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

Optimal. Leaf size=162 \[ \frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{432} \tanh ^{-1}\left (\frac {x}{2}\right ) (19 d+52 f+112 h)-\frac {1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac {1}{54} \log \left (1-x^2\right ) (2 e+5 g+8 i)-\frac {1}{54} \log \left (4-x^2\right ) (2 e+5 g+8 i)+\frac {-\left (x^2 (2 e+5 g+17 i)\right )+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )} \]

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Rubi [A]  time = 0.23, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1673, 1678, 1166, 207, 1663, 1660, 12, 616, 31} \begin {gather*} \frac {x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{432} \tanh ^{-1}\left (\frac {x}{2}\right ) (19 d+52 f+112 h)-\frac {1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac {x^2 (-(2 e+5 g+17 i))+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )}+\frac {1}{54} \log \left (1-x^2\right ) (2 e+5 g+8 i)-\frac {1}{54} \log \left (4-x^2\right ) (2 e+5 g+8 i) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(72*(4 - 5*x^2 + x^4)) + (5*e + 8*g + 20*i - (2*e + 5*g + 17
*i)*x^2)/(18*(4 - 5*x^2 + x^4)) + ((19*d + 52*f + 112*h)*ArcTanh[x/2])/432 - ((d + 7*f + 13*h)*ArcTanh[x])/54
+ ((2*e + 5*g + 8*i)*Log[1 - x^2])/54 - ((2*e + 5*g + 8*i)*Log[4 - x^2])/54

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

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

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 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 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 1660

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[Pq, 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] + Dist[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 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[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] && Inte
gerQ[(m - 1)/2]

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
]

Rubi steps

\begin {align*} \int \frac {d+e x+f x^2+g x^3+h x^4+30 x^5}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {x \left (e+g x^2+30 x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx+\int \frac {d+f x^2+h x^4}{\left (4-5 x^2+x^4\right )^2} \, dx\\ &=\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{72 \left (4-5 x^2+x^4\right )}-\frac {1}{72} \int \frac {-d+20 f+32 h+(5 d+8 f+20 h) x^2}{4-5 x^2+x^4} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e+g x+30 x^2}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {600+5 e+8 g-(510+2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{72 \left (4-5 x^2+x^4\right )}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {240+2 e+5 g}{4-5 x+x^2} \, dx,x,x^2\right )-\frac {1}{54} (-d-7 f-13 h) \int \frac {1}{-1+x^2} \, dx-\frac {1}{216} (19 d+52 f+112 h) \int \frac {1}{-4+x^2} \, dx\\ &=\frac {600+5 e+8 g-(510+2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f+112 h) \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f+13 h) \tanh ^{-1}(x)-\frac {1}{18} (240+2 e+5 g) \operatorname {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac {600+5 e+8 g-(510+2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f+112 h) \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f+13 h) \tanh ^{-1}(x)-\frac {1}{54} (-240-2 e-5 g) \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )-\frac {1}{54} (240+2 e+5 g) \operatorname {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )\\ &=\frac {600+5 e+8 g-(510+2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f+112 h) \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f+13 h) \tanh ^{-1}(x)+\frac {1}{54} (240+2 e+5 g) \log \left (1-x^2\right )-\frac {1}{54} (240+2 e+5 g) \log \left (4-x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 185, normalized size = 1.14 \begin {gather*} \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-68 i x^2+80 i}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{108} \log (1-x) (d+4 e+7 f+10 g+13 h+16 i)+\frac {1}{864} \log (2-x) (-19 d-32 e-52 f-80 g-112 h-128 i)+\frac {1}{108} \log (x+1) (-d+4 e-7 f+10 g-13 h+16 i)+\frac {1}{864} \log (x+2) (19 d-32 e+52 f-80 g+112 h-128 i) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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)
/(72*(4 - 5*x^2 + x^4)) + ((d + 4*e + 7*f + 10*g + 13*h + 16*i)*Log[1 - x])/108 + ((-19*d - 32*e - 52*f - 80*g
- 112*h - 128*i)*Log[2 - x])/864 + ((-d + 4*e - 7*f + 10*g - 13*h + 16*i)*Log[1 + x])/108 + ((19*d - 32*e + 5
2*f - 80*g + 112*h - 128*i)*Log[2 + x])/864

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 29.57, size = 346, normalized size = 2.14 \begin {gather*} -\frac {12 \, {\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 48 \, {\left (2 \, e + 5 \, g + 17 \, i\right )} x^{2} - 12 \, {\left (17 \, d + 20 \, f + 32 \, h\right )} x - {\left ({\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} x^{4} - 5 \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f - 320 \, g + 448 \, h - 512 \, i\right )} \log \left (x + 2\right ) + 8 \, {\left ({\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} x^{4} - 5 \, {\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f - 40 \, g + 52 \, h - 64 \, i\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} x^{4} - 5 \, {\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f + 40 \, g + 52 \, h + 64 \, i\right )} \log \left (x - 1\right ) + {\left ({\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} x^{4} - 5 \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f + 320 \, g + 448 \, h + 512 \, i\right )} \log \left (x - 2\right ) - 240 \, e - 384 \, g - 960 \, i}{864 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*(12*(5*d + 8*f + 20*h)*x^3 + 48*(2*e + 5*g + 17*i)*x^2 - 12*(17*d + 20*f + 32*h)*x - ((19*d - 32*e + 52
*f - 80*g + 112*h - 128*i)*x^4 - 5*(19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*x^2 + 76*d - 128*e + 208*f - 32
0*g + 448*h - 512*i)*log(x + 2) + 8*((d - 4*e + 7*f - 10*g + 13*h - 16*i)*x^4 - 5*(d - 4*e + 7*f - 10*g + 13*h
- 16*i)*x^2 + 4*d - 16*e + 28*f - 40*g + 52*h - 64*i)*log(x + 1) - 8*((d + 4*e + 7*f + 10*g + 13*h + 16*i)*x^
4 - 5*(d + 4*e + 7*f + 10*g + 13*h + 16*i)*x^2 + 4*d + 16*e + 28*f + 40*g + 52*h + 64*i)*log(x - 1) + ((19*d +
32*e + 52*f + 80*g + 112*h + 128*i)*x^4 - 5*(19*d + 32*e + 52*f + 80*g + 112*h + 128*i)*x^2 + 76*d + 128*e +
208*f + 320*g + 448*h + 512*i)*log(x - 2) - 240*e - 384*g - 960*i)/(x^4 - 5*x^2 + 4)

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giac [A]  time = 0.32, size = 179, normalized size = 1.10 \begin {gather*} \frac {1}{864} \, {\left (19 \, d + 52 \, f - 80 \, g + 112 \, h - 128 \, i - 32 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{108} \, {\left (d + 7 \, f - 10 \, g + 13 \, h - 16 \, i - 4 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{108} \, {\left (d + 7 \, f + 10 \, g + 13 \, h + 16 \, i + 4 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{864} \, {\left (19 \, d + 52 \, f + 80 \, g + 112 \, h + 128 \, i + 32 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {5 \, d x^{3} + 8 \, f x^{3} + 20 \, h x^{3} + 20 \, g x^{2} + 68 \, i x^{2} + 8 \, x^{2} e - 17 \, d x - 20 \, f x - 32 \, h x - 32 \, g - 80 \, i - 20 \, e}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864*(19*d + 52*f - 80*g + 112*h - 128*i - 32*e)*log(abs(x + 2)) - 1/108*(d + 7*f - 10*g + 13*h - 16*i - 4*e)
*log(abs(x + 1)) + 1/108*(d + 7*f + 10*g + 13*h + 16*i + 4*e)*log(abs(x - 1)) - 1/864*(19*d + 52*f + 80*g + 11
2*h + 128*i + 32*e)*log(abs(x - 2)) - 1/72*(5*d*x^3 + 8*f*x^3 + 20*h*x^3 + 20*g*x^2 + 68*i*x^2 + 8*x^2*e - 17*
d*x - 20*f*x - 32*h*x - 32*g - 80*i - 20*e)/(x^4 - 5*x^2 + 4)

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maple [B]  time = 0.02, size = 362, normalized size = 2.23 \begin {gather*} -\frac {4 i \ln \left (x +2\right )}{27}+\frac {4 i \ln \left (x -1\right )}{27}+\frac {4 i \ln \left (x +1\right )}{27}-\frac {4 i \ln \left (x -2\right )}{27}+\frac {7 h \ln \left (x +2\right )}{54}+\frac {13 h \ln \left (x -1\right )}{108}-\frac {13 h \ln \left (x +1\right )}{108}-\frac {7 h \ln \left (x -2\right )}{54}+\frac {5 g \ln \left (x -1\right )}{54}-\frac {5 g \ln \left (x +2\right )}{54}-\frac {5 g \ln \left (x -2\right )}{54}+\frac {5 g \ln \left (x +1\right )}{54}+\frac {19 d \ln \left (x +2\right )}{864}-\frac {e \ln \left (x +2\right )}{27}+\frac {e \ln \left (x -1\right )}{27}+\frac {d \ln \left (x -1\right )}{108}+\frac {e \ln \left (x +1\right )}{27}-\frac {d \ln \left (x +1\right )}{108}-\frac {19 d \ln \left (x -2\right )}{864}-\frac {e \ln \left (x -2\right )}{27}-\frac {13 f \ln \left (x -2\right )}{216}-\frac {7 f \ln \left (x +1\right )}{108}+\frac {7 f \ln \left (x -1\right )}{108}+\frac {13 f \ln \left (x +2\right )}{216}+\frac {i}{36 x +36}+\frac {g}{18 x +36}+\frac {g}{36 x +36}+\frac {e}{36 x +36}+\frac {e}{72 x +144}+\frac {2 i}{9 \left (x +2\right )}-\frac {i}{36 \left (x -1\right )}-\frac {2 i}{9 \left (x -2\right )}-\frac {h}{9 \left (x +2\right )}-\frac {h}{36 \left (x +1\right )}-\frac {h}{36 \left (x -1\right )}-\frac {h}{9 \left (x -2\right )}-\frac {g}{36 \left (x -1\right )}-\frac {g}{18 \left (x -2\right )}-\frac {d}{144 \left (x +2\right )}-\frac {d}{144 \left (x -2\right )}-\frac {e}{72 \left (x -2\right )}-\frac {d}{36 \left (x +1\right )}-\frac {d}{36 \left (x -1\right )}-\frac {e}{36 \left (x -1\right )}-\frac {f}{36 \left (x -1\right )}-\frac {f}{36 \left (x +2\right )}-\frac {f}{36 \left (x -2\right )}-\frac {f}{36 \left (x +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/27*i*ln(x+2)+4/27*i*ln(x-1)+4/27*i*ln(x+1)-4/27*i*ln(x-2)+7/54*h*ln(x+2)+13/108*h*ln(x-1)-13/108*h*ln(x+1)-
7/54*h*ln(x-2)+5/54*g*ln(x-1)-5/54*g*ln(x+2)-5/54*g*ln(x-2)+5/54*g*ln(x+1)+19/864*d*ln(x+2)-1/27*e*ln(x+2)+1/2
7*e*ln(x-1)+1/108*d*ln(x-1)+1/27*e*ln(x+1)-1/108*d*ln(x+1)-19/864*d*ln(x-2)-1/27*e*ln(x-2)-13/216*f*ln(x-2)-7/
108*f*ln(x+1)+7/108*f*ln(x-1)+13/216*f*ln(x+2)+2/9/(x+2)*i+1/36/(x+1)*i-1/36/(x-1)*i-2/9/(x-2)*i-1/9/(x+2)*h-1
/36/(x+1)*h-1/36/(x-1)*h-1/9/(x-2)*h+1/18/(x+2)*g+1/36/(x+1)*g-1/36/(x-1)*g-1/18/(x-2)*g-1/144/(x+2)*d+1/72/(x
+2)*e-1/144/(x-2)*d-1/72/(x-2)*e-1/36/(x+1)*d+1/36/(x+1)*e-1/36/(x-1)*d-1/36/(x-1)*e-1/36/(x-1)*f-1/36/(x+2)*f
-1/36/(x-2)*f-1/36/(x+1)*f

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maxima [A]  time = 1.35, size = 163, normalized size = 1.01 \begin {gather*} \frac {1}{864} \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} \log \left (x + 2\right ) - \frac {1}{108} \, {\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} \log \left (x + 1\right ) + \frac {1}{108} \, {\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} \log \left (x - 1\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} \log \left (x - 2\right ) - \frac {{\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 4 \, {\left (2 \, e + 5 \, g + 17 \, i\right )} x^{2} - {\left (17 \, d + 20 \, f + 32 \, h\right )} x - 20 \, e - 32 \, g - 80 \, i}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864*(19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*log(x + 2) - 1/108*(d - 4*e + 7*f - 10*g + 13*h - 16*i)*log(
x + 1) + 1/108*(d + 4*e + 7*f + 10*g + 13*h + 16*i)*log(x - 1) - 1/864*(19*d + 32*e + 52*f + 80*g + 112*h + 12
8*i)*log(x - 2) - 1/72*((5*d + 8*f + 20*h)*x^3 + 4*(2*e + 5*g + 17*i)*x^2 - (17*d + 20*f + 32*h)*x - 20*e - 32
*g - 80*i)/(x^4 - 5*x^2 + 4)

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mupad [B]  time = 0.58, size = 164, normalized size = 1.01 \begin {gather*} \frac {\left (-\frac {5\,d}{72}-\frac {f}{9}-\frac {5\,h}{18}\right )\,x^3+\left (-\frac {e}{9}-\frac {5\,g}{18}-\frac {17\,i}{18}\right )\,x^2+\left (\frac {17\,d}{72}+\frac {5\,f}{18}+\frac {4\,h}{9}\right )\,x+\frac {5\,e}{18}+\frac {4\,g}{9}+\frac {10\,i}{9}}{x^4-5\,x^2+4}+\ln \left (x-1\right )\,\left (\frac {d}{108}+\frac {e}{27}+\frac {7\,f}{108}+\frac {5\,g}{54}+\frac {13\,h}{108}+\frac {4\,i}{27}\right )-\ln \left (x+1\right )\,\left (\frac {d}{108}-\frac {e}{27}+\frac {7\,f}{108}-\frac {5\,g}{54}+\frac {13\,h}{108}-\frac {4\,i}{27}\right )-\ln \left (x-2\right )\,\left (\frac {19\,d}{864}+\frac {e}{27}+\frac {13\,f}{216}+\frac {5\,g}{54}+\frac {7\,h}{54}+\frac {4\,i}{27}\right )+\ln \left (x+2\right )\,\left (\frac {19\,d}{864}-\frac {e}{27}+\frac {13\,f}{216}-\frac {5\,g}{54}+\frac {7\,h}{54}-\frac {4\,i}{27}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*e)/18 + (4*g)/9 + (10*i)/9 + x*((17*d)/72 + (5*f)/18 + (4*h)/9) - x^3*((5*d)/72 + f/9 + (5*h)/18) - x^2*(e
/9 + (5*g)/18 + (17*i)/18))/(x^4 - 5*x^2 + 4) + log(x - 1)*(d/108 + e/27 + (7*f)/108 + (5*g)/54 + (13*h)/108 +
(4*i)/27) - log(x + 1)*(d/108 - e/27 + (7*f)/108 - (5*g)/54 + (13*h)/108 - (4*i)/27) - log(x - 2)*((19*d)/864
+ e/27 + (13*f)/216 + (5*g)/54 + (7*h)/54 + (4*i)/27) + log(x + 2)*((19*d)/864 - e/27 + (13*f)/216 - (5*g)/54
+ (7*h)/54 - (4*i)/27)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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