}+\frac {267 \arctan \left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
-1/4*x*(51*x^2+50)/(x^4+3*x^2+2)^2+1/8*x*(125*x^2+254)/(x^4+3*x^2+2)-369/8 *arctan(x)+267/8*arctan(1/2*x*2^(1/2))*2^(1/2)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {1}{8} \left (\frac {x \left (408+910 x^2+629 x^4+125 x^6\right )}{\left (2+3 x^2+x^4\right )^2}-369 \arctan (x)+267 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )\right ) \]
((x*(408 + 910*x^2 + 629*x^4 + 125*x^6))/(2 + 3*x^2 + x^4)^2 - 369*ArcTan[ x] + 267*Sqrt[2]*ArcTan[x/Sqrt[2]])/8
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2197, 27, 2206, 27, 1480, 216}
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.
\(\displaystyle \int \frac {x^4 \left (5 x^6+3 x^4+x^2+4\right )}{\left (x^4+3 x^2+2\right )^3} \, dx\) |
\(\Big \downarrow \) 2197 |
\(\displaystyle -\frac {1}{8} \int -\frac {2 \left (20 x^6-48 x^4-147 x^2+50\right )}{\left (x^4+3 x^2+2\right )^2}dx-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {20 x^6-48 x^4-147 x^2+50}{\left (x^4+3 x^2+2\right )^2}dx-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 2206 |
\(\displaystyle \frac {1}{4} \left (\frac {x \left (125 x^2+254\right )}{2 \left (x^4+3 x^2+2\right )}-\frac {1}{4} \int \frac {6 \left (68-55 x^2\right )}{x^4+3 x^2+2}dx\right )-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (\frac {x \left (125 x^2+254\right )}{2 \left (x^4+3 x^2+2\right )}-\frac {3}{2} \int \frac {68-55 x^2}{x^4+3 x^2+2}dx\right )-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle \frac {1}{4} \left (\frac {x \left (125 x^2+254\right )}{2 \left (x^4+3 x^2+2\right )}-\frac {3}{2} \left (123 \int \frac {1}{x^2+1}dx-178 \int \frac {1}{x^2+2}dx\right )\right )-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (\frac {x \left (125 x^2+254\right )}{2 \left (x^4+3 x^2+2\right )}-\frac {3}{2} \left (123 \arctan (x)-89 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )\right )\right )-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}\) |
-1/4*(x*(50 + 51*x^2))/(2 + 3*x^2 + x^4)^2 + ((x*(254 + 125*x^2))/(2*(2 + 3*x^2 + x^4)) - (3*(123*ArcTan[x] - 89*Sqrt[2]*ArcTan[x/Sqrt[2]]))/2)/4
3.1.94.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
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[(a + b*x^2 + c*x ^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*Qx + b^2*d*(2*p + 3) - 2* a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; Fre eQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4 *a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, 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[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {\frac {125}{8} x^{7}+\frac {629}{8} x^{5}+\frac {455}{4} x^{3}+51 x}{\left (x^{4}+3 x^{2}+2\right )^{2}}+\frac {267 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {369 \arctan \left (x \right )}{8}\) | \(50\) |
default | \(\frac {\frac {51}{4} x^{3}+\frac {77}{2} x}{\left (x^{2}+2\right )^{2}}+\frac {267 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {-\frac {23}{8} x^{3}-\frac {25}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {369 \arctan \left (x \right )}{8}\) | \(54\) |
(125/8*x^7+629/8*x^5+455/4*x^3+51*x)/(x^4+3*x^2+2)^2+267/8*arctan(1/2*x*2^ (1/2))*2^(1/2)-369/8*arctan(x)
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 267 \, \sqrt {2} {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 369 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 408 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]
1/8*(125*x^7 + 629*x^5 + 910*x^3 + 267*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12* x^2 + 4)*arctan(1/2*sqrt(2)*x) - 369*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*a rctan(x) + 408*x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {125 x^{7} + 629 x^{5} + 910 x^{3} + 408 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac {369 \operatorname {atan}{\left (x \right )}}{8} + \frac {267 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{8} \]
(125*x**7 + 629*x**5 + 910*x**3 + 408*x)/(8*x**8 + 48*x**6 + 104*x**4 + 96 *x**2 + 32) - 369*atan(x)/8 + 267*sqrt(2)*atan(sqrt(2)*x/2)/8
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {267}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 408 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac {369}{8} \, \arctan \left (x\right ) \]
267/8*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/8*(125*x^7 + 629*x^5 + 910*x^3 + 4 08*x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4) - 369/8*arctan(x)
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {267}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 408 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac {369}{8} \, \arctan \left (x\right ) \]
267/8*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/8*(125*x^7 + 629*x^5 + 910*x^3 + 4 08*x)/(x^4 + 3*x^2 + 2)^2 - 369/8*arctan(x)
Time = 8.56 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {267\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{8}-\frac {369\,\mathrm {atan}\left (x\right )}{8}+\frac {\frac {125\,x^7}{8}+\frac {629\,x^5}{8}+\frac {455\,x^3}{4}+51\,x}{x^8+6\,x^6+13\,x^4+12\,x^2+4} \]