\(\int \frac {d+e x+f x^2}{(4-5 x^2+x^4)^3} \, dx\) [27]

Optimal result

Integrand size = 23, antiderivative size = 183 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {e}{108 \left (1-x^2\right )^2}-\frac {e}{54 \left (1-x^2\right )}-\frac {e}{108 \left (4-x^2\right )^2}-\frac {e}{54 \left (4-x^2\right )}+\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f) \text {arctanh}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \text {arctanh}(x)-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right ) \] Output:

1/108*e/(-x^2+1)^2-e/(-54*x^2+54)-1/108*e/(-x^2+4)^2-e/(-54*x^2+216)+1/144 
*x*(17*d+20*f-(5*d+8*f)*x^2)/(x^4-5*x^2+4)^2-x*(59*d+380*f-35*(d+4*f)*x^2) 
/(3456*x^4-17280*x^2+13824)-1/20736*(313*d+820*f)*arctanh(1/2*x)+1/648*(13 
*d+25*f)*arctanh(x)-1/81*e*ln(-x^2+1)+1/81*e*ln(-x^2+4)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {\frac {288 \left (17 d x+20 f x-5 d x^3-8 f x^3+e \left (20-8 x^2\right )\right )}{\left (4-5 x^2+x^4\right )^2}+\frac {12 \left (64 e \left (-5+2 x^2\right )+20 f x \left (-19+7 x^2\right )+d x \left (-59+35 x^2\right )\right )}{4-5 x^2+x^4}-32 (13 d+16 e+25 f) \log (1-x)+(313 d+512 e+820 f) \log (2-x)+32 (13 d-16 e+25 f) \log (1+x)+(-313 d+512 e-820 f) \log (2+x)}{41472} \] Input:

Integrate[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^3,x]
 

Output:

((288*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2)))/(4 - 5*x^2 + 
 x^4)^2 + (12*(64*e*(-5 + 2*x^2) + 20*f*x*(-19 + 7*x^2) + d*x*(-59 + 35*x^ 
2)))/(4 - 5*x^2 + x^4) - 32*(13*d + 16*e + 25*f)*Log[1 - x] + (313*d + 512 
*e + 820*f)*Log[2 - x] + 32*(13*d - 16*e + 25*f)*Log[1 + x] + (-313*d + 51 
2*e - 820*f)*Log[2 + x])/41472
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2202, 27, 1432, 1084, 1492, 25, 1492, 27, 1480, 220, 2009}

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.

\(\displaystyle \int \frac {d+e x+f x^2}{\left (x^4-5 x^2+4\right )^3} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \frac {f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\int \frac {e x}{\left (x^4-5 x^2+4\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+e \int \frac {x}{\left (x^4-5 x^2+4\right )^3}dx\)

\(\Big \downarrow \) 1432

\(\displaystyle \int \frac {f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\frac {1}{2} e \int \frac {1}{\left (x^4-5 x^2+4\right )^3}dx^2\)

\(\Big \downarrow \) 1084

\(\displaystyle \int \frac {f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2\)

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {1}{144} \int -\frac {-5 (5 d+8 f) x^2+19 d-20 f}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{144} \int \frac {-5 (5 d+8 f) x^2+19 d-20 f}{\left (x^4-5 x^2+4\right )^2}dx+\frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{144} \left (-\frac {1}{72} \int -\frac {3 \left (35 (d+4 f) x^2+173 d+260 f\right )}{x^4-5 x^2+4}dx-\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{144} \left (\frac {1}{24} \int \frac {35 (d+4 f) x^2+173 d+260 f}{x^4-5 x^2+4}dx-\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{144} \left (\frac {1}{24} \left (\frac {1}{3} (313 d+820 f) \int \frac {1}{x^2-4}dx-\frac {16}{3} (13 d+25 f) \int \frac {1}{x^2-1}dx\right )-\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{2} e \int \left (-\frac {2}{81 \left (4-x^2\right )}-\frac {1}{27 \left (4-x^2\right )^2}-\frac {1}{27 \left (4-x^2\right )^3}+\frac {2}{81 \left (1-x^2\right )}-\frac {1}{27 \left (1-x^2\right )^2}+\frac {1}{27 \left (1-x^2\right )^3}\right )dx^2+\frac {1}{144} \left (\frac {1}{24} \left (\frac {16}{3} \text {arctanh}(x) (13 d+25 f)-\frac {1}{6} \text {arctanh}\left (\frac {x}{2}\right ) (313 d+820 f)\right )-\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{144} \left (\frac {1}{24} \left (\frac {16}{3} \text {arctanh}(x) (13 d+25 f)-\frac {1}{6} \text {arctanh}\left (\frac {x}{2}\right ) (313 d+820 f)\right )-\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} e \left (-\frac {1}{27 \left (1-x^2\right )}-\frac {1}{27 \left (4-x^2\right )}+\frac {1}{54 \left (1-x^2\right )^2}-\frac {1}{54 \left (4-x^2\right )^2}-\frac {2}{81} \log \left (1-x^2\right )+\frac {2}{81} \log \left (4-x^2\right )\right )\)

Input:

Int[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^3,x]
 

Output:

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (-1/24*(x* 
(59*d + 380*f - 35*(d + 4*f)*x^2))/(4 - 5*x^2 + x^4) + (-1/6*((313*d + 820 
*f)*ArcTanh[x/2]) + (16*(13*d + 25*f)*ArcTanh[x])/3)/24)/144 + (e*(1/(54*( 
1 - x^2)^2) - 1/(27*(1 - x^2)) - 1/(54*(4 - x^2)^2) - 1/(27*(4 - x^2)) - ( 
2*Log[1 - x^2])/81 + (2*Log[4 - x^2])/81))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 1084

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
- 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[(b/2 - q/2 + c*x)^p*(b/2 + q 
/2 + c*x)^p, x], x], x] /;  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c}, 
 x] && IntegerQ[p] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 1432

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]
 

rule 1480

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]
 

rule 1492

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2202

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]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.76

method	result	size
norman	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}\right ) x -\frac {5 e \,x^{4}}{18}+\frac {e \,x^{6}}{27}+\frac {5 e \,x^{2}}{9}-\frac {25 e}{108}}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}\right ) \ln \left (x +2\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}\right ) \ln \left (x -1\right )+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}\right ) \ln \left (1+x \right )+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}\right ) \ln \left (x -2\right )\)	\(139\)
risch	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}\right ) x -\frac {5 e \,x^{4}}{18}+\frac {e \,x^{6}}{27}+\frac {5 e \,x^{2}}{9}-\frac {25 e}{108}}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\frac {13 \ln \left (1+x \right ) d}{1296}-\frac {\ln \left (1+x \right ) e}{81}+\frac {25 \ln \left (1+x \right ) f}{1296}-\frac {313 \ln \left (x +2\right ) d}{41472}+\frac {\ln \left (x +2\right ) e}{81}-\frac {205 \ln \left (x +2\right ) f}{10368}-\frac {13 \ln \left (1-x \right ) d}{1296}-\frac {\ln \left (1-x \right ) e}{81}-\frac {25 \ln \left (1-x \right ) f}{1296}+\frac {313 \ln \left (2-x \right ) d}{41472}+\frac {\ln \left (2-x \right ) e}{81}+\frac {205 \ln \left (2-x \right ) f}{10368}\)	\(175\)
default	\(-\frac {-\frac {19 d}{6912}-\frac {17 e}{3456}-\frac {5 f}{576}}{x -2}-\frac {\frac {d}{1728}+\frac {e}{864}+\frac {f}{432}}{2 \left (x -2\right )^{2}}+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}\right ) \ln \left (x -2\right )-\frac {-\frac {d}{432}+\frac {e}{144}-\frac {5 f}{432}}{1+x}-\frac {\frac {d}{216}-\frac {e}{216}+\frac {f}{216}}{2 \left (1+x \right )^{2}}+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}\right ) \ln \left (1+x \right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}\right ) \ln \left (x -1\right )-\frac {-\frac {d}{432}-\frac {e}{144}-\frac {5 f}{432}}{x -1}-\frac {-\frac {d}{216}-\frac {e}{216}-\frac {f}{216}}{2 \left (x -1\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}\right ) \ln \left (x +2\right )-\frac {-\frac {19 d}{6912}+\frac {17 e}{3456}-\frac {5 f}{576}}{x +2}-\frac {-\frac {d}{1728}+\frac {e}{864}-\frac {f}{432}}{2 \left (x +2\right )^{2}}\)	\(198\)
parallelrisch	\(\frac {2064 d x -12480 f x -9600 e -12960 f \,x^{5}+1680 f \,x^{7}+1536 e \,x^{6}+8192 \ln \left (x -2\right ) e +420 d \,x^{7}+8192 \ln \left (x +2\right ) e +16640 \ln \left (x -1\right ) x^{2} d -416 \ln \left (x -1\right ) x^{8} d -512 \ln \left (x -1\right ) x^{8} e +20480 \ln \left (x -1\right ) x^{2} e +3780 d \,x^{3}-512 \ln \left (1+x \right ) x^{8} e +23040 e \,x^{2}+13120 \ln \left (x -2\right ) f +26400 \ln \left (1+x \right ) x^{4} f -10329 \ln \left (x +2\right ) x^{4} d +6656 \ln \left (1+x \right ) d -13120 \ln \left (x +2\right ) f -6656 \ln \left (x -1\right ) d -800 \ln \left (x -1\right ) x^{8} f +416 \ln \left (1+x \right ) x^{8} d +5120 \ln \left (1+x \right ) x^{6} e -8000 \ln \left (1+x \right ) x^{6} f +3130 \ln \left (x +2\right ) x^{6} d -8200 \ln \left (x -2\right ) x^{6} f -5008 \ln \left (x +2\right ) d +5008 \ln \left (x -2\right ) d +313 \ln \left (x -2\right ) x^{8} d -5120 \ln \left (x +2\right ) x^{6} e +8200 \ln \left (x +2\right ) x^{6} f -8192 \ln \left (x -1\right ) e +800 \ln \left (1+x \right ) x^{8} f -313 \ln \left (x +2\right ) x^{8} d +512 \ln \left (x +2\right ) x^{8} e -820 \ln \left (x +2\right ) x^{8} f -3130 \ln \left (x -2\right ) x^{6} d +12800 \ln \left (1+x \right ) f -8192 \ln \left (1+x \right ) e -12800 \ln \left (x -1\right ) f +16896 \ln \left (x -2\right ) x^{4} e +27060 \ln \left (x -2\right ) x^{4} f -16896 \ln \left (x -1\right ) x^{4} e -26400 \ln \left (x -1\right ) x^{4} f +13728 \ln \left (1+x \right ) x^{4} d -16896 \ln \left (1+x \right ) x^{4} e -2808 d \,x^{5}+27216 f \,x^{3}+32000 \ln \left (x -1\right ) x^{2} f -16640 \ln \left (1+x \right ) x^{2} d +20480 \ln \left (1+x \right ) x^{2} e -12520 \ln \left (x -2\right ) x^{2} d -20480 \ln \left (x -2\right ) x^{2} e +10329 \ln \left (x -2\right ) x^{4} d -32000 \ln \left (1+x \right ) x^{2} f +12520 \ln \left (x +2\right ) x^{2} d -20480 \ln \left (x +2\right ) x^{2} e +32800 \ln \left (x +2\right ) x^{2} f -5120 \ln \left (x -2\right ) x^{6} e +512 \ln \left (x -2\right ) x^{8} e +820 \ln \left (x -2\right ) x^{8} f -13728 \ln \left (x -1\right ) x^{4} d -27060 \ln \left (x +2\right ) x^{4} f -32800 \ln \left (x -2\right ) x^{2} f -11520 e \,x^{4}+16896 \ln \left (x +2\right ) x^{4} e +4160 \ln \left (x -1\right ) x^{6} d +5120 \ln \left (x -1\right ) x^{6} e +8000 \ln \left (x -1\right ) x^{6} f -4160 \ln \left (1+x \right ) x^{6} d}{41472 \left (x^{4}-5 x^{2}+4\right )^{2}}\)	\(645\)

Input:

int((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x,method=_RETURNVERBOSE)
 

Output:

((-13/192*d-5/16*f)*x^5+(35/384*d+21/32*f)*x^3+(35/3456*d+35/864*f)*x^7+(4 
3/864*d-65/216*f)*x-5/18*e*x^4+1/27*e*x^6+5/9*e*x^2-25/108*e)/(x^4-5*x^2+4 
)^2+(-313/41472*d+1/81*e-205/10368*f)*ln(x+2)+(-13/1296*d-1/81*e-25/1296*f 
)*ln(x-1)+(13/1296*d-1/81*e+25/1296*f)*ln(1+x)+(313/41472*d+1/81*e+205/103 
68*f)*ln(x-2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (152) = 304\).

Time = 0.14 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.13 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {420 \, {\left (d + 4 \, f\right )} x^{7} + 1536 \, e x^{6} - 216 \, {\left (13 \, d + 60 \, f\right )} x^{5} - 11520 \, e x^{4} + 756 \, {\left (5 \, d + 36 \, f\right )} x^{3} + 23040 \, e x^{2} + 48 \, {\left (43 \, d - 260 \, f\right )} x - {\left ({\left (313 \, d - 512 \, e + 820 \, f\right )} x^{8} - 10 \, {\left (313 \, d - 512 \, e + 820 \, f\right )} x^{6} + 33 \, {\left (313 \, d - 512 \, e + 820 \, f\right )} x^{4} - 40 \, {\left (313 \, d - 512 \, e + 820 \, f\right )} x^{2} + 5008 \, d - 8192 \, e + 13120 \, f\right )} \log \left (x + 2\right ) + 32 \, {\left ({\left (13 \, d - 16 \, e + 25 \, f\right )} x^{8} - 10 \, {\left (13 \, d - 16 \, e + 25 \, f\right )} x^{6} + 33 \, {\left (13 \, d - 16 \, e + 25 \, f\right )} x^{4} - 40 \, {\left (13 \, d - 16 \, e + 25 \, f\right )} x^{2} + 208 \, d - 256 \, e + 400 \, f\right )} \log \left (x + 1\right ) - 32 \, {\left ({\left (13 \, d + 16 \, e + 25 \, f\right )} x^{8} - 10 \, {\left (13 \, d + 16 \, e + 25 \, f\right )} x^{6} + 33 \, {\left (13 \, d + 16 \, e + 25 \, f\right )} x^{4} - 40 \, {\left (13 \, d + 16 \, e + 25 \, f\right )} x^{2} + 208 \, d + 256 \, e + 400 \, f\right )} \log \left (x - 1\right ) + {\left ({\left (313 \, d + 512 \, e + 820 \, f\right )} x^{8} - 10 \, {\left (313 \, d + 512 \, e + 820 \, f\right )} x^{6} + 33 \, {\left (313 \, d + 512 \, e + 820 \, f\right )} x^{4} - 40 \, {\left (313 \, d + 512 \, e + 820 \, f\right )} x^{2} + 5008 \, d + 8192 \, e + 13120 \, f\right )} \log \left (x - 2\right ) - 9600 \, e}{41472 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \] Input:

integrate((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="fricas")
 

Output:

1/41472*(420*(d + 4*f)*x^7 + 1536*e*x^6 - 216*(13*d + 60*f)*x^5 - 11520*e* 
x^4 + 756*(5*d + 36*f)*x^3 + 23040*e*x^2 + 48*(43*d - 260*f)*x - ((313*d - 
 512*e + 820*f)*x^8 - 10*(313*d - 512*e + 820*f)*x^6 + 33*(313*d - 512*e + 
 820*f)*x^4 - 40*(313*d - 512*e + 820*f)*x^2 + 5008*d - 8192*e + 13120*f)* 
log(x + 2) + 32*((13*d - 16*e + 25*f)*x^8 - 10*(13*d - 16*e + 25*f)*x^6 + 
33*(13*d - 16*e + 25*f)*x^4 - 40*(13*d - 16*e + 25*f)*x^2 + 208*d - 256*e 
+ 400*f)*log(x + 1) - 32*((13*d + 16*e + 25*f)*x^8 - 10*(13*d + 16*e + 25* 
f)*x^6 + 33*(13*d + 16*e + 25*f)*x^4 - 40*(13*d + 16*e + 25*f)*x^2 + 208*d 
 + 256*e + 400*f)*log(x - 1) + ((313*d + 512*e + 820*f)*x^8 - 10*(313*d + 
512*e + 820*f)*x^6 + 33*(313*d + 512*e + 820*f)*x^4 - 40*(313*d + 512*e + 
820*f)*x^2 + 5008*d + 8192*e + 13120*f)*log(x - 2) - 9600*e)/(x^8 - 10*x^6 
 + 33*x^4 - 40*x^2 + 16)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2822 vs. \(2 (148) = 296\).

Time = 72.87 (sec) , antiderivative size = 2822, normalized size of antiderivative = 15.42 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x**2+e*x+d)/(x**4-5*x**2+4)**3,x)
 

Output:

(13*d - 16*e + 25*f)*log(x + (-1106258459719280*d**5*e - 13113710954343*d* 
*5*(13*d - 16*e + 25*f) - 12929482401572800*d**4*e*f - 107063904267900*d** 
4*f*(13*d - 16*e + 25*f) - 817263343042560*d**3*e**3 + 153628968222720*d** 
3*e**2*(13*d - 16*e + 25*f) - 59478343838144000*d**3*e*f**2 + 953019755724 
8*d**3*e*(13*d - 16*e + 25*f)**2 - 324891412840800*d**3*f**2*(13*d - 16*e 
+ 25*f) + 88038005760*d**3*(13*d - 16*e + 25*f)**3 - 2885705898393600*d**2 
*e**3*f + 1014848673546240*d**2*e**2*f*(13*d - 16*e + 25*f) - 134905286808 
320000*d**2*e*f**3 + 63469758382080*d**2*e*f*(13*d - 16*e + 25*f)**2 - 422 
972724528000*d**2*f**3*(13*d - 16*e + 25*f) + 364616847360*d**2*f*(13*d - 
16*e + 25*f)**3 + 5035763255214080*d*e**5 + 142661633703936*d*e**4*(13*d - 
 16*e + 25*f) - 2138314899456000*d*e**3*f**2 - 19670950215680*d*e**3*(13*d 
 - 16*e + 25*f)**2 + 2257033730457600*d*e**2*f**2*(13*d - 16*e + 25*f) - 5 
57272006656*d*e**2*(13*d - 16*e + 25*f)**3 - 151082645593600000*d*e*f**4 + 
 141056507904000*d*e*f**2*(13*d - 16*e + 25*f)**2 - 167683154400000*d*f**4 
*(13*d - 16*e + 25*f) + 339373670400*d*f**2*(13*d - 16*e + 25*f)**3 + 1064 
3272556871680*e**5*f + 214404767416320*e**4*f*(13*d - 16*e + 25*f) + 52999 
2253440000*e**3*f**3 - 41575283425280*e**3*f*(13*d - 16*e + 25*f)**2 + 167 
1759396864000*e**2*f**3*(13*d - 16*e + 25*f) - 837518622720*e**2*f*(13*d - 
 16*e + 25*f)**3 - 66895452108800000*e*f**5 + 104485486592000*e*f**3*(13*d 
 - 16*e + 25*f)**2 + 51041923200000*f**5*(13*d - 16*e + 25*f) - 8028979...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.85 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f\right )} \log \left (x + 2\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f\right )} \log \left (x + 1\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f\right )} \log \left (x - 1\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f\right )} \log \left (x - 2\right ) + \frac {35 \, {\left (d + 4 \, f\right )} x^{7} + 128 \, e x^{6} - 18 \, {\left (13 \, d + 60 \, f\right )} x^{5} - 960 \, e x^{4} + 63 \, {\left (5 \, d + 36 \, f\right )} x^{3} + 1920 \, e x^{2} + 4 \, {\left (43 \, d - 260 \, f\right )} x - 800 \, e}{3456 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \] Input:

integrate((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="maxima")
 

Output:

-1/41472*(313*d - 512*e + 820*f)*log(x + 2) + 1/1296*(13*d - 16*e + 25*f)* 
log(x + 1) - 1/1296*(13*d + 16*e + 25*f)*log(x - 1) + 1/41472*(313*d + 512 
*e + 820*f)*log(x - 2) + 1/3456*(35*(d + 4*f)*x^7 + 128*e*x^6 - 18*(13*d + 
 60*f)*x^5 - 960*e*x^4 + 63*(5*d + 36*f)*x^3 + 1920*e*x^2 + 4*(43*d - 260* 
f)*x - 800*e)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {35 \, d x^{7} + 140 \, f x^{7} + 128 \, e x^{6} - 234 \, d x^{5} - 1080 \, f x^{5} - 960 \, e x^{4} + 315 \, d x^{3} + 2268 \, f x^{3} + 1920 \, e x^{2} + 172 \, d x - 1040 \, f x - 800 \, e}{3456 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \] Input:

integrate((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="giac")
 

Output:

-1/41472*(313*d - 512*e + 820*f)*log(abs(x + 2)) + 1/1296*(13*d - 16*e + 2 
5*f)*log(abs(x + 1)) - 1/1296*(13*d + 16*e + 25*f)*log(abs(x - 1)) + 1/414 
72*(313*d + 512*e + 820*f)*log(abs(x - 2)) + 1/3456*(35*d*x^7 + 140*f*x^7 
+ 128*e*x^6 - 234*d*x^5 - 1080*f*x^5 - 960*e*x^4 + 315*d*x^3 + 2268*f*x^3 
+ 1920*e*x^2 + 172*d*x - 1040*f*x - 800*e)/(x^4 - 5*x^2 + 4)^2
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 18.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.83 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\ln \left (x+1\right )\,\left (\frac {13\,d}{1296}-\frac {e}{81}+\frac {25\,f}{1296}\right )-\ln \left (x-1\right )\,\left (\frac {13\,d}{1296}+\frac {e}{81}+\frac {25\,f}{1296}\right )+\ln \left (x-2\right )\,\left (\frac {313\,d}{41472}+\frac {e}{81}+\frac {205\,f}{10368}\right )-\ln \left (x+2\right )\,\left (\frac {313\,d}{41472}-\frac {e}{81}+\frac {205\,f}{10368}\right )+\frac {\left (\frac {35\,d}{3456}+\frac {35\,f}{864}\right )\,x^7+\frac {e\,x^6}{27}+\left (-\frac {13\,d}{192}-\frac {5\,f}{16}\right )\,x^5-\frac {5\,e\,x^4}{18}+\left (\frac {35\,d}{384}+\frac {21\,f}{32}\right )\,x^3+\frac {5\,e\,x^2}{9}+\left (\frac {43\,d}{864}-\frac {65\,f}{216}\right )\,x-\frac {25\,e}{108}}{x^8-10\,x^6+33\,x^4-40\,x^2+16} \] Input:

int((d + e*x + f*x^2)/(x^4 - 5*x^2 + 4)^3,x)
 

Output:

log(x + 1)*((13*d)/1296 - e/81 + (25*f)/1296) - log(x - 1)*((13*d)/1296 + 
e/81 + (25*f)/1296) + log(x - 2)*((313*d)/41472 + e/81 + (205*f)/10368) - 
log(x + 2)*((313*d)/41472 - e/81 + (205*f)/10368) + (x^3*((35*d)/384 + (21 
*f)/32) - x^5*((13*d)/192 + (5*f)/16) - (25*e)/108 + x^7*((35*d)/3456 + (3 
5*f)/864) + (5*e*x^2)/9 - (5*e*x^4)/18 + (e*x^6)/27 + x*((43*d)/864 - (65* 
f)/216))/(33*x^4 - 40*x^2 - 10*x^6 + x^8 + 16)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 655, normalized size of antiderivative = 3.58 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x)
 

Output:

(1565*log(x - 2)*d*x**8 - 15650*log(x - 2)*d*x**6 + 51645*log(x - 2)*d*x** 
4 - 62600*log(x - 2)*d*x**2 + 25040*log(x - 2)*d + 2560*log(x - 2)*e*x**8 
- 25600*log(x - 2)*e*x**6 + 84480*log(x - 2)*e*x**4 - 102400*log(x - 2)*e* 
x**2 + 40960*log(x - 2)*e + 4100*log(x - 2)*f*x**8 - 41000*log(x - 2)*f*x* 
*6 + 135300*log(x - 2)*f*x**4 - 164000*log(x - 2)*f*x**2 + 65600*log(x - 2 
)*f - 2080*log(x - 1)*d*x**8 + 20800*log(x - 1)*d*x**6 - 68640*log(x - 1)* 
d*x**4 + 83200*log(x - 1)*d*x**2 - 33280*log(x - 1)*d - 2560*log(x - 1)*e* 
x**8 + 25600*log(x - 1)*e*x**6 - 84480*log(x - 1)*e*x**4 + 102400*log(x - 
1)*e*x**2 - 40960*log(x - 1)*e - 4000*log(x - 1)*f*x**8 + 40000*log(x - 1) 
*f*x**6 - 132000*log(x - 1)*f*x**4 + 160000*log(x - 1)*f*x**2 - 64000*log( 
x - 1)*f - 1565*log(x + 2)*d*x**8 + 15650*log(x + 2)*d*x**6 - 51645*log(x 
+ 2)*d*x**4 + 62600*log(x + 2)*d*x**2 - 25040*log(x + 2)*d + 2560*log(x + 
2)*e*x**8 - 25600*log(x + 2)*e*x**6 + 84480*log(x + 2)*e*x**4 - 102400*log 
(x + 2)*e*x**2 + 40960*log(x + 2)*e - 4100*log(x + 2)*f*x**8 + 41000*log(x 
 + 2)*f*x**6 - 135300*log(x + 2)*f*x**4 + 164000*log(x + 2)*f*x**2 - 65600 
*log(x + 2)*f + 2080*log(x + 1)*d*x**8 - 20800*log(x + 1)*d*x**6 + 68640*l 
og(x + 1)*d*x**4 - 83200*log(x + 1)*d*x**2 + 33280*log(x + 1)*d - 2560*log 
(x + 1)*e*x**8 + 25600*log(x + 1)*e*x**6 - 84480*log(x + 1)*e*x**4 + 10240 
0*log(x + 1)*e*x**2 - 40960*log(x + 1)*e + 4000*log(x + 1)*f*x**8 - 40000* 
log(x + 1)*f*x**6 + 132000*log(x + 1)*f*x**4 - 160000*log(x + 1)*f*x**2...