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

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

[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

_______________________________________________________________________________________

Rubi [A]  time = 0.470195, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} \tanh ^{-1}\left (\frac{x}{2}\right ) (19 d+52 f+112 h)-\frac{1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac{1}{54} \log \left (1-x^2\right ) (2 e+5 g+8 i)-\frac{1}{54} \log \left (4-x^2\right ) (2 e+5 g+8 i)+\frac{x^2 (-(2 e+5 g+17 i))+5 e+8 g+20 i}{18 \left (x^4-5 x^2+4\right )} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 69.3777, size = 139, normalized size = 0.86 \[ \frac{x \left (2125 d + 2500 f + 4000 h - x^{3} \left (625 e + 1000 g + 2500\right ) - x^{2} \left (625 d + 1000 f + 2500 h\right ) + x \left (2125 e + 2500 g + 4000\right )\right )}{9000 \left (x^{4} - 5 x^{2} + 4\right )} - \left (\frac{d}{54} + \frac{7 f}{54} + \frac{13 h}{54}\right ) \operatorname{atanh}{\left (x \right )} + \left (\frac{19 d}{432} + \frac{13 f}{108} + \frac{7 h}{27}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{e}{27} + \frac{5 g}{54} - \frac{i}{6} + \frac{17}{54}\right ) \log{\left (- x^{2} + 1 \right )} - \left (\frac{e}{27} + \frac{5 g}{54} - \frac{i}{6} + \frac{17}{54}\right ) \log{\left (- x^{2} + 4 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

x*(2125*d + 2500*f + 4000*h - x**3*(625*e + 1000*g + 2500) - x**2*(625*d + 1000*
f + 2500*h) + x*(2125*e + 2500*g + 4000))/(9000*(x**4 - 5*x**2 + 4)) - (d/54 + 7
*f/54 + 13*h/54)*atanh(x) + (19*d/432 + 13*f/108 + 7*h/27)*atanh(x/2) + (e/27 +
5*g/54 - i/6 + 17/54)*log(-x**2 + 1) - (e/27 + 5*g/54 - i/6 + 17/54)*log(-x**2 +
4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.188378, size = 185, normalized size = 1.14 \[ \frac{-5 d x^3+17 d x-8 e x^2+20 e-8 f x^3+20 f x-20 g x^2+32 g-20 h x^3+32 h x-68 i x^2+80 i}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{108} \log (1-x) (d+4 e+7 f+10 g+13 h+16 i)+\frac{1}{864} \log (2-x) (-19 d-32 e-52 f-80 g-112 h-128 i)+\frac{1}{108} \log (x+1) (-d+4 e-7 f+10 g-13 h+16 i)+\frac{1}{864} \log (x+2) (19 d-32 e+52 f-80 g+112 h-128 i) \]

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)*Lo
g[2 - x])/864 + ((-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

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 362, normalized size = 2.2 \[ -{\frac{2\,i}{9\,x-18}}+{\frac{i}{36+36\,x}}-{\frac{i}{-36+36\,x}}+{\frac{2\,i}{18+9\,x}}-{\frac{h}{9\,x-18}}-{\frac{h}{36+36\,x}}-{\frac{h}{-36+36\,x}}-{\frac{h}{18+9\,x}}+{\frac{g}{36+36\,x}}-{\frac{g}{-36+36\,x}}+{\frac{g}{36+18\,x}}-{\frac{g}{18\,x-36}}-{\frac{f}{36+36\,x}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{d}{144\,x-288}}-{\frac{e}{72\,x-144}}-{\frac{f}{36\,x-72}}-{\frac{f}{-36+36\,x}}-{\frac{d}{288+144\,x}}+{\frac{e}{144+72\,x}}-{\frac{d}{-36+36\,x}}-{\frac{e}{-36+36\,x}}-{\frac{f}{72+36\,x}}-{\frac{\ln \left ( 1+x \right ) d}{108}}+{\frac{\ln \left ( 1+x \right ) e}{27}}+{\frac{\ln \left ( -1+x \right ) d}{108}}+{\frac{\ln \left ( -1+x \right ) e}{27}}-{\frac{4\,\ln \left ( x-2 \right ) i}{27}}+{\frac{4\,\ln \left ( 1+x \right ) i}{27}}+{\frac{4\,\ln \left ( -1+x \right ) i}{27}}-{\frac{4\,\ln \left ( 2+x \right ) i}{27}}-{\frac{7\,\ln \left ( x-2 \right ) h}{54}}-{\frac{13\,\ln \left ( 1+x \right ) h}{108}}+{\frac{7\,\ln \left ( 2+x \right ) h}{54}}+{\frac{13\,\ln \left ( -1+x \right ) h}{108}}+{\frac{5\,\ln \left ( 1+x \right ) g}{54}}-{\frac{5\,\ln \left ( x-2 \right ) g}{54}}+{\frac{5\,\ln \left ( -1+x \right ) g}{54}}-{\frac{5\,\ln \left ( 2+x \right ) g}{54}}-{\frac{19\,\ln \left ( x-2 \right ) d}{864}}-{\frac{\ln \left ( x-2 \right ) e}{27}}-{\frac{\ln \left ( 2+x \right ) e}{27}}-{\frac{13\,\ln \left ( x-2 \right ) f}{216}}+{\frac{19\,\ln \left ( 2+x \right ) d}{864}}-{\frac{7\,\ln \left ( 1+x \right ) f}{108}}+{\frac{7\,\ln \left ( -1+x \right ) f}{108}}+{\frac{13\,\ln \left ( 2+x \right ) f}{216}} \]

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]

-2/9/(x-2)*i+1/36/(1+x)*i-1/36/(-1+x)*i+2/9/(2+x)*i-1/9/(x-2)*h-1/36/(1+x)*h-1/3
6/(-1+x)*h-1/9/(2+x)*h+1/36/(1+x)*g-1/36/(-1+x)*g+1/18/(2+x)*g-1/18/(x-2)*g-1/36
/(1+x)*f-1/36/(1+x)*d+1/36/(1+x)*e-1/144/(x-2)*d-1/72/(x-2)*e-1/36/(x-2)*f-1/36/
(-1+x)*f-1/144/(2+x)*d+1/72/(2+x)*e-1/36/(-1+x)*d-1/36/(-1+x)*e-1/36/(2+x)*f-1/1
08*ln(1+x)*d+1/27*ln(1+x)*e+1/108*ln(-1+x)*d+1/27*ln(-1+x)*e-4/27*ln(x-2)*i+4/27
*ln(1+x)*i+4/27*ln(-1+x)*i-4/27*ln(2+x)*i-7/54*ln(x-2)*h-13/108*ln(1+x)*h+7/54*l
n(2+x)*h+13/108*ln(-1+x)*h+5/54*ln(1+x)*g-5/54*ln(x-2)*g+5/54*ln(-1+x)*g-5/54*ln
(2+x)*g-19/864*ln(x-2)*d-1/27*ln(x-2)*e-1/27*ln(2+x)*e-13/216*ln(x-2)*f+19/864*l
n(2+x)*d-7/108*ln(1+x)*f+7/108*ln(-1+x)*f+13/216*ln(2+x)*f

_______________________________________________________________________________________

Maxima [A]  time = 0.710376, size = 220, normalized size = 1.36 \[ \frac{1}{864} \,{\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} \log \left (x + 2\right ) - \frac{1}{108} \,{\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} \log \left (x + 1\right ) + \frac{1}{108} \,{\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} \log \left (x - 1\right ) - \frac{1}{864} \,{\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 4 \,{\left (2 \, e + 5 \, g + 17 \, i\right )} x^{2} -{\left (17 \, d + 20 \, f + 32 \, h\right )} x - 20 \, e - 32 \, g - 80 \, i}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

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 + 128*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)

_______________________________________________________________________________________

Fricas [A]  time = 8.51822, size = 467, normalized size = 2.88 \[ -\frac{12 \,{\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 48 \,{\left (2 \, e + 5 \, g + 17 \, i\right )} x^{2} - 12 \,{\left (17 \, d + 20 \, f + 32 \, h\right )} x -{\left ({\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} x^{4} - 5 \,{\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h - 128 \, i\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f - 320 \, g + 448 \, h - 512 \, i\right )} \log \left (x + 2\right ) + 8 \,{\left ({\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} x^{4} - 5 \,{\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h - 16 \, i\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f - 40 \, g + 52 \, h - 64 \, i\right )} \log \left (x + 1\right ) - 8 \,{\left ({\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} x^{4} - 5 \,{\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h + 16 \, i\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f + 40 \, g + 52 \, h + 64 \, i\right )} \log \left (x - 1\right ) +{\left ({\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} x^{4} - 5 \,{\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h + 128 \, i\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f + 320 \, g + 448 \, h + 512 \, i\right )} \log \left (x - 2\right ) - 240 \, e - 384 \, g - 960 \, i}{864 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

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 - 320*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 - 1
0*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)

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.275813, size = 242, normalized size = 1.49 \[ \frac{1}{864} \,{\left (19 \, d + 52 \, f - 80 \, g + 112 \, h - 128 \, i - 32 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{108} \,{\left (d + 7 \, f - 10 \, g + 13 \, h - 16 \, i - 4 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{108} \,{\left (d + 7 \, f + 10 \, g + 13 \, h + 16 \, i + 4 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{864} \,{\left (19 \, d + 52 \, f + 80 \, g + 112 \, h + 128 \, i + 32 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{5 \, d x^{3} + 8 \, f x^{3} + 20 \, h x^{3} + 20 \, g x^{2} + 68 \, i x^{2} + 8 \, x^{2} e - 17 \, d x - 20 \, f x - 32 \, h x - 32 \, g - 80 \, i - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

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)*ln(abs(x + 2)) - 1/108*(d + 7*
f - 10*g + 13*h - 16*i - 4*e)*ln(abs(x + 1)) + 1/108*(d + 7*f + 10*g + 13*h + 16
*i + 4*e)*ln(abs(x - 1)) - 1/864*(19*d + 52*f + 80*g + 112*h + 128*i + 32*e)*ln(
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)