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

Optimal. Leaf size=150 \[ \frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} \tanh ^{-1}\left (\frac{x}{2}\right ) (19 d+52 f+112 h)-\frac{1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac{1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac{1}{54} (2 e+5 g) \log \left (4-x^2\right )+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{18 \left (x^4-5 x^2+4\right )} \]

[Out]

(5*e + 8*g - (2*e + 5*g)*x^2)/(18*(4 - 5*x^2 + x^4)) + (x*(17*d + 20*f + 32*h -
(5*d + 8*f + 20*h)*x^2))/(72*(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)*Log[1 - x^2])/54 -
((2*e + 5*g)*Log[4 - x^2])/54

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Rubi [A]  time = 0.431328, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242 \[ \frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} \tanh ^{-1}\left (\frac{x}{2}\right ) (19 d+52 f+112 h)-\frac{1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac{1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac{1}{54} (2 e+5 g) \log \left (4-x^2\right )+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{18 \left (x^4-5 x^2+4\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*e + 8*g - (2*e + 5*g)*x^2)/(18*(4 - 5*x^2 + x^4)) + (x*(17*d + 20*f + 32*h -
(5*d + 8*f + 20*h)*x^2))/(72*(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)*Log[1 - x^2])/54 -
((2*e + 5*g)*Log[4 - x^2])/54

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Rubi in Sympy [A]  time = 58.549, size = 122, normalized size = 0.81 \[ \frac{x \left (425 d + 500 f + 800 h - x^{3} \left (125 e + 200 g\right ) - x^{2} \left (125 d + 200 f + 500 h\right ) + x \left (425 e + 500 g\right )\right )}{1800 \left (x^{4} - 5 x^{2} + 4\right )} + \left (\frac{e}{27} + \frac{5 g}{54}\right ) \log{\left (- x^{2} + 1 \right )} - \left (\frac{e}{27} + \frac{5 g}{54}\right ) \log{\left (- x^{2} + 4 \right )} - \left (\frac{d}{54} + \frac{7 f}{54} + \frac{13 h}{54}\right ) \operatorname{atanh}{\left (x \right )} + \left (\frac{19 d}{432} + \frac{13 f}{108} + \frac{7 h}{27}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(425*d + 500*f + 800*h - x**3*(125*e + 200*g) - x**2*(125*d + 200*f + 500*h) +
x*(425*e + 500*g))/(1800*(x**4 - 5*x**2 + 4)) + (e/27 + 5*g/54)*log(-x**2 + 1)
- (e/27 + 5*g/54)*log(-x**2 + 4) - (d/54 + 7*f/54 + 13*h/54)*atanh(x) + (19*d/43
2 + 13*f/108 + 7*h/27)*atanh(x/2)

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Mathematica [A]  time = 0.140405, size = 159, normalized size = 1.06 \[ \frac{1}{864} \left (-\frac{12 \left (x \left (d \left (5 x^2-17\right )+4 f \left (2 x^2-5\right )+4 h \left (5 x^2-8\right )\right )+4 e \left (2 x^2-5\right )+4 g \left (5 x^2-8\right )\right )}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f+10 g+13 h)-\log (2-x) (19 d+32 e+52 f+80 g+112 h)-8 \log (x+1) (d-4 e+7 f-10 g+13 h)+\log (x+2) (19 d-32 e+52 f-80 g+112 h)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-12*(4*e*(-5 + 2*x^2) + 4*g*(-8 + 5*x^2) + x*(4*f*(-5 + 2*x^2) + d*(-17 + 5*x^
2) + 4*h*(-8 + 5*x^2))))/(4 - 5*x^2 + x^4) + 8*(d + 4*e + 7*f + 10*g + 13*h)*Log
[1 - x] - (19*d + 32*e + 52*f + 80*g + 112*h)*Log[2 - x] - 8*(d - 4*e + 7*f - 10
*g + 13*h)*Log[1 + x] + (19*d - 32*e + 52*f - 80*g + 112*h)*Log[2 + x])/864

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Maple [B]  time = 0.028, size = 302, normalized size = 2. \[ -{\frac{h}{9\,x-18}}-{\frac{h}{36+36\,x}}-{\frac{h}{-36+36\,x}}-{\frac{h}{18+9\,x}}+{\frac{g}{36+36\,x}}-{\frac{g}{-36+36\,x}}+{\frac{g}{36+18\,x}}-{\frac{g}{18\,x-36}}-{\frac{f}{36+36\,x}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{d}{144\,x-288}}-{\frac{e}{72\,x-144}}-{\frac{f}{36\,x-72}}-{\frac{f}{-36+36\,x}}-{\frac{d}{288+144\,x}}+{\frac{e}{144+72\,x}}-{\frac{d}{-36+36\,x}}-{\frac{e}{-36+36\,x}}-{\frac{f}{72+36\,x}}-{\frac{\ln \left ( 1+x \right ) d}{108}}+{\frac{\ln \left ( 1+x \right ) e}{27}}+{\frac{\ln \left ( -1+x \right ) d}{108}}+{\frac{\ln \left ( -1+x \right ) e}{27}}-{\frac{7\,\ln \left ( x-2 \right ) h}{54}}-{\frac{13\,\ln \left ( 1+x \right ) h}{108}}+{\frac{7\,\ln \left ( 2+x \right ) h}{54}}+{\frac{13\,\ln \left ( -1+x \right ) h}{108}}+{\frac{5\,\ln \left ( 1+x \right ) g}{54}}-{\frac{5\,\ln \left ( x-2 \right ) g}{54}}+{\frac{5\,\ln \left ( -1+x \right ) g}{54}}-{\frac{5\,\ln \left ( 2+x \right ) g}{54}}-{\frac{19\,\ln \left ( x-2 \right ) d}{864}}-{\frac{\ln \left ( x-2 \right ) e}{27}}-{\frac{\ln \left ( 2+x \right ) e}{27}}-{\frac{13\,\ln \left ( x-2 \right ) f}{216}}+{\frac{19\,\ln \left ( 2+x \right ) d}{864}}-{\frac{7\,\ln \left ( 1+x \right ) f}{108}}+{\frac{7\,\ln \left ( -1+x \right ) f}{108}}+{\frac{13\,\ln \left ( 2+x \right ) f}{216}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9/(x-2)*h-1/36/(1+x)*h-1/36/(-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-1/108*ln(1+x)*d+1/27*ln(1+x)*e+1/108*ln(-1+x)*d+1/27*l
n(-1+x)*e-7/54*ln(x-2)*h-13/108*ln(1+x)*h+7/54*ln(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*ln(2+x)*d-7/108*ln(1+x)*f+7/108*l
n(-1+x)*f+13/216*ln(2+x)*f

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Maxima [A]  time = 0.698831, size = 196, normalized size = 1.31 \[ \frac{1}{864} \,{\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h\right )} \log \left (x + 2\right ) - \frac{1}{108} \,{\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h\right )} \log \left (x + 1\right ) + \frac{1}{108} \,{\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h\right )} \log \left (x - 1\right ) - \frac{1}{864} \,{\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 4 \,{\left (2 \, e + 5 \, g\right )} x^{2} -{\left (17 \, d + 20 \, f + 32 \, h\right )} x - 20 \, e - 32 \, g}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76552, size = 410, normalized size = 2.73 \[ -\frac{12 \,{\left (5 \, d + 8 \, f + 20 \, h\right )} x^{3} + 48 \,{\left (2 \, e + 5 \, g\right )} x^{2} - 12 \,{\left (17 \, d + 20 \, f + 32 \, h\right )} x -{\left ({\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h\right )} x^{4} - 5 \,{\left (19 \, d - 32 \, e + 52 \, f - 80 \, g + 112 \, h\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f - 320 \, g + 448 \, h\right )} \log \left (x + 2\right ) + 8 \,{\left ({\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h\right )} x^{4} - 5 \,{\left (d - 4 \, e + 7 \, f - 10 \, g + 13 \, h\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f - 40 \, g + 52 \, h\right )} \log \left (x + 1\right ) - 8 \,{\left ({\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h\right )} x^{4} - 5 \,{\left (d + 4 \, e + 7 \, f + 10 \, g + 13 \, h\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f + 40 \, g + 52 \, h\right )} \log \left (x - 1\right ) +{\left ({\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h\right )} x^{4} - 5 \,{\left (19 \, d + 32 \, e + 52 \, f + 80 \, g + 112 \, h\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f + 320 \, g + 448 \, h\right )} \log \left (x - 2\right ) - 240 \, e - 384 \, g}{864 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

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

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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((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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GIAC/XCAS [A]  time = 0.277695, size = 213, normalized size = 1.42 \[ \frac{1}{864} \,{\left (19 \, d + 52 \, f - 80 \, g + 112 \, h - 32 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{108} \,{\left (d + 7 \, f - 10 \, g + 13 \, h - 4 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{108} \,{\left (d + 7 \, f + 10 \, g + 13 \, h + 4 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{864} \,{\left (19 \, d + 52 \, f + 80 \, g + 112 \, h + 32 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{5 \, d x^{3} + 8 \, f x^{3} + 20 \, h x^{3} + 20 \, g x^{2} + 8 \, x^{2} e - 17 \, d x - 20 \, f x - 32 \, h x - 32 \, g - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

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)^2,x, algorithm="giac")

[Out]

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