Integrand size = 26, antiderivative size = 243 \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=\frac {x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac {(2 e-g) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac {x \left (2 d+3 f-7 (d-f) x^2\right )}{24 \left (1+x^2+x^4\right )}-\frac {(13 d+2 f) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {(13 d+2 f) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {(2 e-g) \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{32} (9 d-4 f) \log \left (1-x+x^2\right )+\frac {1}{32} (9 d-4 f) \log \left (1+x+x^2\right ) \]
1/12*x*(d+f-(d-2*f)*x^2)/(x^4+x^2+1)^2+1/12*(e-2*g+(2*e-g)*x^2)/(x^4+x^2+1 )^2+1/12*(2*e-g)*(2*x^2+1)/(x^4+x^2+1)+1/24*x*(2*d+3*f-7*(d-f)*x^2)/(x^4+x ^2+1)-1/32*(9*d-4*f)*ln(x^2-x+1)+1/32*(9*d-4*f)*ln(x^2+x+1)-1/144*(13*d+2* f)*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/144*(13*d+2*f)*arctan(1/3*(1+2*x) *3^(1/2))*3^(1/2)+1/9*(2*e-g)*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.07 \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \left (\frac {6 \left (2 d x+3 f x-7 d x^3+7 f x^3-2 g \left (1+2 x^2\right )+e \left (4+8 x^2\right )\right )}{1+x^2+x^4}+\frac {12 \left (e+2 e x^2-g \left (2+x^2\right )+x \left (d+f-d x^2+2 f x^2\right )\right )}{\left (1+x^2+x^4\right )^2}-\frac {\left (\left (-47 i+7 \sqrt {3}\right ) d+\left (17 i-7 \sqrt {3}\right ) f\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (\left (47 i+7 \sqrt {3}\right ) d-\left (17 i+7 \sqrt {3}\right ) f\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-16 \sqrt {3} (2 e-g) \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )\right ) \]
((6*(2*d*x + 3*f*x - 7*d*x^3 + 7*f*x^3 - 2*g*(1 + 2*x^2) + e*(4 + 8*x^2))) /(1 + x^2 + x^4) + (12*(e + 2*e*x^2 - g*(2 + x^2) + x*(d + f - d*x^2 + 2*f *x^2)))/(1 + x^2 + x^4)^2 - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f )*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt [3])*d - (17*I + 7*Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sq rt[3])/6] - 16*Sqrt[3]*(2*e - g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144
Time = 0.61 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {2202, 1492, 1492, 27, 1483, 1142, 25, 1083, 217, 1103, 1576, 1159, 1086, 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.
\(\displaystyle \int \frac {d+e x+f x^2+g x^3}{\left (x^4+x^2+1\right )^3} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx\) |
\(\Big \downarrow \) 1492 |
\(\displaystyle \frac {1}{12} \int \frac {-5 (d-2 f) x^2+11 d-f}{\left (x^4+x^2+1\right )^2}dx+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1492 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{6} \int \frac {3 \left (5 (4 d-f)-7 (d-f) x^2\right )}{x^4+x^2+1}dx+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \int \frac {5 (4 d-f)-7 (d-f) x^2}{x^4+x^2+1}dx+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1483 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {5 (4 d-f)-3 (9 d-4 f) x}{x^2-x+1}dx+\frac {1}{2} \int \frac {5 (4 d-f)+3 (9 d-4 f) x}{x^2+x+1}dx\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (13 d+2 f) \int \frac {1}{x^2-x+1}dx-\frac {3}{2} (9 d-4 f) \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} (13 d+2 f) \int \frac {1}{x^2+x+1}dx+\frac {3}{2} (9 d-4 f) \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (13 d+2 f) \int \frac {1}{x^2-x+1}dx+\frac {3}{2} (9 d-4 f) \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} (13 d+2 f) \int \frac {1}{x^2+x+1}dx+\frac {3}{2} (9 d-4 f) \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {3}{2} (9 d-4 f) \int \frac {1-2 x}{x^2-x+1}dx-(13 d+2 f) \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{2} \left (\frac {3}{2} (9 d-4 f) \int \frac {2 x+1}{x^2+x+1}dx-(13 d+2 f) \int \frac {1}{-(2 x+1)^2-3}d(2 x+1)\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {3}{2} (9 d-4 f) \int \frac {1-2 x}{x^2-x+1}dx+\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}\right )+\frac {1}{2} \left (\frac {3}{2} (9 d-4 f) \int \frac {2 x+1}{x^2+x+1}dx+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \int \frac {x \left (g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}-\frac {3}{2} (9 d-4 f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}+\frac {3}{2} (9 d-4 f) \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {1}{2} \int \frac {g x^2+e}{\left (x^4+x^2+1\right )^3}dx^2+\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}-\frac {3}{2} (9 d-4 f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}+\frac {3}{2} (9 d-4 f) \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} (2 e-g) \int \frac {1}{\left (x^4+x^2+1\right )^2}dx^2+\frac {x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}-\frac {3}{2} (9 d-4 f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}+\frac {3}{2} (9 d-4 f) \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} (2 e-g) \left (\frac {2}{3} \int \frac {1}{x^4+x^2+1}dx^2+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}-\frac {3}{2} (9 d-4 f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}+\frac {3}{2} (9 d-4 f) \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} (2 e-g) \left (\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}-\frac {4}{3} \int \frac {1}{-x^4-3}d\left (2 x^2+1\right )\right )+\frac {x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}-\frac {3}{2} (9 d-4 f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}+\frac {3}{2} (9 d-4 f) \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}-\frac {3}{2} (9 d-4 f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f)}{\sqrt {3}}+\frac {3}{2} (9 d-4 f) \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (-7 x^2 (d-f)+2 d+3 f\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g)+\frac {x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{12 \left (x^4+x^2+1\right )^2}\) |
(x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^2 + x^4)^2) + ((e - 2*g + (2*e - g) *x^2)/(6*(1 + x^2 + x^4)^2) + ((2*e - g)*((1 + 2*x^2)/(3*(1 + x^2 + x^4)) + (4*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3])))/2)/2 + ((x*(2*d + 3*f - 7* (d - f)*x^2))/(2*(1 + x^2 + x^4)) + ((((13*d + 2*f)*ArcTan[(-1 + 2*x)/Sqrt [3]])/Sqrt[3] - (3*(9*d - 4*f)*Log[1 - x + x^2])/2)/2 + (((13*d + 2*f)*Arc Tan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] + (3*(9*d - 4*f)*Log[1 + x + x^2])/2)/2)/2 )/12
3.1.49.3.1 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[(-(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[((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]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {\left (\frac {7 d}{3}-\frac {7 f}{3}-\frac {4 e}{3}-\frac {g}{3}\right ) x^{3}+\left (-6 d +4 f +2 g \right ) x^{2}+\left (\frac {20 d}{3}-\frac {13 f}{3}+\frac {e}{3}-\frac {8 g}{3}\right ) x -4 d +\frac {4 f}{3}-2 e +2 g}{16 \left (x^{2}-x +1\right )^{2}}-\frac {\left (27 d -12 f \right ) \ln \left (x^{2}-x +1\right )}{96}-\frac {\left (-\frac {13 d}{2}-16 e -f +8 g \right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{72}+\frac {\left (-\frac {7 d}{3}+\frac {7 f}{3}-\frac {4 e}{3}-\frac {g}{3}\right ) x^{3}+\left (-6 d +4 f -2 g \right ) x^{2}+\left (-\frac {20 d}{3}+\frac {13 f}{3}+\frac {e}{3}-\frac {8 g}{3}\right ) x -4 d +\frac {4 f}{3}+2 e -2 g}{16 \left (x^{2}+x +1\right )^{2}}+\frac {\left (27 d -12 f \right ) \ln \left (x^{2}+x +1\right )}{96}+\frac {\left (\frac {13 d}{2}-16 e +f +8 g \right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{72}\) | \(232\) |
risch | \(\text {Expression too large to display}\) | \(28371\) |
-1/16*((7/3*d-7/3*f-4/3*e-1/3*g)*x^3+(-6*d+4*f+2*g)*x^2+(20/3*d-13/3*f+1/3 *e-8/3*g)*x-4*d+4/3*f-2*e+2*g)/(x^2-x+1)^2-1/96*(27*d-12*f)*ln(x^2-x+1)-1/ 72*(-13/2*d-16*e-f+8*g)*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/16*((-7/3*d+ 7/3*f-4/3*e-1/3*g)*x^3+(-6*d+4*f-2*g)*x^2+(-20/3*d+13/3*f+1/3*e-8/3*g)*x-4 *d+4/3*f+2*e-2*g)/(x^2+x+1)^2+1/96*(27*d-12*f)*ln(x^2+x+1)+1/72*(13/2*d-16 *e+f+8*g)*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)
Time = 0.49 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.79 \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=-\frac {84 \, {\left (d - f\right )} x^{7} - 48 \, {\left (2 \, e - g\right )} x^{6} + 60 \, {\left (d - 2 \, f\right )} x^{5} - 72 \, {\left (2 \, e - g\right )} x^{4} + 84 \, {\left (d - 2 \, f\right )} x^{3} - 96 \, {\left (2 \, e - g\right )} x^{2} - 2 \, \sqrt {3} {\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{8} + 2 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{6} + 3 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{4} + 2 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{8} + 2 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{6} + 3 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{4} + 2 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (4 \, d + 5 \, f\right )} x - 9 \, {\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \, {\left (9 \, d - 4 \, f\right )} x^{6} + 3 \, {\left (9 \, d - 4 \, f\right )} x^{4} + 2 \, {\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \, {\left (9 \, d - 4 \, f\right )} x^{6} + 3 \, {\left (9 \, d - 4 \, f\right )} x^{4} + 2 \, {\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e + 72 \, g}{288 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
-1/288*(84*(d - f)*x^7 - 48*(2*e - g)*x^6 + 60*(d - 2*f)*x^5 - 72*(2*e - g )*x^4 + 84*(d - 2*f)*x^3 - 96*(2*e - g)*x^2 - 2*sqrt(3)*((13*d - 32*e + 2* f + 16*g)*x^8 + 2*(13*d - 32*e + 2*f + 16*g)*x^6 + 3*(13*d - 32*e + 2*f + 16*g)*x^4 + 2*(13*d - 32*e + 2*f + 16*g)*x^2 + 13*d - 32*e + 2*f + 16*g)*a rctan(1/3*sqrt(3)*(2*x + 1)) - 2*sqrt(3)*((13*d + 32*e + 2*f - 16*g)*x^8 + 2*(13*d + 32*e + 2*f - 16*g)*x^6 + 3*(13*d + 32*e + 2*f - 16*g)*x^4 + 2*( 13*d + 32*e + 2*f - 16*g)*x^2 + 13*d + 32*e + 2*f - 16*g)*arctan(1/3*sqrt( 3)*(2*x - 1)) - 12*(4*d + 5*f)*x - 9*((9*d - 4*f)*x^8 + 2*(9*d - 4*f)*x^6 + 3*(9*d - 4*f)*x^4 + 2*(9*d - 4*f)*x^2 + 9*d - 4*f)*log(x^2 + x + 1) + 9* ((9*d - 4*f)*x^8 + 2*(9*d - 4*f)*x^6 + 3*(9*d - 4*f)*x^4 + 2*(9*d - 4*f)*x ^2 + 9*d - 4*f)*log(x^2 - x + 1) - 72*e + 72*g)/(x^8 + 2*x^6 + 3*x^4 + 2*x ^2 + 1)
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{32} \, {\left (9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{32} \, {\left (9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - \frac {7 \, {\left (d - f\right )} x^{7} - 4 \, {\left (2 \, e - g\right )} x^{6} + 5 \, {\left (d - 2 \, f\right )} x^{5} - 6 \, {\left (2 \, e - g\right )} x^{4} + 7 \, {\left (d - 2 \, f\right )} x^{3} - 8 \, {\left (2 \, e - g\right )} x^{2} - {\left (4 \, d + 5 \, f\right )} x - 6 \, e + 6 \, g}{24 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
1/144*sqrt(3)*(13*d - 32*e + 2*f + 16*g)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1 /144*sqrt(3)*(13*d + 32*e + 2*f - 16*g)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/ 32*(9*d - 4*f)*log(x^2 + x + 1) - 1/32*(9*d - 4*f)*log(x^2 - x + 1) - 1/24 *(7*(d - f)*x^7 - 4*(2*e - g)*x^6 + 5*(d - 2*f)*x^5 - 6*(2*e - g)*x^4 + 7* (d - 2*f)*x^3 - 8*(2*e - g)*x^2 - (4*d + 5*f)*x - 6*e + 6*g)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)
Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.79 \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{32} \, {\left (9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{32} \, {\left (9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 7 \, f x^{7} - 8 \, e x^{6} + 4 \, g x^{6} + 5 \, d x^{5} - 10 \, f x^{5} - 12 \, e x^{4} + 6 \, g x^{4} + 7 \, d x^{3} - 14 \, f x^{3} - 16 \, e x^{2} + 8 \, g x^{2} - 4 \, d x - 5 \, f x - 6 \, e + 6 \, g}{24 \, {\left (x^{4} + x^{2} + 1\right )}^{2}} \]
1/144*sqrt(3)*(13*d - 32*e + 2*f + 16*g)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1 /144*sqrt(3)*(13*d + 32*e + 2*f - 16*g)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/ 32*(9*d - 4*f)*log(x^2 + x + 1) - 1/32*(9*d - 4*f)*log(x^2 - x + 1) - 1/24 *(7*d*x^7 - 7*f*x^7 - 8*e*x^6 + 4*g*x^6 + 5*d*x^5 - 10*f*x^5 - 12*e*x^4 + 6*g*x^4 + 7*d*x^3 - 14*f*x^3 - 16*e*x^2 + 8*g*x^2 - 4*d*x - 5*f*x - 6*e + 6*g)/(x^4 + x^2 + 1)^2
Time = 8.07 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.21 \[ \int \frac {d+e x+f x^2+g x^3}{\left (1+x^2+x^4\right )^3} \, dx=\frac {\left (\frac {7\,f}{24}-\frac {7\,d}{24}\right )\,x^7+\left (\frac {e}{3}-\frac {g}{6}\right )\,x^6+\left (\frac {5\,f}{12}-\frac {5\,d}{24}\right )\,x^5+\left (\frac {e}{2}-\frac {g}{4}\right )\,x^4+\left (\frac {7\,f}{12}-\frac {7\,d}{24}\right )\,x^3+\left (\frac {2\,e}{3}-\frac {g}{3}\right )\,x^2+\left (\frac {d}{6}+\frac {5\,f}{24}\right )\,x+\frac {e}{4}-\frac {g}{4}}{x^8+2\,x^6+3\,x^4+2\,x^2+1}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}-\frac {f}{8}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{144}-\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{18}\right )-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{8}-\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{144}+\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{18}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{8}-\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{144}-\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{18}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}-\frac {f}{8}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{144}+\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{18}\right ) \]
(e/4 - g/4 - x^5*((5*d)/24 - (5*f)/12) - x^3*((7*d)/24 - (7*f)/12) - x^7*( (7*d)/24 - (7*f)/24) + x^2*((2*e)/3 - g/3) + x^4*(e/2 - g/4) + x^6*(e/3 - g/6) + x*(d/6 + (5*f)/24))/(2*x^2 + 3*x^4 + 2*x^6 + x^8 + 1) - log(x - (3^ (1/2)*1i)/2 - 1/2)*((9*d)/32 - f/8 + (3^(1/2)*d*13i)/288 + (3^(1/2)*e*1i)/ 9 + (3^(1/2)*f*1i)/144 - (3^(1/2)*g*1i)/18) - log(x - (3^(1/2)*1i)/2 + 1/2 )*(f/8 - (9*d)/32 + (3^(1/2)*d*13i)/288 - (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i )/144 + (3^(1/2)*g*1i)/18) + log(x + (3^(1/2)*1i)/2 - 1/2)*(f/8 - (9*d)/32 + (3^(1/2)*d*13i)/288 + (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/144 - (3^(1/2)* g*1i)/18) + log(x + (3^(1/2)*1i)/2 + 1/2)*((9*d)/32 - f/8 + (3^(1/2)*d*13i )/288 - (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/144 + (3^(1/2)*g*1i)/18)