Integrand size = 24, antiderivative size = 349 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\frac {1+(-1+x)^2}{8 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {3 \left (1+(-1+x)^2\right )}{16 (4+a)^2 \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {\left ((6+a) (25+7 a)+6 (7+2 a) (-1+x)^2\right ) (-1+x)}{32 (3+a)^2 (4+a)^2 \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {3 \left (80+7 a^2+14 \sqrt {4+a}+a \left (47+4 \sqrt {4+a}\right )\right ) \arctan \left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^{5/2} \sqrt {1-\sqrt {4+a}}}-\frac {3 \left (14+4 a-\frac {80+47 a+7 a^2}{\sqrt {4+a}}\right ) \arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^2 \sqrt {1+\sqrt {4+a}}}+\frac {3 \text {arctanh}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{16 (4+a)^{5/2}} \]
1/8*(1+(-1+x)^2)/(4+a)/(3+a-2*(-1+x)^2-(-1+x)^4)^2+3/16*(1+(-1+x)^2)/(4+a) ^2/(3+a-2*(-1+x)^2-(-1+x)^4)+1/8*(5+a+(-1+x)^2)*(-1+x)/(a^2+7*a+12)/(3+a-2 *(-1+x)^2-(-1+x)^4)^2+1/32*((6+a)*(25+7*a)+6*(7+2*a)*(-1+x)^2)*(-1+x)/(a^2 +7*a+12)^2/(3+a-2*(-1+x)^2-(-1+x)^4)+3/16*arctanh((1+(-1+x)^2)/(4+a)^(1/2) )/(4+a)^(5/2)-3/64*arctan((-1+x)/(1-(4+a)^(1/2))^(1/2))*(80+7*a^2+14*(4+a) ^(1/2)+a*(47+4*(4+a)^(1/2)))/(3+a)^2/(4+a)^(5/2)/(1-(4+a)^(1/2))^(1/2)-3/6 4*arctan((-1+x)/(1+(4+a)^(1/2))^(1/2))*(14+4*a+(-7*a^2-47*a-80)/(4+a)^(1/2 ))/(3+a)^2/(4+a)^2/(1+(4+a)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\frac {1}{128} \left (\frac {16 \left (a+2 x-a x+a x^2+x^3\right )}{(3+a) (4+a) \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )^2}+\frac {4 \left (a^2 \left (5-5 x+6 x^2\right )+6 \left (-14+28 x-12 x^2+7 x^3\right )+a \left (-7+31 x+12 x^3\right )\right )}{(3+a)^2 (4+a)^2 \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )}-\frac {3 \text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {72 \log (x-\text {$\#$1})+31 a \log (x-\text {$\#$1})+3 a^2 \log (x-\text {$\#$1})+8 \log (x-\text {$\#$1}) \text {$\#$1}+16 a \log (x-\text {$\#$1}) \text {$\#$1}+4 a^2 \log (x-\text {$\#$1}) \text {$\#$1}+14 \log (x-\text {$\#$1}) \text {$\#$1}^2+4 a \log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]}{\left (12+7 a+a^2\right )^2}\right ) \]
((16*(a + 2*x - a*x + a*x^2 + x^3))/((3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4* x^2 + x^3))^2) + (4*(a^2*(5 - 5*x + 6*x^2) + 6*(-14 + 28*x - 12*x^2 + 7*x^ 3) + a*(-7 + 31*x + 12*x^3)))/((3 + a)^2*(4 + a)^2*(a - x*(-8 + 8*x - 4*x^ 2 + x^3))) - (3*RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (72*Log[x - #1] + 31*a*Log[x - #1] + 3*a^2*Log[x - #1] + 8*Log[x - #1]*#1 + 16*a*Log[x - #1]*#1 + 4*a^2*Log[x - #1]*#1 + 14*Log[x - #1]*#1^2 + 4*a*Log[x - #1]*# 1^2)/(-2 + 4*#1 - 3*#1^2 + #1^3) & ])/(12 + 7*a + a^2)^2)/128
Time = 0.67 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2459, 2202, 1405, 27, 1432, 1086, 1086, 1083, 219, 1492, 27, 1480, 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.
| \(\displaystyle \int \frac {x}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^3} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {x}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^3}d(x-1)\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^3}d(x-1)+\int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^3}d(x-1)\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle -\frac {\int -\frac {2 \left (5 (x-1)^2+7 a+27\right )}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{16 \left (a^2+7 a+12\right )}+\int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^3}d(x-1)+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^3}d(x-1)+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {1}{2} \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^3}d(x-1)^2+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {1}{2} \left (\frac {3 \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)^2}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {1}{2} \left (\frac {3 \left (\frac {\int \frac {1}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2}{2 (a+4)}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {1}{2} \left (\frac {3 \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\int \frac {1}{4 (a+4)-(x-1)^4}d\left (-2 (x-1)^2-2\right )}{a+4}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac {1}{2} \left (\frac {3 \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\int -\frac {6 \left (7 a^2+51 a+2 (2 a+7) (x-1)^2+94\right )}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{8 \left (a^2+7 a+12\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac {1}{2} \left (\frac {3 \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {7 a^2+51 a+2 (2 a+7) (x-1)^2+94}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac {1}{2} \left (\frac {3 \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \left (-\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \int \frac {1}{-(x-1)^2-\sqrt {a+4}-1}d(x-1)+\frac {1}{2} \left (\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \int \frac {1}{-(x-1)^2+\sqrt {a+4}-1}d(x-1)\right )}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac {1}{2} \left (\frac {3 \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\left (\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \arctan \left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\left (-\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}\right )}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac {1}{2} \left (\frac {3 \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )}{4 (a+4)}+\frac {x}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\right )\) |
((5 + a + (-1 + x)^2)*(-1 + x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + ((((6 + a)*(25 + 7*a) + 6*(7 + 2*a)*(-1 + x)^2)*(-1 + x ))/(4*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (3*(-1/2*((1 4 + 4*a + (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]])/Sqrt[1 - Sqrt[4 + a]] - ((14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(2*Sqrt[1 + Sqrt[4 + a]])))/( 4*(12 + 7*a + a^2)))/(8*(12 + 7*a + a^2)) + (x/(4*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + (3*(x/(2*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x) ^4)) - ArcTanh[(-2 - 2*(-1 + x)^2)/(2*Sqrt[4 + a])]/(2*(4 + a)^(3/2))))/(4 *(4 + a)))/2
3.2.29.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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[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[((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[(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[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(-\frac {\frac {3 \left (7+2 a \right ) x^{7}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {3 \left (a^{2}-8 a -40\right ) x^{6}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {\left (29 a^{2}-127 a -792\right ) x^{5}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (73 a^{2}-227 a -1668\right ) x^{4}}{32 a^{4}+448 a^{3}+2336 a^{2}+5376 a +4608}-\frac {\left (62 a^{2}-103 a -1104\right ) x^{3}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {\left (5 a^{3}-26 a^{2}+140 a +1008\right ) x^{2}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {3 \left (3 a^{3}-17 a^{2}-40 a +192\right ) x}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {3 a \left (3 a^{2}+7 a -12\right )}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}}{\left (-x^{4}+4 x^{3}-8 x^{2}+a +8 x \right )^{2}}-\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (-72+2 \left (-2 a -7\right ) \textit {\_R}^{2}+4 \left (-a^{2}-4 a -2\right ) \textit {\_R} -3 a^{2}-31 a \right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{128 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}\) | \(409\) |
| risch | \(\frac {-\frac {3 \left (7+2 a \right ) x^{7}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {3 \left (a^{2}-8 a -40\right ) x^{6}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (29 a^{2}-127 a -792\right ) x^{5}}{32 a^{4}+448 a^{3}+2336 a^{2}+5376 a +4608}-\frac {\left (73 a^{2}-227 a -1668\right ) x^{4}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (62 a^{2}-103 a -1104\right ) x^{3}}{16 a^{4}+224 a^{3}+1168 a^{2}+2688 a +2304}+\frac {\left (5 a^{3}-26 a^{2}+140 a +1008\right ) x^{2}}{16 a^{4}+224 a^{3}+1168 a^{2}+2688 a +2304}-\frac {3 \left (3 a^{3}-17 a^{2}-40 a +192\right ) x}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {3 a \left (3 a^{2}+7 a -12\right )}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}}{\left (-x^{4}+4 x^{3}-8 x^{2}+a +8 x \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {2 \left (7+2 a \right ) \textit {\_R}^{2}}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}+\frac {4 \left (a^{2}+4 a +2\right ) \textit {\_R}}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}+\frac {3 a^{2}+31 a +72}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{128}\) | \(448\) |
-(3/16*(7+2*a)/(a^4+14*a^3+73*a^2+168*a+144)*x^7+3/16*(a^2-8*a-40)/(a^4+14 *a^3+73*a^2+168*a+144)*x^6-1/32*(29*a^2-127*a-792)/(a^4+14*a^3+73*a^2+168* a+144)*x^5+1/32*(73*a^2-227*a-1668)/(a^4+14*a^3+73*a^2+168*a+144)*x^4-1/16 *(62*a^2-103*a-1104)/(a^4+14*a^3+73*a^2+168*a+144)*x^3-1/16*(5*a^3-26*a^2+ 140*a+1008)/(a^4+14*a^3+73*a^2+168*a+144)*x^2+3/32*(3*a^3-17*a^2-40*a+192) /(a^4+14*a^3+73*a^2+168*a+144)*x-3/32*a*(3*a^2+7*a-12)/(a^4+14*a^3+73*a^2+ 168*a+144))/(-x^4+4*x^3-8*x^2+a+8*x)^2-3/128/(a^4+14*a^3+73*a^2+168*a+144) *sum((-72+2*(-2*a-7)*_R^2+4*(-a^2-4*a-2)*_R-3*a^2-31*a)/(-_R^3+3*_R^2-4*_R +2)*ln(x-_R),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a))
Timed out. \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (318) = 636\).
Time = 42.84 (sec) , antiderivative size = 1102, normalized size of antiderivative = 3.16 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\text {Too large to display} \]
-(-9*a**3 - 21*a**2 + 36*a + x**7*(12*a + 42) + x**6*(6*a**2 - 48*a - 240) + x**5*(-29*a**2 + 127*a + 792) + x**4*(73*a**2 - 227*a - 1668) + x**3*(- 124*a**2 + 206*a + 2208) + x**2*(-10*a**3 + 52*a**2 - 280*a - 2016) + x*(9 *a**3 - 51*a**2 - 120*a + 576))/(32*a**6 + 448*a**5 + 2336*a**4 + 5376*a** 3 + 4608*a**2 + x**8*(32*a**4 + 448*a**3 + 2336*a**2 + 5376*a + 4608) + x* *7*(-256*a**4 - 3584*a**3 - 18688*a**2 - 43008*a - 36864) + x**6*(1024*a** 4 + 14336*a**3 + 74752*a**2 + 172032*a + 147456) + x**5*(-2560*a**4 - 3584 0*a**3 - 186880*a**2 - 430080*a - 368640) + x**4*(-64*a**5 + 3200*a**4 + 5 2672*a**3 + 288256*a**2 + 678912*a + 589824) + x**3*(256*a**5 - 512*a**4 - 38656*a**3 - 256000*a**2 - 651264*a - 589824) + x**2*(-512*a**5 - 5120*a* *4 - 8704*a**3 + 63488*a**2 + 270336*a + 294912) + x*(512*a**5 + 7168*a**4 + 37376*a**3 + 86016*a**2 + 73728*a)) - RootSum(_t**4*(268435456*a**15 + 14763950080*a**14 + 378493992960*a**13 + 5999532441600*a**12 + 65757291479 040*a**11 + 527875908304896*a**10 + 3206246773555200*a**9 + 15003759578972 160*a**8 + 54537151127224320*a**7 + 153980418717122560*a**6 + 334927734494 986240*a**5 + 551152193655275520*a**4 + 664192984106926080*a**3 + 55336221 2027105280*a**2 + 284993413919539200*a + 68398419340689408) + _t**2*(-4718 592*a**10 - 196116480*a**9 - 3648061440*a**8 - 40022212608*a**7 - 28693993 8816*a**6 - 1405437345792*a**5 - 4764645457920*a**4 - 11043392716800*a**3 - 16752587046912*a**2 - 15023392948224*a - 6049461436416) + _t*(-270950...
\[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\int { -\frac {x}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{3}} \,d x } \]
-1/32*(6*(2*a + 7)*x^7 + 6*(a^2 - 8*a - 40)*x^6 - (29*a^2 - 127*a - 792)*x ^5 + (73*a^2 - 227*a - 1668)*x^4 - 2*(62*a^2 - 103*a - 1104)*x^3 - 9*a^3 - 2*(5*a^3 - 26*a^2 + 140*a + 1008)*x^2 - 21*a^2 + 3*(3*a^3 - 17*a^2 - 40*a + 192)*x + 36*a)/((a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^8 - 8*(a^4 + 14 *a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144 )*x^6 + a^6 - 80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a ^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*x^3 + 168*a^3 - 16*(a^5 + 10* a^4 + 17*a^3 - 124*a^2 - 528*a - 576)*x^2 + 144*a^2 + 16*(a^5 + 14*a^4 + 7 3*a^3 + 168*a^2 + 144*a)*x) - 3/32*integrate((2*(2*a + 7)*x^2 + 3*a^2 + 4* (a^2 + 4*a + 2)*x + 31*a + 72)/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)/(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)
\[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\int { -\frac {x}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{3}} \,d x } \]
Time = 9.93 (sec) , antiderivative size = 2200, normalized size of antiderivative = 6.30 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\text {Too large to display} \]
symsum(log(root(15003759578972160*a^8*z^4 + 54537151127224320*a^7*z^4 + 15 3980418717122560*a^6*z^4 + 334927734494986240*a^5*z^4 + 551152193655275520 *a^4*z^4 + 664192984106926080*a^3*z^4 + 553362212027105280*a^2*z^4 + 59995 32441600*a^12*z^4 + 527875908304896*a^10*z^4 + 284993413919539200*a*z^4 + 3206246773555200*a^9*z^4 + 14763950080*a^14*z^4 + 65757291479040*a^11*z^4 + 378493992960*a^13*z^4 + 268435456*a^15*z^4 + 68398419340689408*z^4 - 471 8592*a^10*z^2 - 3648061440*a^8*z^2 - 286939938816*a^6*z^2 - 15023392948224 *a*z^2 - 16752587046912*a^2*z^2 - 4764645457920*a^4*z^2 - 40022212608*a^7* z^2 - 11043392716800*a^3*z^2 - 1405437345792*a^5*z^2 - 196116480*a^9*z^2 - 6049461436416*z^2 + 5375877120*a^4*z + 839890944*a^5*z + 47542173696*a^2* z + 72880128*a^6*z + 2709504*a^7*z + 20640890880*a^3*z + 60827369472*a*z + 33351008256*z - 74027520*a - 29249424*a^2 - 4706424*a^3 - 155601*a^4 + 20 736*a^5 - 68345856, z, k)*((242823168*a + 170044416*a^2 + 63509760*a^3 + 1 3340736*a^4 + 1494144*a^5 + 69696*a^6 + 144506880)/(16384*(940032*a + 1195 776*a^2 + 899328*a^3 + 442864*a^4 + 149208*a^5 + 34833*a^6 + 5564*a^7 + 58 2*a^8 + 36*a^9 + a^10 + 331776)) + root(15003759578972160*a^8*z^4 + 545371 51127224320*a^7*z^4 + 153980418717122560*a^6*z^4 + 334927734494986240*a^5* z^4 + 551152193655275520*a^4*z^4 + 664192984106926080*a^3*z^4 + 5533622120 27105280*a^2*z^4 + 5999532441600*a^12*z^4 + 527875908304896*a^10*z^4 + 284 993413919539200*a*z^4 + 3206246773555200*a^9*z^4 + 14763950080*a^14*z^4...