\(\int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{(4-5 x^2+x^4)^3} \, dx\) [43]

Optimal result

Integrand size = 38, antiderivative size = 252 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {2 e+5 g+11 i}{108 \left (1-x^2\right )}-\frac {2 e+5 g+11 i}{108 \left (4-x^2\right )}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac {5 e+8 g+20 i-(2 e+5 g+17 i) x^2}{36 \left (4-5 x^2+x^4\right )^2}-\frac {x \left (59 d+380 f+848 h-5 (7 d+28 f+64 h) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f+1936 h) \text {arctanh}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f+61 h) \text {arctanh}(x)-\frac {1}{162} (2 e+5 g+11 i) \log \left (1-x^2\right )+\frac {1}{162} (2 e+5 g+11 i) \log \left (4-x^2\right ) \] Output:

-1/108*(2*e+5*g+11*i)/(-x^2+1)-(2*e+5*g+11*i)/(-108*x^2+432)+1/144*x*(17*d 
+20*f+32*h-(5*d+8*f+20*h)*x^2)/(x^4-5*x^2+4)^2+1/36*(5*e+8*g+20*i-(2*e+5*g 
+17*i)*x^2)/(x^4-5*x^2+4)^2-x*(59*d+380*f+848*h-5*(7*d+28*f+64*h)*x^2)/(34 
56*x^4-17280*x^2+13824)-1/20736*(313*d+820*f+1936*h)*arctanh(1/2*x)+1/648* 
(13*d+25*f+61*h)*arctanh(x)-1/162*(2*e+5*g+11*i)*ln(-x^2+1)+1/162*(2*e+5*g 
+11*i)*ln(-x^2+4)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {20 e+32 g+80 i+17 d x+20 f x+32 h x-8 e x^2-20 g x^2-68 i x^2-5 d x^3-8 f x^3-20 h x^3}{144 \left (4-5 x^2+x^4\right )^2}+\frac {-320 e-800 g-1760 i-59 d x-380 f x-848 h x+128 e x^2+320 g x^2+704 i x^2+35 d x^3+140 f x^3+320 h x^3}{3456 \left (4-5 x^2+x^4\right )}+\frac {(-13 d-16 e-25 f-40 g-61 h-88 i) \log (1-x)}{1296}+\frac {(313 d+512 e+820 f+1280 g+1936 h+2816 i) \log (2-x)}{41472}+\frac {(13 d-16 e+25 f-40 g+61 h-88 i) \log (1+x)}{1296}+\frac {(-313 d+512 e-820 f+1280 g-1936 h+2816 i) \log (2+x)}{41472} \] Input:

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

Output:

(20*e + 32*g + 80*i + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 68*i 
*x^2 - 5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(144*(4 - 5*x^2 + x^4)^2) + (-320*e - 
 800*g - 1760*i - 59*d*x - 380*f*x - 848*h*x + 128*e*x^2 + 320*g*x^2 + 704 
*i*x^2 + 35*d*x^3 + 140*f*x^3 + 320*h*x^3)/(3456*(4 - 5*x^2 + x^4)) + ((-1 
3*d - 16*e - 25*f - 40*g - 61*h - 88*i)*Log[1 - x])/1296 + ((313*d + 512*e 
 + 820*f + 1280*g + 1936*h + 2816*i)*Log[2 - x])/41472 + ((13*d - 16*e + 2 
5*f - 40*g + 61*h - 88*i)*Log[1 + x])/1296 + ((-313*d + 512*e - 820*f + 12 
80*g - 1936*h + 2816*i)*Log[2 + x])/41472
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2202, 2194, 2191, 27, 1084, 2009, 2206, 25, 1492, 27, 1480, 220}

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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 2194

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

\(\Big \downarrow \) 2191

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{18} \int \frac {3 (2 e+5 g+11 i)}{\left (x^4-5 x^2+4\right )^2}dx^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} (2 e+5 g+11 i) \int \frac {1}{\left (x^4-5 x^2+4\right )^2}dx^2\right )\)

\(\Big \downarrow \) 1084

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} (2 e+5 g+11 i) \int \left (\frac {2}{27 \left (4-x^2\right )}+\frac {1}{9 \left (4-x^2\right )^2}-\frac {2}{27 \left (1-x^2\right )}+\frac {1}{9 \left (1-x^2\right )^2}\right )dx^2\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4-5 x^2+4\right )^3}dx+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {1}{144} \int -\frac {-5 (5 d+8 f+20 h) x^2+19 d-20 f-32 h}{\left (x^4-5 x^2+4\right )^2}dx+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{144} \int \frac {-5 (5 d+8 f+20 h) x^2+19 d-20 f-32 h}{\left (x^4-5 x^2+4\right )^2}dx+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{144} \left (-\frac {1}{72} \int -\frac {3 \left (5 (7 d+28 f+64 h) x^2+173 d+260 f+656 h\right )}{x^4-5 x^2+4}dx-\frac {x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{144} \left (\frac {1}{24} \int \frac {5 (7 d+28 f+64 h) x^2+173 d+260 f+656 h}{x^4-5 x^2+4}dx-\frac {x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{144} \left (\frac {1}{24} \left (\frac {1}{3} (313 d+820 f+1936 h) \int \frac {1}{x^2-4}dx-\frac {16}{3} (13 d+25 f+61 h) \int \frac {1}{x^2-1}dx\right )-\frac {x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{144} \left (\frac {1}{24} \left (\frac {16}{3} \text {arctanh}(x) (13 d+25 f+61 h)-\frac {1}{6} \text {arctanh}\left (\frac {x}{2}\right ) (313 d+820 f+1936 h)\right )-\frac {x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{24 \left (x^4-5 x^2+4\right )}\right )+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}+\frac {1}{2} \left (\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )^2}-\frac {1}{6} \left (\frac {1}{9 \left (1-x^2\right )}+\frac {1}{9 \left (4-x^2\right )}+\frac {2}{27} \log \left (1-x^2\right )-\frac {2}{27} \log \left (4-x^2\right )\right ) (2 e+5 g+11 i)\right )\)

Input:

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

Output:

(x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(144*(4 - 5*x^2 + x^4)^2 
) + (-1/24*(x*(59*d + 380*f + 848*h - 5*(7*d + 28*f + 64*h)*x^2))/(4 - 5*x 
^2 + x^4) + (-1/6*((313*d + 820*f + 1936*h)*ArcTanh[x/2]) + (16*(13*d + 25 
*f + 61*h)*ArcTanh[x])/3)/24)/144 + ((5*e + 8*g + 20*i - (2*e + 5*g + 17*i 
)*x^2)/(18*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g + 11*i)*(1/(9*(1 - x^2)) + 1 
/(9*(4 - x^2)) + (2*Log[1 - x^2])/27 - (2*Log[4 - x^2])/27))/6)/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 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 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 2194

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : 
> Simp[1/2   Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) 
^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ 
[(m - 1)/2]
 

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]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.86

method	result	size
norman	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}-\frac {17 h}{24}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}+\frac {35 h}{24}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}+\frac {5 h}{54}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}-\frac {41 h}{54}\right ) x +\left (-\frac {5 e}{18}-\frac {25 g}{36}-\frac {55 i}{36}\right ) x^{4}+\left (\frac {e}{27}+\frac {5 g}{54}+\frac {11 i}{54}\right ) x^{6}+\left (\frac {5 e}{9}+\frac {25 g}{18}+\frac {26 i}{9}\right ) x^{2}-\frac {25 e}{108}-\frac {19 g}{27}-\frac {40 i}{27}}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}+\frac {5 g}{162}-\frac {121 h}{2592}+\frac {11 i}{162}\right ) \ln \left (x +2\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}-\frac {5 g}{162}-\frac {61 h}{1296}-\frac {11 i}{162}\right ) \ln \left (x -1\right )+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}-\frac {5 g}{162}+\frac {61 h}{1296}-\frac {11 i}{162}\right ) \ln \left (1+x \right )+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}+\frac {5 g}{162}+\frac {121 h}{2592}+\frac {11 i}{162}\right ) \ln \left (x -2\right )\)	\(217\)
default	\(-\frac {-\frac {19 d}{6912}-\frac {17 e}{3456}-\frac {5 f}{576}-\frac {13 g}{864}-\frac {11 h}{432}-\frac {i}{24}}{x -2}-\frac {\frac {d}{1728}+\frac {e}{864}+\frac {f}{432}+\frac {g}{216}+\frac {h}{108}+\frac {i}{54}}{2 \left (x -2\right )^{2}}+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}+\frac {5 g}{162}+\frac {121 h}{2592}+\frac {11 i}{162}\right ) \ln \left (x -2\right )-\frac {-\frac {d}{432}+\frac {e}{144}-\frac {5 f}{432}+\frac {7 g}{432}-\frac {h}{48}+\frac {11 i}{432}}{1+x}-\frac {\frac {d}{216}-\frac {e}{216}+\frac {f}{216}-\frac {g}{216}+\frac {h}{216}-\frac {i}{216}}{2 \left (1+x \right )^{2}}+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}-\frac {5 g}{162}+\frac {61 h}{1296}-\frac {11 i}{162}\right ) \ln \left (1+x \right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}-\frac {5 g}{162}-\frac {61 h}{1296}-\frac {11 i}{162}\right ) \ln \left (x -1\right )-\frac {-\frac {d}{432}-\frac {e}{144}-\frac {5 f}{432}-\frac {7 g}{432}-\frac {h}{48}-\frac {11 i}{432}}{x -1}-\frac {-\frac {d}{216}-\frac {e}{216}-\frac {f}{216}-\frac {g}{216}-\frac {h}{216}-\frac {i}{216}}{2 \left (x -1\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}+\frac {5 g}{162}-\frac {121 h}{2592}+\frac {11 i}{162}\right ) \ln \left (x +2\right )-\frac {-\frac {19 d}{6912}+\frac {17 e}{3456}-\frac {5 f}{576}+\frac {13 g}{864}-\frac {11 h}{432}+\frac {i}{24}}{x +2}-\frac {-\frac {d}{1728}+\frac {e}{864}-\frac {f}{432}+\frac {g}{216}-\frac {h}{108}+\frac {i}{54}}{2 \left (x +2\right )^{2}}\)	\(306\)
risch	\(\frac {\ln \left (2-x \right ) e}{81}+\frac {\ln \left (x +2\right ) e}{81}-\frac {13 \ln \left (1-x \right ) d}{1296}+\frac {313 \ln \left (2-x \right ) d}{41472}+\frac {205 \ln \left (2-x \right ) f}{10368}+\frac {13 \ln \left (1+x \right ) d}{1296}+\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}-\frac {17 h}{24}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}+\frac {35 h}{24}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}+\frac {5 h}{54}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}-\frac {41 h}{54}\right ) x +\left (-\frac {5 e}{18}-\frac {25 g}{36}-\frac {55 i}{36}\right ) x^{4}+\left (\frac {e}{27}+\frac {5 g}{54}+\frac {11 i}{54}\right ) x^{6}+\left (\frac {5 e}{9}+\frac {25 g}{18}+\frac {26 i}{9}\right ) x^{2}-\frac {25 e}{108}-\frac {19 g}{27}-\frac {40 i}{27}}{\left (x^{4}-5 x^{2}+4\right )^{2}}-\frac {205 \ln \left (x +2\right ) f}{10368}-\frac {313 \ln \left (x +2\right ) d}{41472}-\frac {25 \ln \left (1-x \right ) f}{1296}-\frac {\ln \left (1-x \right ) e}{81}+\frac {25 \ln \left (1+x \right ) f}{1296}-\frac {\ln \left (1+x \right ) e}{81}+\frac {61 \ln \left (1+x \right ) h}{1296}-\frac {11 \ln \left (1+x \right ) i}{162}-\frac {5 \ln \left (1+x \right ) g}{162}+\frac {5 \ln \left (x +2\right ) g}{162}-\frac {121 \ln \left (x +2\right ) h}{2592}+\frac {11 \ln \left (x +2\right ) i}{162}-\frac {5 \ln \left (1-x \right ) g}{162}-\frac {61 \ln \left (1-x \right ) h}{1296}-\frac {11 \ln \left (1-x \right ) i}{162}+\frac {5 \ln \left (2-x \right ) g}{162}+\frac {121 \ln \left (2-x \right ) h}{2592}+\frac {11 \ln \left (2-x \right ) i}{162}\)	\(313\)
parallelrisch	\(\text {Expression too large to display}\)	\(1273\)

Input:

int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x,method=_RETURNVERBOS 
E)
 

Output:

((-13/192*d-5/16*f-17/24*h)*x^5+(35/384*d+21/32*f+35/24*h)*x^3+(35/3456*d+ 
35/864*f+5/54*h)*x^7+(43/864*d-65/216*f-41/54*h)*x+(-5/18*e-25/36*g-55/36* 
i)*x^4+(1/27*e+5/54*g+11/54*i)*x^6+(5/9*e+25/18*g+26/9*i)*x^2-25/108*e-19/ 
27*g-40/27*i)/(x^4-5*x^2+4)^2+(-313/41472*d+1/81*e-205/10368*f+5/162*g-121 
/2592*h+11/162*i)*ln(x+2)+(-13/1296*d-1/81*e-25/1296*f-5/162*g-61/1296*h-1 
1/162*i)*ln(x-1)+(13/1296*d-1/81*e+25/1296*f-5/162*g+61/1296*h-11/162*i)*l 
n(1+x)+(313/41472*d+1/81*e+205/10368*f+5/162*g+121/2592*h+11/162*i)*ln(x-2 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (226) = 452\).

Time = 6.46 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.44 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="fr 
icas")
 

Output:

1/41472*(60*(7*d + 28*f + 64*h)*x^7 + 768*(2*e + 5*g + 11*i)*x^6 - 216*(13 
*d + 60*f + 136*h)*x^5 - 5760*(2*e + 5*g + 11*i)*x^4 + 756*(5*d + 36*f + 8 
0*h)*x^3 + 2304*(10*e + 25*g + 52*i)*x^2 + 48*(43*d - 260*f - 656*h)*x - ( 
(313*d - 512*e + 820*f - 1280*g + 1936*h - 2816*i)*x^8 - 10*(313*d - 512*e 
 + 820*f - 1280*g + 1936*h - 2816*i)*x^6 + 33*(313*d - 512*e + 820*f - 128 
0*g + 1936*h - 2816*i)*x^4 - 40*(313*d - 512*e + 820*f - 1280*g + 1936*h - 
 2816*i)*x^2 + 5008*d - 8192*e + 13120*f - 20480*g + 30976*h - 45056*i)*lo 
g(x + 2) + 32*((13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*x^8 - 10*(13*d - 
16*e + 25*f - 40*g + 61*h - 88*i)*x^6 + 33*(13*d - 16*e + 25*f - 40*g + 61 
*h - 88*i)*x^4 - 40*(13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*x^2 + 208*d 
- 256*e + 400*f - 640*g + 976*h - 1408*i)*log(x + 1) - 32*((13*d + 16*e + 
25*f + 40*g + 61*h + 88*i)*x^8 - 10*(13*d + 16*e + 25*f + 40*g + 61*h + 88 
*i)*x^6 + 33*(13*d + 16*e + 25*f + 40*g + 61*h + 88*i)*x^4 - 40*(13*d + 16 
*e + 25*f + 40*g + 61*h + 88*i)*x^2 + 208*d + 256*e + 400*f + 640*g + 976* 
h + 1408*i)*log(x - 1) + ((313*d + 512*e + 820*f + 1280*g + 1936*h + 2816* 
i)*x^8 - 10*(313*d + 512*e + 820*f + 1280*g + 1936*h + 2816*i)*x^6 + 33*(3 
13*d + 512*e + 820*f + 1280*g + 1936*h + 2816*i)*x^4 - 40*(313*d + 512*e + 
 820*f + 1280*g + 1936*h + 2816*i)*x^2 + 5008*d + 8192*e + 13120*f + 20480 
*g + 30976*h + 45056*i)*log(x - 2) - 9600*e - 29184*g - 61440*i)/(x^8 - 10 
*x^6 + 33*x^4 - 40*x^2 + 16)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h - 2816 \, i\right )} \log \left (x + 2\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h - 88 \, i\right )} \log \left (x + 1\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h + 88 \, i\right )} \log \left (x - 1\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h + 2816 \, i\right )} \log \left (x - 2\right ) + \frac {5 \, {\left (7 \, d + 28 \, f + 64 \, h\right )} x^{7} + 64 \, {\left (2 \, e + 5 \, g + 11 \, i\right )} x^{6} - 18 \, {\left (13 \, d + 60 \, f + 136 \, h\right )} x^{5} - 480 \, {\left (2 \, e + 5 \, g + 11 \, i\right )} x^{4} + 63 \, {\left (5 \, d + 36 \, f + 80 \, h\right )} x^{3} + 192 \, {\left (10 \, e + 25 \, g + 52 \, i\right )} x^{2} + 4 \, {\left (43 \, d - 260 \, f - 656 \, h\right )} x - 800 \, e - 2432 \, g - 5120 \, i}{3456 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \] Input:

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="ma 
xima")
 

Output:

-1/41472*(313*d - 512*e + 820*f - 1280*g + 1936*h - 2816*i)*log(x + 2) + 1 
/1296*(13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*log(x + 1) - 1/1296*(13*d 
+ 16*e + 25*f + 40*g + 61*h + 88*i)*log(x - 1) + 1/41472*(313*d + 512*e + 
820*f + 1280*g + 1936*h + 2816*i)*log(x - 2) + 1/3456*(5*(7*d + 28*f + 64* 
h)*x^7 + 64*(2*e + 5*g + 11*i)*x^6 - 18*(13*d + 60*f + 136*h)*x^5 - 480*(2 
*e + 5*g + 11*i)*x^4 + 63*(5*d + 36*f + 80*h)*x^3 + 192*(10*e + 25*g + 52* 
i)*x^2 + 4*(43*d - 260*f - 656*h)*x - 800*e - 2432*g - 5120*i)/(x^8 - 10*x 
^6 + 33*x^4 - 40*x^2 + 16)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.99 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h - 2816 \, i\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h - 88 \, i\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h + 88 \, i\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h + 2816 \, i\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {35 \, d x^{7} + 140 \, f x^{7} + 320 \, h x^{7} + 128 \, e x^{6} + 320 \, g x^{6} + 704 \, i x^{6} - 234 \, d x^{5} - 1080 \, f x^{5} - 2448 \, h x^{5} - 960 \, e x^{4} - 2400 \, g x^{4} - 5280 \, i x^{4} + 315 \, d x^{3} + 2268 \, f x^{3} + 5040 \, h x^{3} + 1920 \, e x^{2} + 4800 \, g x^{2} + 9984 \, i x^{2} + 172 \, d x - 1040 \, f x - 2624 \, h x - 800 \, e - 2432 \, g - 5120 \, i}{3456 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \] Input:

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="gi 
ac")
 

Output:

-1/41472*(313*d - 512*e + 820*f - 1280*g + 1936*h - 2816*i)*log(abs(x + 2) 
) + 1/1296*(13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*log(abs(x + 1)) - 1/1 
296*(13*d + 16*e + 25*f + 40*g + 61*h + 88*i)*log(abs(x - 1)) + 1/41472*(3 
13*d + 512*e + 820*f + 1280*g + 1936*h + 2816*i)*log(abs(x - 2)) + 1/3456* 
(35*d*x^7 + 140*f*x^7 + 320*h*x^7 + 128*e*x^6 + 320*g*x^6 + 704*i*x^6 - 23 
4*d*x^5 - 1080*f*x^5 - 2448*h*x^5 - 960*e*x^4 - 2400*g*x^4 - 5280*i*x^4 + 
315*d*x^3 + 2268*f*x^3 + 5040*h*x^3 + 1920*e*x^2 + 4800*g*x^2 + 9984*i*x^2 
 + 172*d*x - 1040*f*x - 2624*h*x - 800*e - 2432*g - 5120*i)/(x^4 - 5*x^2 + 
 4)^2
 

Mupad [B] (verification not implemented)

Time = 18.53 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx=\ln \left (x+1\right )\,\left (\frac {13\,d}{1296}-\frac {e}{81}+\frac {25\,f}{1296}-\frac {5\,g}{162}+\frac {61\,h}{1296}-\frac {11\,i}{162}\right )-\ln \left (x-1\right )\,\left (\frac {13\,d}{1296}+\frac {e}{81}+\frac {25\,f}{1296}+\frac {5\,g}{162}+\frac {61\,h}{1296}+\frac {11\,i}{162}\right )-\frac {\left (-\frac {35\,d}{3456}-\frac {35\,f}{864}-\frac {5\,h}{54}\right )\,x^7+\left (-\frac {e}{27}-\frac {5\,g}{54}-\frac {11\,i}{54}\right )\,x^6+\left (\frac {13\,d}{192}+\frac {5\,f}{16}+\frac {17\,h}{24}\right )\,x^5+\left (\frac {5\,e}{18}+\frac {25\,g}{36}+\frac {55\,i}{36}\right )\,x^4+\left (-\frac {35\,d}{384}-\frac {21\,f}{32}-\frac {35\,h}{24}\right )\,x^3+\left (-\frac {5\,e}{9}-\frac {25\,g}{18}-\frac {26\,i}{9}\right )\,x^2+\left (\frac {65\,f}{216}-\frac {43\,d}{864}+\frac {41\,h}{54}\right )\,x+\frac {25\,e}{108}+\frac {19\,g}{27}+\frac {40\,i}{27}}{x^8-10\,x^6+33\,x^4-40\,x^2+16}+\ln \left (x-2\right )\,\left (\frac {313\,d}{41472}+\frac {e}{81}+\frac {205\,f}{10368}+\frac {5\,g}{162}+\frac {121\,h}{2592}+\frac {11\,i}{162}\right )-\ln \left (x+2\right )\,\left (\frac {313\,d}{41472}-\frac {e}{81}+\frac {205\,f}{10368}-\frac {5\,g}{162}+\frac {121\,h}{2592}-\frac {11\,i}{162}\right ) \] Input:

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

Output:

log(x + 1)*((13*d)/1296 - e/81 + (25*f)/1296 - (5*g)/162 + (61*h)/1296 - ( 
11*i)/162) - log(x - 1)*((13*d)/1296 + e/81 + (25*f)/1296 + (5*g)/162 + (6 
1*h)/1296 + (11*i)/162) - ((25*e)/108 + (19*g)/27 + (40*i)/27 + x*((65*f)/ 
216 - (43*d)/864 + (41*h)/54) + x^5*((13*d)/192 + (5*f)/16 + (17*h)/24) - 
x^3*((35*d)/384 + (21*f)/32 + (35*h)/24) - x^7*((35*d)/3456 + (35*f)/864 + 
 (5*h)/54) - x^2*((5*e)/9 + (25*g)/18 + (26*i)/9) - x^6*(e/27 + (5*g)/54 + 
 (11*i)/54) + x^4*((5*e)/18 + (25*g)/36 + (55*i)/36))/(33*x^4 - 40*x^2 - 1 
0*x^6 + x^8 + 16) + log(x - 2)*((313*d)/41472 + e/81 + (205*f)/10368 + (5* 
g)/162 + (121*h)/2592 + (11*i)/162) - log(x + 2)*((313*d)/41472 - e/81 + ( 
205*f)/10368 - (5*g)/162 + (121*h)/2592 - (11*i)/162)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1283, normalized size of antiderivative = 5.09 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (4-5 x^2+x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((i*x^5+h*x^4+g*x^3+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 + 6400*log(x - 2)*g*x**8 - 64000*log(x - 2)*g*x**6 + 211200*log(x - 2) 
*g*x**4 - 256000*log(x - 2)*g*x**2 + 102400*log(x - 2)*g + 9680*log(x - 2) 
*h*x**8 - 96800*log(x - 2)*h*x**6 + 319440*log(x - 2)*h*x**4 - 387200*log( 
x - 2)*h*x**2 + 154880*log(x - 2)*h + 14080*log(x - 2)*i*x**8 - 140800*log 
(x - 2)*i*x**6 + 464640*log(x - 2)*i*x**4 - 563200*log(x - 2)*i*x**2 + 225 
280*log(x - 2)*i - 2080*log(x - 1)*d*x**8 + 20800*log(x - 1)*d*x**6 - 6864 
0*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 + 10 
2400*log(x - 1)*e*x**2 - 40960*log(x - 1)*e - 4000*log(x - 1)*f*x**8 + 400 
00*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 - 6400*log(x - 1)*g*x**8 + 64000*log(x - 1)*g*x**6 - 
 211200*log(x - 1)*g*x**4 + 256000*log(x - 1)*g*x**2 - 102400*log(x - 1)*g 
 - 9760*log(x - 1)*h*x**8 + 97600*log(x - 1)*h*x**6 - 322080*log(x - 1)*h* 
x**4 + 390400*log(x - 1)*h*x**2 - 156160*log(x - 1)*h - 14080*log(x - 1)*i 
*x**8 + 140800*log(x - 1)*i*x**6 - 464640*log(x - 1)*i*x**4 + 563200*lo...