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

Optimal. Leaf size=138 \[ \frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac{1}{54} (2 e+5 g) \log \left (4-x^2\right )+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{18 \left (x^4-5 x^2+4\right )} \]

[Out]

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(72*(4 - 5*x^2 + x^4)) + (5*e + 8*g - (2*e +
5*g)*x^2)/(18*(4 - 5*x^2 + x^4)) + ((19*d + 52*f)*ArcTanh[x/2])/432 - ((d + 7*f
)*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 = 0.351792, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac{1}{54} (2 e+5 g) \log \left (4-x^2\right )+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{18 \left (x^4-5 x^2+4\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(72*(4 - 5*x^2 + x^4)) + (5*e + 8*g - (2*e +
5*g)*x^2)/(18*(4 - 5*x^2 + x^4)) + ((19*d + 52*f)*ArcTanh[x/2])/432 - ((d + 7*f
)*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 = 45.9873, size = 105, normalized size = 0.76 \[ \frac{x \left (17 d + 20 f - x^{3} \left (5 e + 8 g\right ) - x^{2} \left (5 d + 8 f\right ) + x \left (17 e + 20 g\right )\right )}{72 \left (x^{4} - 5 x^{2} + 4\right )} - \left (\frac{d}{54} + \frac{7 f}{54}\right ) \operatorname{atanh}{\left (x \right )} + \left (\frac{19 d}{432} + \frac{13 f}{108}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{e}{27} + \frac{5 g}{54}\right ) \log{\left (- x^{2} + 1 \right )} - \left (\frac{e}{27} + \frac{5 g}{54}\right ) \log{\left (- x^{2} + 4 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(17*d + 20*f - x**3*(5*e + 8*g) - x**2*(5*d + 8*f) + x*(17*e + 20*g))/(72*(x**
4 - 5*x**2 + 4)) - (d/54 + 7*f/54)*atanh(x) + (19*d/432 + 13*f/108)*atanh(x/2) +
(e/27 + 5*g/54)*log(-x**2 + 1) - (e/27 + 5*g/54)*log(-x**2 + 4)

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Mathematica [A]  time = 0.101133, size = 134, normalized size = 0.97 \[ \frac{1}{864} \left (\frac{12 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x-4 g \left (5 x^2-8\right )\right )}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f+10 g)-\log (2-x) (19 d+32 e+52 f+80 g)-8 \log (x+1) (d-4 e+7 f-10 g)+\log (x+2) (19 d-32 e+52 f-80 g)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((12*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2) - 4*g*(-8 + 5*x^2)))/
(4 - 5*x^2 + x^4) + 8*(d + 4*e + 7*f + 10*g)*Log[1 - x] - (19*d + 32*e + 52*f +
80*g)*Log[2 - x] - 8*(d - 4*e + 7*f - 10*g)*Log[1 + x] + (19*d - 32*e + 52*f - 8
0*g)*Log[2 + x])/864

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Maple [A]  time = 0.027, size = 242, normalized size = 1.8 \[{\frac{g}{36+36\,x}}-{\frac{g}{-36+36\,x}}+{\frac{g}{36+18\,x}}-{\frac{g}{18\,x-36}}-{\frac{f}{36+36\,x}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{d}{144\,x-288}}-{\frac{e}{72\,x-144}}-{\frac{f}{36\,x-72}}-{\frac{f}{-36+36\,x}}-{\frac{d}{288+144\,x}}+{\frac{e}{144+72\,x}}-{\frac{d}{-36+36\,x}}-{\frac{e}{-36+36\,x}}-{\frac{f}{72+36\,x}}-{\frac{\ln \left ( 1+x \right ) d}{108}}+{\frac{\ln \left ( 1+x \right ) e}{27}}+{\frac{\ln \left ( -1+x \right ) d}{108}}+{\frac{\ln \left ( -1+x \right ) e}{27}}+{\frac{5\,\ln \left ( 1+x \right ) g}{54}}-{\frac{5\,\ln \left ( x-2 \right ) g}{54}}+{\frac{5\,\ln \left ( -1+x \right ) g}{54}}-{\frac{5\,\ln \left ( 2+x \right ) g}{54}}-{\frac{19\,\ln \left ( x-2 \right ) d}{864}}-{\frac{\ln \left ( x-2 \right ) e}{27}}-{\frac{\ln \left ( 2+x \right ) e}{27}}-{\frac{13\,\ln \left ( x-2 \right ) f}{216}}+{\frac{19\,\ln \left ( 2+x \right ) d}{864}}-{\frac{7\,\ln \left ( 1+x \right ) f}{108}}+{\frac{7\,\ln \left ( -1+x \right ) f}{108}}+{\frac{13\,\ln \left ( 2+x \right ) f}{216}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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*ln(-1+x)*e+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*ln(-1+x)*f+13/216*ln(2+x)*f

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Maxima [A]  time = 0.703605, size = 171, normalized size = 1.24 \[ \frac{1}{864} \,{\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} \log \left (x + 2\right ) - \frac{1}{108} \,{\left (d - 4 \, e + 7 \, f - 10 \, g\right )} \log \left (x + 1\right ) + \frac{1}{108} \,{\left (d + 4 \, e + 7 \, f + 10 \, g\right )} \log \left (x - 1\right ) - \frac{1}{864} \,{\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 8 \, f\right )} x^{3} + 4 \,{\left (2 \, e + 5 \, g\right )} x^{2} -{\left (17 \, d + 20 \, f\right )} x - 20 \, e - 32 \, g}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)*log(x + 2) - 1/108*(d - 4*e + 7*f - 10*g)*log(
x + 1) + 1/108*(d + 4*e + 7*f + 10*g)*log(x - 1) - 1/864*(19*d + 32*e + 52*f + 8
0*g)*log(x - 2) - 1/72*((5*d + 8*f)*x^3 + 4*(2*e + 5*g)*x^2 - (17*d + 20*f)*x -
20*e - 32*g)/(x^4 - 5*x^2 + 4)

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Fricas [A]  time = 0.561459, size = 354, normalized size = 2.57 \[ -\frac{12 \,{\left (5 \, d + 8 \, f\right )} x^{3} + 48 \,{\left (2 \, e + 5 \, g\right )} x^{2} - 12 \,{\left (17 \, d + 20 \, f\right )} x -{\left ({\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} x^{4} - 5 \,{\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f - 320 \, g\right )} \log \left (x + 2\right ) + 8 \,{\left ({\left (d - 4 \, e + 7 \, f - 10 \, g\right )} x^{4} - 5 \,{\left (d - 4 \, e + 7 \, f - 10 \, g\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f - 40 \, g\right )} \log \left (x + 1\right ) - 8 \,{\left ({\left (d + 4 \, e + 7 \, f + 10 \, g\right )} x^{4} - 5 \,{\left (d + 4 \, e + 7 \, f + 10 \, g\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f + 40 \, g\right )} \log \left (x - 1\right ) +{\left ({\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} x^{4} - 5 \,{\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f + 320 \, g\right )} \log \left (x - 2\right ) - 240 \, e - 384 \, g}{864 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)*x^3 + 48*(2*e + 5*g)*x^2 - 12*(17*d + 20*f)*x - ((19*d -
32*e + 52*f - 80*g)*x^4 - 5*(19*d - 32*e + 52*f - 80*g)*x^2 + 76*d - 128*e + 208
*f - 320*g)*log(x + 2) + 8*((d - 4*e + 7*f - 10*g)*x^4 - 5*(d - 4*e + 7*f - 10*g
)*x^2 + 4*d - 16*e + 28*f - 40*g)*log(x + 1) - 8*((d + 4*e + 7*f + 10*g)*x^4 - 5
*(d + 4*e + 7*f + 10*g)*x^2 + 4*d + 16*e + 28*f + 40*g)*log(x - 1) + ((19*d + 32
*e + 52*f + 80*g)*x^4 - 5*(19*d + 32*e + 52*f + 80*g)*x^2 + 76*d + 128*e + 208*f
+ 320*g)*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((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 = 0.273555, size = 184, normalized size = 1.33 \[ \frac{1}{864} \,{\left (19 \, d + 52 \, f - 80 \, g - 32 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{108} \,{\left (d + 7 \, f - 10 \, g - 4 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{108} \,{\left (d + 7 \, f + 10 \, g + 4 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{864} \,{\left (19 \, d + 52 \, f + 80 \, g + 32 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{5 \, d x^{3} + 8 \, f x^{3} + 20 \, g x^{2} + 8 \, x^{2} e - 17 \, d x - 20 \, f x - 32 \, g - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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