Integrand size = 17, antiderivative size = 268 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=-\frac {\arctan \left (\frac {3-\left (1+\frac {4}{x}\right )^2}{6 \sqrt {7}}\right )}{12 \sqrt {7}}-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {2-\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )-\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}-\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )+\frac {1}{24} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}+\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right ) \]
-1/84*arctan(1/42*(3-(1+4/x)^2)*7^(1/2))*7^(1/2)-1/29232*ln((1+4/x)^2+3*29 ^(1/2)-(1+4/x)*(6+6*29^(1/2))^(1/2))*(-132762+81606*29^(1/2))^(1/2)+1/2923 2*ln((1+4/x)^2+3*29^(1/2)+(1+4/x)*(6+6*29^(1/2))^(1/2))*(-132762+81606*29^ (1/2))^(1/2)-1/14616*arctan((2+8/x-(6+6*29^(1/2))^(1/2))/(-6+6*29^(1/2))^( 1/2))*(132762+81606*29^(1/2))^(1/2)-1/14616*arctan((2+8/x+(6+6*29^(1/2))^( 1/2))/(-6+6*29^(1/2))^(1/2))*(132762+81606*29^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.17 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\text {RootSum}\left [8+8 \text {$\#$1}-\text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{8-3 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ] \]
Time = 0.66 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.20, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {2504, 27, 2202, 27, 1432, 1083, 217, 1483, 27, 1142, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{8 x^4-x^3+8 x+8} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -1024 \int \frac {\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^2}{512 \left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )}d\left (\frac {1}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int \frac {\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^2}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )+\int -\frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )-8 \int \frac {\frac {1}{4}+\frac {1}{x}}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )-4 \int \frac {1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -2 \left (8 \int \frac {1}{-\left (\frac {1}{4}+\frac {1}{x}\right )^4-258048}d\left (512 \left (\frac {1}{4}+\frac {1}{x}\right )^2-96\right )+\int \frac {16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -2 \left (\frac {\int \frac {8 \left (\sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}-2 \left (1-3 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{24 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\int \frac {8 \left (2 \left (1-3 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{24 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {\int \frac {\sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}-2 \left (1-3 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\int \frac {2 \left (1-3 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -2 \left (\frac {\sqrt {\frac {3}{2} \left (109+67 \sqrt {29}\right )} \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )-\frac {1}{16} \left (1-3 \sqrt {29}\right ) \int -\frac {4 \left (\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\sqrt {\frac {3}{2} \left (109+67 \sqrt {29}\right )} \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{16} \left (1-3 \sqrt {29}\right ) \int \frac {4 \left (8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {\sqrt {\frac {3}{2} \left (109+67 \sqrt {29}\right )} \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (1-3 \sqrt {29}\right ) \int \frac {\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\sqrt {\frac {3}{2} \left (109+67 \sqrt {29}\right )} \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (1-3 \sqrt {29}\right ) \int \frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -2 \left (\frac {\frac {1}{4} \left (1-3 \sqrt {29}\right ) \int \frac {\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )-\sqrt {6 \left (109+67 \sqrt {29}\right )} \int \frac {1}{96 \left (1-\sqrt {29}\right )-\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )^2}d\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{4} \left (1-3 \sqrt {29}\right ) \int \frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )-\sqrt {6 \left (109+67 \sqrt {29}\right )} \int \frac {1}{96 \left (1-\sqrt {29}\right )-\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )^2}d\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 \left (\frac {\frac {1}{4} \left (1-3 \sqrt {29}\right ) \int \frac {\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{4} \sqrt {\frac {109+67 \sqrt {29}}{\sqrt {29}-1}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{4} \left (1-3 \sqrt {29}\right ) \int \frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{4} \sqrt {\frac {109+67 \sqrt {29}}{\sqrt {29}-1}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -2 \left (-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}+\frac {\frac {1}{4} \sqrt {\frac {109+67 \sqrt {29}}{\sqrt {29}-1}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )-\frac {1}{16} \left (1-3 \sqrt {29}\right ) \log \left (16 \left (\frac {1}{x}+\frac {1}{4}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{x}+\frac {1}{4}\right )+3 \sqrt {29}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{4} \sqrt {\frac {109+67 \sqrt {29}}{\sqrt {29}-1}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )+\frac {1}{16} \left (1-3 \sqrt {29}\right ) \log \left (16 \left (\frac {1}{x}+\frac {1}{4}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{x}+\frac {1}{4}\right )+3 \sqrt {29}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}\right )\) |
-2*(-1/24*ArcTan[(-96 + 512*(1/4 + x^(-1))^2)/(192*Sqrt[7])]/Sqrt[7] + ((S qrt[(109 + 67*Sqrt[29])/(-1 + Sqrt[29])]*ArcTan[(-4*Sqrt[6*(1 + Sqrt[29])] + 32*(1/4 + x^(-1)))/(4*Sqrt[6*(-1 + Sqrt[29])])])/4 - ((1 - 3*Sqrt[29])* Log[3*Sqrt[29] - 4*Sqrt[6*(1 + Sqrt[29])]*(1/4 + x^(-1)) + 16*(1/4 + x^(-1 ))^2])/16)/(3*Sqrt[174*(1 + Sqrt[29])]) + ((Sqrt[(109 + 67*Sqrt[29])/(-1 + Sqrt[29])]*ArcTan[(4*Sqrt[6*(1 + Sqrt[29])] + 32*(1/4 + x^(-1)))/(4*Sqrt[ 6*(-1 + Sqrt[29])])])/4 + ((1 - 3*Sqrt[29])*Log[3*Sqrt[29] + 4*Sqrt[6*(1 + Sqrt[29])]*(1/4 + x^(-1)) + 16*(1/4 + x^(-1))^2])/16)/(3*Sqrt[174*(1 + Sq rt[29])]))
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[((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[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
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[(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]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15
| method | result | size |
| default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\) | \(41\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\) | \(41\) |
Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 1015, normalized size of antiderivative = 3.79 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\text {Too large to display} \]
-1/168*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696))*log(28731419 5392*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^3 - 1203 8906880*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 + 1 6878104*x + 4897683*I*sqrt(7) - 411405372*sqrt(65/43848*I*sqrt(7) - 109/87 696) + 6055613) - 1/168*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/876 96))*log(-35914274424*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109 /87696))^3 + 16443*(-1/168*I*sqrt(7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 109/ 87696))^2*(-13001*I*sqrt(7) + 1092084*sqrt(65/43848*I*sqrt(7) - 109/87696) - 91520) + 609*(351027*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 1 09/87696))^2 - 613)*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696)) + 2109763*x - 1911147/8*I*sqrt(7) + 40134087/2*sqrt(65/43848*I*sqrt(7) - 109/87696) - 1461344) + 1/1044*(sqrt(174)*sqrt(-4698*(1/168*I*sqrt(7) - 1/ 2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 - 4698*(-1/168*I*sqrt(7) - 1/2*s qrt(-65/43848*I*sqrt(7) - 109/87696))^2 - 87/784*(I*sqrt(7) + 84*sqrt(-65/ 43848*I*sqrt(7) - 109/87696))*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 1 09/87696)) - 7) + 261*sqrt(65/43848*I*sqrt(7) - 109/87696) + 261*sqrt(-65/ 43848*I*sqrt(7) - 109/87696))*log(-16443/2*(-1/168*I*sqrt(7) - 1/2*sqrt(-6 5/43848*I*sqrt(7) - 109/87696))^2*(-13001*I*sqrt(7) + 1092084*sqrt(65/4384 8*I*sqrt(7) - 109/87696) - 91520) - 609/2*(351027*(1/168*I*sqrt(7) - 1/2*s qrt(65/43848*I*sqrt(7) - 109/87696))^2 - 613)*(I*sqrt(7) + 84*sqrt(-65/...
Time = 0.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\operatorname {RootSum} {\left (66298176 t^{4} + 74088 t^{2} + 4095 t + 64, \left ( t \mapsto t \log {\left (\frac {35914274424 t^{3}}{2109763} - \frac {1504863360 t^{2}}{2109763} + \frac {102851343 t}{2109763} + x + \frac {6055613}{16878104} \right )} \right )\right )} \]
RootSum(66298176*_t**4 + 74088*_t**2 + 4095*_t + 64, Lambda(_t, _t*log(359 14274424*_t**3/2109763 - 1504863360*_t**2/2109763 + 102851343*_t/2109763 + x + 6055613/16878104)))
\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - x^{3} + 8 \, x + 8} \,d x } \]
\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - x^{3} + 8 \, x + 8} \,d x } \]
Time = 10.58 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.46 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\sum _{k=1}^4\ln \left (-\frac {\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )\,\left (8064\,\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )+256\,x+\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )\,x\,12285+{\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )}^2\,x\,148176+198072\,{\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )}^2-8\right )}{4096}\right )\,\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right ) \]
symsum(log(-(root(z^4 + (7*z^2)/6264 + (65*z)/1052352 + 1/1035909, z, k)*( 8064*root(z^4 + (7*z^2)/6264 + (65*z)/1052352 + 1/1035909, z, k) + 256*x + 12285*root(z^4 + (7*z^2)/6264 + (65*z)/1052352 + 1/1035909, z, k)*x + 148 176*root(z^4 + (7*z^2)/6264 + (65*z)/1052352 + 1/1035909, z, k)^2*x + 1980 72*root(z^4 + (7*z^2)/6264 + (65*z)/1052352 + 1/1035909, z, k)^2 - 8))/409 6)*root(z^4 + (7*z^2)/6264 + (65*z)/1052352 + 1/1035909, z, k), k, 1, 4)