Integrand size = 17, antiderivative size = 211 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=-\frac {\arctan \left (\frac {3-\left (1+\frac {4}{x}\right )^2}{6 \sqrt {7}}\right )}{12 \sqrt {7}}-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )-\frac {1}{12} \sqrt {\frac {109+67 \sqrt {29}}{1218}} \arctan \left (\frac {8+\left (2-\sqrt {6 \left (1+\sqrt {29}\right )}\right ) x}{\sqrt {6 \left (-1+\sqrt {29}\right )} x}\right )+\frac {1}{12} \sqrt {\frac {-109+67 \sqrt {29}}{1218}} \text {arctanh}\left (\frac {\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )}{3 \sqrt {29}+\left (1+\frac {4}{x}\right )^2}\right ) \] Output:
-1/84*arctan(1/42*(3-(1+4/x)^2)*7^(1/2))*7^(1/2)-1/14616*(132762+81606*29^ (1/2))^(1/2)*arctan((2+(6+6*29^(1/2))^(1/2)+8/x)/(-6+6*29^(1/2))^(1/2))-1/ 14616*(132762+81606*29^(1/2))^(1/2)*arctan((8+(2-(6+6*29^(1/2))^(1/2))*x)/ (-6+6*29^(1/2))^(1/2)/x)+1/14616*(-132762+81606*29^(1/2))^(1/2)*arctanh((6 +6*29^(1/2))^(1/2)*(1+4/x)/(3*29^(1/2)+(1+4/x)^2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.21 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\text {RootSum}\left [8+8 \text {$\#$1}-\text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{8-3 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(8 + 8*x - x^3 + 8*x^4)^(-1),x]
Output:
RootSum[8 + 8*#1 - #1^3 + 8*#1^4 & , Log[x - #1]/(8 - 3*#1^2 + 32*#1^3) & ]
Time = 1.07 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.52, 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 |
| \(\displaystyle -2 \left (-\frac {\arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{24 \sqrt {7}}+\frac {\frac {1}{4} \sqrt {\frac {109+67 \sqrt {29}}{\sqrt {29}-1}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )-\frac {1}{16} \left (1-3 \sqrt {29}\right ) \log \left (16 \left (\frac {1}{x}+\frac {1}{4}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{x}+\frac {1}{4}\right )+3 \sqrt {29}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{4} \sqrt {\frac {109+67 \sqrt {29}}{\sqrt {29}-1}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )+\frac {1}{16} \left (1-3 \sqrt {29}\right ) \log \left (16 \left (\frac {1}{x}+\frac {1}{4}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{x}+\frac {1}{4}\right )+3 \sqrt {29}\right )}{3 \sqrt {174 \left (1+\sqrt {29}\right )}}\right )\) |
Input:
Int[(8 + 8*x - x^3 + 8*x^4)^(-1),x]
Output:
-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])]))
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 = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.19
| 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 | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\) | \(41\) |
Input:
int(1/(8*x^4-x^3+8*x+8),x,method=_RETURNVERBOSE)
Output:
sum(1/(32*_R^3-3*_R^2+8)*ln(x-_R),_R=RootOf(8*_Z^4-_Z^3+8*_Z+8))
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (148) = 296\).
Time = 0.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.44 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=-\frac {1}{12} \, \sqrt {\frac {1}{455} \, {\left (67 \, \sqrt {29} + 109\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} + \frac {67}{1218} \, \sqrt {29} + \frac {283}{1218}} \arctan \left (-\frac {1}{8187140} \, {\left (65 \, \sqrt {29} {\left (62891 \, x - 4738\right )} - 174 \, {\left (\sqrt {29} {\left (15701 \, x - 33858\right )} + 25102 \, x - 74206\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} + 5310045 \, x - 5553210\right )} \sqrt {\frac {1}{455} \, {\left (67 \, \sqrt {29} + 109\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} + \frac {67}{1218} \, \sqrt {29} + \frac {283}{1218}}\right ) + \frac {1}{12} \, \sqrt {-\frac {1}{455} \, {\left (67 \, \sqrt {29} + 109\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} + \frac {67}{1218} \, \sqrt {29} + \frac {283}{1218}} \arctan \left (\frac {1}{8187140} \, {\left (65 \, \sqrt {29} {\left (62891 \, x - 4738\right )} + 174 \, {\left (\sqrt {29} {\left (15701 \, x - 33858\right )} + 25102 \, x - 74206\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} + 5310045 \, x - 5553210\right )} \sqrt {-\frac {1}{455} \, {\left (67 \, \sqrt {29} + 109\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} + \frac {67}{1218} \, \sqrt {29} + \frac {283}{1218}}\right ) + \frac {1}{24} \, \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} \log \left (2080 \, x^{2} + 3 \, {\left (\sqrt {29} {\left (65 \, x - 88\right )} + 1885 \, x - 116\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} - 130 \, x + 390 \, \sqrt {29} + 130\right ) - \frac {1}{24} \, \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} \log \left (2080 \, x^{2} - 3 \, {\left (\sqrt {29} {\left (65 \, x - 88\right )} + 1885 \, x - 116\right )} \sqrt {\frac {67}{1218} \, \sqrt {29} - \frac {109}{1218}} - 130 \, x + 390 \, \sqrt {29} + 130\right ) \] Input:
integrate(1/(8*x^4-x^3+8*x+8),x, algorithm="fricas")
Output:
-1/12*sqrt(1/455*(67*sqrt(29) + 109)*sqrt(67/1218*sqrt(29) - 109/1218) + 6 7/1218*sqrt(29) + 283/1218)*arctan(-1/8187140*(65*sqrt(29)*(62891*x - 4738 ) - 174*(sqrt(29)*(15701*x - 33858) + 25102*x - 74206)*sqrt(67/1218*sqrt(2 9) - 109/1218) + 5310045*x - 5553210)*sqrt(1/455*(67*sqrt(29) + 109)*sqrt( 67/1218*sqrt(29) - 109/1218) + 67/1218*sqrt(29) + 283/1218)) + 1/12*sqrt(- 1/455*(67*sqrt(29) + 109)*sqrt(67/1218*sqrt(29) - 109/1218) + 67/1218*sqrt (29) + 283/1218)*arctan(1/8187140*(65*sqrt(29)*(62891*x - 4738) + 174*(sqr t(29)*(15701*x - 33858) + 25102*x - 74206)*sqrt(67/1218*sqrt(29) - 109/121 8) + 5310045*x - 5553210)*sqrt(-1/455*(67*sqrt(29) + 109)*sqrt(67/1218*sqr t(29) - 109/1218) + 67/1218*sqrt(29) + 283/1218)) + 1/24*sqrt(67/1218*sqrt (29) - 109/1218)*log(2080*x^2 + 3*(sqrt(29)*(65*x - 88) + 1885*x - 116)*sq rt(67/1218*sqrt(29) - 109/1218) - 130*x + 390*sqrt(29) + 130) - 1/24*sqrt( 67/1218*sqrt(29) - 109/1218)*log(2080*x^2 - 3*(sqrt(29)*(65*x - 88) + 1885 *x - 116)*sqrt(67/1218*sqrt(29) - 109/1218) - 130*x + 390*sqrt(29) + 130)
Time = 0.57 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.19 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\operatorname {RootSum} {\left (66298176 t^{4} + 74088 t^{2} + 4095 t + 64, \left ( t \mapsto t \log {\left (\frac {35914274424 t^{3}}{2109763} - \frac {1504863360 t^{2}}{2109763} + \frac {102851343 t}{2109763} + x + \frac {6055613}{16878104} \right )} \right )\right )} \] Input:
integrate(1/(8*x**4-x**3+8*x+8),x)
Output:
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)))
\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - x^{3} + 8 \, x + 8} \,d x } \] Input:
integrate(1/(8*x^4-x^3+8*x+8),x, algorithm="maxima")
Output:
integrate(1/(8*x^4 - x^3 + 8*x + 8), x)
\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - x^{3} + 8 \, x + 8} \,d x } \] Input:
integrate(1/(8*x^4-x^3+8*x+8),x, algorithm="giac")
Output:
integrate(1/(8*x^4 - x^3 + 8*x + 8), x)
Time = 23.73 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.58 \[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\sum _{k=1}^4\ln \left (-\frac {\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )\,\left (8064\,\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )+256\,x+\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )\,x\,12285+{\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )}^2\,x\,148176+198072\,{\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right )}^2-8\right )}{4096}\right )\,\mathrm {root}\left (z^4+\frac {7\,z^2}{6264}+\frac {65\,z}{1052352}+\frac {1}{1035909},z,k\right ) \] Input:
int(1/(8*x - x^3 + 8*x^4 + 8),x)
Output:
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)
\[ \int \frac {1}{8+8 x-x^3+8 x^4} \, dx=\int \frac {1}{8 x^{4}-x^{3}+8 x +8}d x \] Input:
int(1/(8*x^4-x^3+8*x+8),x)
Output:
int(1/(8*x**4 - x**3 + 8*x + 8),x)