Integrand size = 22, antiderivative size = 208 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \arctan \left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \arctan \left (\frac {8+\left (6-\sqrt {2 \left (19+\sqrt {517}\right )}\right ) x}{\sqrt {2 \left (-19+\sqrt {517}\right )} x}\right )+\frac {1}{4} \sqrt {\frac {3}{13}} \arctan \left (\frac {8+12 x-5 x^2}{\sqrt {39} x^2}\right )+\frac {1}{4} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \text {arctanh}\left (\frac {\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )}{\sqrt {517}+\left (3+\frac {4}{x}\right )^2}\right ) \] Output:
-1/161304*(208364442+9476610*517^(1/2))^(1/2)*arctan((6+(38+2*517^(1/2))^( 1/2)+8/x)/(-38+2*517^(1/2))^(1/2))-1/161304*(208364442+9476610*517^(1/2))^ (1/2)*arctan((8+(6-(38+2*517^(1/2))^(1/2))*x)/(-38+2*517^(1/2))^(1/2)/x)+1 /52*39^(1/2)*arctan(1/39*(-5*x^2+12*x+8)*39^(1/2)/x^2)+1/161304*(-20836444 2+9476610*517^(1/2))^(1/2)*arctanh((38+2*517^(1/2))^(1/2)*(3+4/x)/(517^(1/ 2)+(3+4/x)^2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.26 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\text {RootSum}\left [8+24 \text {$\#$1}+8 \text {$\#$1}^2-15 \text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{24+16 \text {$\#$1}-45 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-1),x]
Output:
RootSum[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , Log[x - #1]/(24 + 16*#1 - 45*#1^2 + 32*#1^3) & ]
Time = 1.13 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.50, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, 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.
| \(\displaystyle \int \frac {1}{8 x^4-15 x^3+8 x^2+24 x+8} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -1024 \int \frac {\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2}{512 \left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int \frac {\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )+\int -\frac {24 \left (\frac {3}{4}+\frac {1}{x}\right )}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )-24 \int \frac {\frac {3}{4}+\frac {1}{x}}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )-12 \int \frac {1}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -2 \left (24 \int \frac {1}{-\left (\frac {3}{4}+\frac {1}{x}\right )^4-159744}d\left (512 \left (\frac {3}{4}+\frac {1}{x}\right )^2-608\right )+\int \frac {16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 \left (\int \frac {16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9}{256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}d\left (\frac {3}{4}+\frac {1}{x}\right )-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -2 \left (\frac {\int \frac {8 \left (9 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}-2 \left (9-\sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{8 \sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\int \frac {8 \left (2 \left (9-\sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )+9 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{8 \sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {\int \frac {9 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}-2 \left (9-\sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\int \frac {2 \left (9-\sqrt {517}\right ) \left (\frac {3}{4}+\frac {1}{x}\right )+9 \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -2 \left (\frac {\sqrt {\frac {1}{2} \left (5167+235 \sqrt {517}\right )} \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )-\frac {1}{16} \left (9-\sqrt {517}\right ) \int -\frac {4 \left (\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\sqrt {\frac {1}{2} \left (5167+235 \sqrt {517}\right )} \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{16} \left (9-\sqrt {517}\right ) \int \frac {4 \left (8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {\sqrt {\frac {1}{2} \left (5167+235 \sqrt {517}\right )} \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (9-\sqrt {517}\right ) \int \frac {\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\sqrt {\frac {1}{2} \left (5167+235 \sqrt {517}\right )} \int \frac {1}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (9-\sqrt {517}\right ) \int \frac {8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -2 \left (\frac {\frac {1}{4} \left (9-\sqrt {517}\right ) \int \frac {\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )-\sqrt {2 \left (5167+235 \sqrt {517}\right )} \int \frac {1}{32 \left (19-\sqrt {517}\right )-\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )^2}d\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{4} \left (9-\sqrt {517}\right ) \int \frac {8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )-\sqrt {2 \left (5167+235 \sqrt {517}\right )} \int \frac {1}{32 \left (19-\sqrt {517}\right )-\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )^2}d\left (32 \left (\frac {3}{4}+\frac {1}{x}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 \left (\frac {\frac {1}{4} \left (9-\sqrt {517}\right ) \int \frac {\sqrt {2 \left (19+\sqrt {517}\right )}-8 \left (\frac {3}{4}+\frac {1}{x}\right )}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{\sqrt {517}-19}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{4} \left (9-\sqrt {517}\right ) \int \frac {8 \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {2 \left (19+\sqrt {517}\right )}}{16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {3}{4}+\frac {1}{x}\right )+\sqrt {517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{\sqrt {517}-19}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -2 \left (-\frac {1}{8} \sqrt {\frac {3}{13}} \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {3}{4}\right )^2-608}{64 \sqrt {39}}\right )+\frac {\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{\sqrt {517}-19}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )-4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )-\frac {1}{16} \left (9-\sqrt {517}\right ) \log \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2-4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {1}{x}+\frac {3}{4}\right )+\sqrt {517}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}+\frac {\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{\sqrt {517}-19}} \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {3}{4}\right )+4 \sqrt {2 \left (19+\sqrt {517}\right )}}{4 \sqrt {2 \left (\sqrt {517}-19\right )}}\right )+\frac {1}{16} \left (9-\sqrt {517}\right ) \log \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+4 \sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {1}{x}+\frac {3}{4}\right )+\sqrt {517}\right )}{\sqrt {1034 \left (19+\sqrt {517}\right )}}\right )\) |
Input:
Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-1),x]
Output:
-2*(-1/8*(Sqrt[3/13]*ArcTan[(-608 + 512*(3/4 + x^(-1))^2)/(64*Sqrt[39])]) + ((Sqrt[(5167 + 235*Sqrt[517])/(-19 + Sqrt[517])]*ArcTan[(-4*Sqrt[2*(19 + Sqrt[517])] + 32*(3/4 + x^(-1)))/(4*Sqrt[2*(-19 + Sqrt[517])])])/4 - ((9 - Sqrt[517])*Log[Sqrt[517] - 4*Sqrt[2*(19 + Sqrt[517])]*(3/4 + x^(-1)) + 1 6*(3/4 + x^(-1))^2])/16)/Sqrt[1034*(19 + Sqrt[517])] + ((Sqrt[(5167 + 235* Sqrt[517])/(-19 + Sqrt[517])]*ArcTan[(4*Sqrt[2*(19 + Sqrt[517])] + 32*(3/4 + x^(-1)))/(4*Sqrt[2*(-19 + Sqrt[517])])])/4 + ((9 - Sqrt[517])*Log[Sqrt[ 517] + 4*Sqrt[2*(19 + Sqrt[517])]*(3/4 + x^(-1)) + 16*(3/4 + x^(-1))^2])/1 6)/Sqrt[1034*(19 + Sqrt[517])])
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 = 49, normalized size of antiderivative = 0.24
| method | result | size |
| default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\) | \(49\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\) | \(49\) |
Input:
int(1/(8*x^4-15*x^3+8*x^2+24*x+8),x,method=_RETURNVERBOSE)
Output:
sum(1/(32*_R^3-45*_R^2+16*_R+24)*ln(x-_R),_R=RootOf(8*_Z^4-15*_Z^3+8*_Z^2+ 24*_Z+8))
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (148) = 296\).
Time = 0.11 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.47 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8),x, algorithm="fricas")
Output:
-1/4*sqrt(1/1417*(2585*sqrt(47/11) + 5167)*sqrt(5/78*sqrt(47/11) - 5167/40 326) + 5/78*sqrt(47/11) + 14473/40326)*arctan(-11/414926812*(109*sqrt(47/1 1)*(1650521*x - 1323522) - 94*(11*sqrt(47/11)*(1048281*x - 2936950) + 2272 7406*x - 65658598)*sqrt(5/78*sqrt(47/11) - 5167/40326) + 329311563*x - 314 296050)*sqrt(1/1417*(2585*sqrt(47/11) + 5167)*sqrt(5/78*sqrt(47/11) - 5167 /40326) + 5/78*sqrt(47/11) + 14473/40326)) + 1/4*sqrt(-1/1417*(2585*sqrt(4 7/11) + 5167)*sqrt(5/78*sqrt(47/11) - 5167/40326) + 5/78*sqrt(47/11) + 144 73/40326)*arctan(11/414926812*(109*sqrt(47/11)*(1650521*x - 1323522) + 94* (11*sqrt(47/11)*(1048281*x - 2936950) + 22727406*x - 65658598)*sqrt(5/78*s qrt(47/11) - 5167/40326) + 329311563*x - 314296050)*sqrt(-1/1417*(2585*sqr t(47/11) + 5167)*sqrt(5/78*sqrt(47/11) - 5167/40326) + 5/78*sqrt(47/11) + 14473/40326)) + 1/8*sqrt(5/78*sqrt(47/11) - 5167/40326)*log(3488*x^2 + 11* (sqrt(47/11)*(2071*x - 2064) + 5123*x - 3948)*sqrt(5/78*sqrt(47/11) - 5167 /40326) - 3270*x + 2398*sqrt(47/11) + 1962) - 1/8*sqrt(5/78*sqrt(47/11) - 5167/40326)*log(3488*x^2 - 11*(sqrt(47/11)*(2071*x - 2064) + 5123*x - 3948 )*sqrt(5/78*sqrt(47/11) - 5167/40326) - 3270*x + 2398*sqrt(47/11) + 1962)
Time = 1.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.20 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\operatorname {RootSum} {\left (50326848 t^{4} + 765960 t^{2} + 12753 t + 64, \left ( t \mapsto t \log {\left (\frac {100785893208 t^{3}}{4758335} - \frac {1430512512 t^{2}}{4758335} + \frac {72982352521 t}{223641745} + x + \frac {2270349121}{1789133960} \right )} \right )\right )} \] Input:
integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8),x)
Output:
RootSum(50326848*_t**4 + 765960*_t**2 + 12753*_t + 64, Lambda(_t, _t*log(1 00785893208*_t**3/4758335 - 1430512512*_t**2/4758335 + 72982352521*_t/2236 41745 + x + 2270349121/1789133960)))
\[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8} \,d x } \] Input:
integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8),x, algorithm="maxima")
Output:
integrate(1/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)
\[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8} \,d x } \] Input:
integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8),x, algorithm="giac")
Output:
integrate(1/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)
Time = 22.77 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\sum _{k=1}^4\ln \left (-\frac {\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )\,\left (2184\,\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )+256\,x+\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )\,x\,38259+{\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )}^2\,x\,1531920+805896\,{\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )}^2-120\right )}{4096}\right )\,\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right ) \] Input:
int(1/(24*x + 8*x^2 - 15*x^3 + 8*x^4 + 8),x)
Output:
symsum(log(-(root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z, k)*(2184*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z, k) + 256*x + 38259*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z , k)*x + 1531920*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z, k)^2*x + 805896*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/7863 57, z, k)^2 - 120))/4096)*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z, k), k, 1, 4)
\[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\int \frac {1}{8 x^{4}-15 x^{3}+8 x^{2}+24 x +8}d x \] Input:
int(1/(8*x^4-15*x^3+8*x^2+24*x+8),x)
Output:
int(1/(8*x**4 - 15*x**3 + 8*x**2 + 24*x + 8),x)