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

Optimal. Leaf size=224 \[ -\frac{x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{\tanh ^{-1}\left (\frac{x}{2}\right ) (313 d+820 f+1936 h)}{20736}+\frac{1}{648} \tanh ^{-1}(x) (13 d+25 f+61 h)-\frac{\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right ) \]

[Out]

(5*e + 8*g - (2*e + 5*g)*x^2)/(36*(4 - 5*x^2 + x^4)^2) + (x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(14
4*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*f + 848*h - 5*(7*d
+ 28*f + 64*h)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f + 1936*h)*ArcTanh[x/2])/20736 + ((13*d + 25*f
+ 61*h)*ArcTanh[x])/648 - ((2*e + 5*g)*Log[1 - x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

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Rubi [A]  time = 0.306831, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {1673, 1678, 1178, 1166, 207, 1247, 638, 614, 616, 31} \[ -\frac{x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{\tanh ^{-1}\left (\frac{x}{2}\right ) (313 d+820 f+1936 h)}{20736}+\frac{1}{648} \tanh ^{-1}(x) (13 d+25 f+61 h)-\frac{\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*e + 8*g - (2*e + 5*g)*x^2)/(36*(4 - 5*x^2 + x^4)^2) + (x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(14
4*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*f + 848*h - 5*(7*d
+ 28*f + 64*h)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f + 1936*h)*ArcTanh[x/2])/20736 + ((13*d + 25*f
+ 61*h)*ArcTanh[x])/648 - ((2*e + 5*g)*Log[1 - x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, 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] + Dist[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)*PolynomialQuot
ient[Pq, 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[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> 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] + Dist[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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[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 207

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

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 614

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] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx &=\int \frac{x \left (e+g x^2\right )}{\left (4-5 x^2+x^4\right )^3} \, dx+\int \frac{d+f x^2+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx\\ &=\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{1}{144} \int \frac{-19 d+20 f+32 h+5 (5 d+8 f+20 h) x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\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{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{\int \frac{3 (173 d+260 f+656 h)+15 (7 d+28 f+64 h) x^2}{4-5 x^2+x^4} \, dx}{10368}+\frac{1}{12} (-2 e-5 g) \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\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{(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\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{1}{54} (2 e+5 g) \operatorname{Subst}\left (\int \frac{1}{4-5 x+x^2} \, dx,x,x^2\right )+\frac{1}{648} (-13 d-25 f-61 h) \int \frac{1}{-1+x^2} \, dx+\frac{(313 d+820 f+1936 h) \int \frac{1}{-4+x^2} \, dx}{10368}\\ &=\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\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{(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\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) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f+61 h) \tanh ^{-1}(x)+\frac{1}{162} (-2 e-5 g) \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )+\frac{1}{162} (2 e+5 g) \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )\\ &=\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\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{(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\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) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f+61 h) \tanh ^{-1}(x)-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.125163, size = 231, normalized size = 1.03 \[ \frac{-5 d x^3+17 d x-8 e x^2+20 e-8 f x^3+20 f x-20 g x^2+32 g-20 h x^3+32 h x}{144 \left (x^4-5 x^2+4\right )^2}+\frac{35 d x^3-59 d x+128 e x^2-320 e+140 f x^3-380 f x+320 g x^2-800 g+320 h x^3-848 h x}{3456 \left (x^4-5 x^2+4\right )}+\frac{\log (1-x) (-13 d-16 e-25 f-40 g-61 h)}{1296}+\frac{\log (2-x) (313 d+512 e+820 f+1280 g+1936 h)}{41472}+\frac{\log (x+1) (13 d-16 e+25 f-40 g+61 h)}{1296}+\frac{\log (x+2) (-313 d+512 e-820 f+1280 g-1936 h)}{41472} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e + 32*g + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*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 - 59*d*x - 380*f*x - 848*h*x + 128*e*x^2 + 320*g*x^2 + 35*d*x^3 + 140*f*x^3 + 320*h
*x^3)/(3456*(4 - 5*x^2 + x^4)) + ((-13*d - 16*e - 25*f - 40*g - 61*h)*Log[1 - x])/1296 + ((313*d + 512*e + 820
*f + 1280*g + 1936*h)*Log[2 - x])/41472 + ((13*d - 16*e + 25*f - 40*g + 61*h)*Log[1 + x])/1296 + ((-313*d + 51
2*e - 820*f + 1280*g - 1936*h)*Log[2 + x])/41472

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Maple [B]  time = 0.02, size = 462, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-313/41472*ln(2+x)*d+1/81*ln(2+x)*e+13/1296*ln(1+x)*d-1/81*ln(1+x)*e+313/41472*ln(x-2)*d+1/81*ln(x-2)*e-13/129
6*ln(x-1)*d-1/81*ln(x-1)*e+1/432/(x-1)^2*h+1/216/(2+x)^2*h-1/432/(1+x)^2*h-1/216/(x-2)^2*h-1/432/(2+x)^2*g+1/4
32/(1+x)^2*g-1/432/(x-2)^2*g+1/432/(x-1)^2*g-1/432/(1+x)^2*f+1/864/(2+x)^2*f+1/432/(x-1)^2*d+1/432/(x-1)^2*e+1
/3456/(2+x)^2*d-1/1728/(2+x)^2*e-1/864/(x-2)^2*f-1/432/(1+x)^2*d+1/432/(1+x)^2*e+1/432/(x-1)^2*f-1/3456/(x-2)^
2*d-1/1728/(x-2)^2*e+11/432/(x-2)*h+1/48/(x-1)*h+1/48/(1+x)*h+11/432/(2+x)*h-13/864/(2+x)*g+1/432/(1+x)*d-1/14
4/(1+x)*e+13/864/(x-2)*g+19/6912/(x-2)*d+17/3456/(x-2)*e+7/432/(x-1)*g+1/432/(x-1)*d+1/144/(x-1)*e+19/6912/(2+
x)*d-17/3456/(2+x)*e-7/432/(1+x)*g+5/432/(1+x)*f+5/576/(x-2)*f+5/432/(x-1)*f+5/576/(2+x)*f+5/162*ln(2+x)*g-5/1
62*ln(1+x)*g+5/162*ln(x-2)*g-5/162*ln(x-1)*g-121/2592*ln(2+x)*h+61/1296*ln(1+x)*h+121/2592*ln(x-2)*h-61/1296*l
n(x-1)*h+205/10368*ln(x-2)*f-25/1296*ln(x-1)*f-205/10368*ln(2+x)*f+25/1296*ln(1+x)*f

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Maxima [A]  time = 0.950256, size = 289, normalized size = 1.29 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} \log \left (x + 2\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} \log \left (x + 1\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} \log \left (x - 1\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} \log \left (x - 2\right ) + \frac{5 \,{\left (7 \, d + 28 \, f + 64 \, h\right )} x^{7} + 64 \,{\left (2 \, e + 5 \, g\right )} x^{6} - 18 \,{\left (13 \, d + 60 \, f + 136 \, h\right )} x^{5} - 480 \,{\left (2 \, e + 5 \, g\right )} x^{4} + 63 \,{\left (5 \, d + 36 \, f + 80 \, h\right )} x^{3} + 960 \,{\left (2 \, e + 5 \, g\right )} x^{2} + 4 \,{\left (43 \, d - 260 \, f - 656 \, h\right )} x - 800 \, e - 2432 \, g}{3456 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/41472*(313*d - 512*e + 820*f - 1280*g + 1936*h)*log(x + 2) + 1/1296*(13*d - 16*e + 25*f - 40*g + 61*h)*log(
x + 1) - 1/1296*(13*d + 16*e + 25*f + 40*g + 61*h)*log(x - 1) + 1/41472*(313*d + 512*e + 820*f + 1280*g + 1936
*h)*log(x - 2) + 1/3456*(5*(7*d + 28*f + 64*h)*x^7 + 64*(2*e + 5*g)*x^6 - 18*(13*d + 60*f + 136*h)*x^5 - 480*(
2*e + 5*g)*x^4 + 63*(5*d + 36*f + 80*h)*x^3 + 960*(2*e + 5*g)*x^2 + 4*(43*d - 260*f - 656*h)*x - 800*e - 2432*
g)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

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Fricas [B]  time = 12.9746, size = 1671, normalized size = 7.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/41472*(60*(7*d + 28*f + 64*h)*x^7 + 768*(2*e + 5*g)*x^6 - 216*(13*d + 60*f + 136*h)*x^5 - 5760*(2*e + 5*g)*x
^4 + 756*(5*d + 36*f + 80*h)*x^3 + 11520*(2*e + 5*g)*x^2 + 48*(43*d - 260*f - 656*h)*x - ((313*d - 512*e + 820
*f - 1280*g + 1936*h)*x^8 - 10*(313*d - 512*e + 820*f - 1280*g + 1936*h)*x^6 + 33*(313*d - 512*e + 820*f - 128
0*g + 1936*h)*x^4 - 40*(313*d - 512*e + 820*f - 1280*g + 1936*h)*x^2 + 5008*d - 8192*e + 13120*f - 20480*g + 3
0976*h)*log(x + 2) + 32*((13*d - 16*e + 25*f - 40*g + 61*h)*x^8 - 10*(13*d - 16*e + 25*f - 40*g + 61*h)*x^6 +
33*(13*d - 16*e + 25*f - 40*g + 61*h)*x^4 - 40*(13*d - 16*e + 25*f - 40*g + 61*h)*x^2 + 208*d - 256*e + 400*f
- 640*g + 976*h)*log(x + 1) - 32*((13*d + 16*e + 25*f + 40*g + 61*h)*x^8 - 10*(13*d + 16*e + 25*f + 40*g + 61*
h)*x^6 + 33*(13*d + 16*e + 25*f + 40*g + 61*h)*x^4 - 40*(13*d + 16*e + 25*f + 40*g + 61*h)*x^2 + 208*d + 256*e
+ 400*f + 640*g + 976*h)*log(x - 1) + ((313*d + 512*e + 820*f + 1280*g + 1936*h)*x^8 - 10*(313*d + 512*e + 82
0*f + 1280*g + 1936*h)*x^6 + 33*(313*d + 512*e + 820*f + 1280*g + 1936*h)*x^4 - 40*(313*d + 512*e + 820*f + 12
80*g + 1936*h)*x^2 + 5008*d + 8192*e + 13120*f + 20480*g + 30976*h)*log(x - 2) - 9600*e - 29184*g)/(x^8 - 10*x
^6 + 33*x^4 - 40*x^2 + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.13631, size = 302, normalized size = 1.35 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d + 820 \, f - 1280 \, g + 1936 \, h - 512 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d + 25 \, f - 40 \, g + 61 \, h - 16 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 25 \, f + 40 \, g + 61 \, h + 16 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 820 \, f + 1280 \, g + 1936 \, h + 512 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac{35 \, d x^{7} + 140 \, f x^{7} + 320 \, h x^{7} + 320 \, g x^{6} + 128 \, x^{6} e - 234 \, d x^{5} - 1080 \, f x^{5} - 2448 \, h x^{5} - 2400 \, g x^{4} - 960 \, x^{4} e + 315 \, d x^{3} + 2268 \, f x^{3} + 5040 \, h x^{3} + 4800 \, g x^{2} + 1920 \, x^{2} e + 172 \, d x - 1040 \, f x - 2624 \, h x - 2432 \, g - 800 \, e}{3456 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/41472*(313*d + 820*f - 1280*g + 1936*h - 512*e)*log(abs(x + 2)) + 1/1296*(13*d + 25*f - 40*g + 61*h - 16*e)
*log(abs(x + 1)) - 1/1296*(13*d + 25*f + 40*g + 61*h + 16*e)*log(abs(x - 1)) + 1/41472*(313*d + 820*f + 1280*g
+ 1936*h + 512*e)*log(abs(x - 2)) + 1/3456*(35*d*x^7 + 140*f*x^7 + 320*h*x^7 + 320*g*x^6 + 128*x^6*e - 234*d*
x^5 - 1080*f*x^5 - 2448*h*x^5 - 2400*g*x^4 - 960*x^4*e + 315*d*x^3 + 2268*f*x^3 + 5040*h*x^3 + 4800*g*x^2 + 19
20*x^2*e + 172*d*x - 1040*f*x - 2624*h*x - 2432*g - 800*e)/(x^4 - 5*x^2 + 4)^2