3.11 \(\int \frac{d+e x+f x^2}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=51 \[ -\frac{1}{6} (d+4 f) \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} (d+f) \tanh ^{-1}(x)-\frac{1}{6} e \log \left (1-x^2\right )+\frac{1}{6} e \log \left (4-x^2\right ) \]

[Out]

-((d + 4*f)*ArcTanh[x/2])/6 + ((d + f)*ArcTanh[x])/3 - (e*Log[1 - x^2])/6 + (e*L
og[4 - x^2])/6

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Rubi [A]  time = 0.116943, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{1}{6} (d+4 f) \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} (d+f) \tanh ^{-1}(x)-\frac{1}{6} e \log \left (1-x^2\right )+\frac{1}{6} e \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((d + 4*f)*ArcTanh[x/2])/6 + ((d + f)*ArcTanh[x])/3 - (e*Log[1 - x^2])/6 + (e*L
og[4 - x^2])/6

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Rubi in Sympy [A]  time = 23.5199, size = 42, normalized size = 0.82 \[ - \frac{e \log{\left (- x^{2} + 1 \right )}}{6} + \frac{e \log{\left (- x^{2} + 4 \right )}}{6} - \left (\frac{d}{6} + \frac{2 f}{3}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{d}{3} + \frac{f}{3}\right ) \operatorname{atanh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*log(-x**2 + 1)/6 + e*log(-x**2 + 4)/6 - (d/6 + 2*f/3)*atanh(x/2) + (d/3 + f/3
)*atanh(x)

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Mathematica [A]  time = 0.0502888, size = 58, normalized size = 1.14 \[ \frac{1}{12} (-2 \log (1-x) (d+e+f)+\log (2-x) (d+2 e+4 f)+2 \log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e + f)*Log[1 - x] + (d + 2*e + 4*f)*Log[2 - x] + 2*(d - e + f)*Log[1 +
x] - (d - 2*e + 4*f)*Log[2 + x])/12

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Maple [B]  time = 0.011, size = 86, normalized size = 1.7 \[ -{\frac{\ln \left ( 2+x \right ) d}{12}}+{\frac{\ln \left ( 2+x \right ) e}{6}}-{\frac{\ln \left ( 2+x \right ) f}{3}}-{\frac{\ln \left ( -1+x \right ) d}{6}}-{\frac{\ln \left ( -1+x \right ) e}{6}}-{\frac{\ln \left ( -1+x \right ) f}{6}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) f}{6}}+{\frac{\ln \left ( x-2 \right ) d}{12}}+{\frac{\ln \left ( x-2 \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) f}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*ln(2+x)*d+1/6*ln(2+x)*e-1/3*ln(2+x)*f-1/6*ln(-1+x)*d-1/6*ln(-1+x)*e-1/6*ln
(-1+x)*f+1/6*ln(1+x)*d-1/6*ln(1+x)*e+1/6*ln(1+x)*f+1/12*ln(x-2)*d+1/6*ln(x-2)*e+
1/3*ln(x-2)*f

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Maxima [A]  time = 0.702025, size = 69, normalized size = 1.35 \[ -\frac{1}{12} \,{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e + f\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")

[Out]

-1/12*(d - 2*e + 4*f)*log(x + 2) + 1/6*(d - e + f)*log(x + 1) - 1/6*(d + e + f)*
log(x - 1) + 1/12*(d + 2*e + 4*f)*log(x - 2)

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Fricas [A]  time = 0.324102, size = 69, normalized size = 1.35 \[ -\frac{1}{12} \,{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e + f\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")

[Out]

-1/12*(d - 2*e + 4*f)*log(x + 2) + 1/6*(d - e + f)*log(x + 1) - 1/6*(d + e + f)*
log(x - 1) + 1/12*(d + 2*e + 4*f)*log(x - 2)

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Sympy [A]  time = 88.8178, size = 2195, normalized size = 43.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d - 2*e + 4*f)*log(x + (-35*d**5*e + 51*d**5*(d - 2*e + 4*f)/2 - 820*d**4*e*f
+ 90*d**4*f*(d - 2*e + 4*f) - 180*d**3*e**3 - 90*d**3*e**2*(d - 2*e + 4*f) - 410
0*d**3*e*f**2 + 41*d**3*e*(d - 2*e + 4*f)**2 + 42*d**3*f**2*(d - 2*e + 4*f) - 15
*d**3*(d - 2*e + 4*f)**3/2 - 432*d**2*e**2*f*(d - 2*e + 4*f) - 8000*d**2*e*f**3
+ 240*d**2*e*f*(d - 2*e + 4*f)**2 - 240*d**2*f**3*(d - 2*e + 4*f) - 12*d**2*f*(d
- 2*e + 4*f)**3 + 320*d*e**5 - 96*d*e**4*(d - 2*e + 4*f) + 720*d*e**3*f**2 - 80
*d*e**3*(d - 2*e + 4*f)**2 - 1080*d*e**2*f**2*(d - 2*e + 4*f) + 24*d*e**2*(d - 2
*e + 4*f)**3 - 6400*d*e*f**4 + 492*d*e*f**2*(d - 2*e + 4*f)**2 - 576*d*f**4*(d -
2*e + 4*f) + 30*d*f**2*(d - 2*e + 4*f)**3 + 512*e**5*f - 128*e**3*f*(d - 2*e +
4*f)**2 - 576*e**2*f**3*(d - 2*e + 4*f) - 1472*e*f**5 + 320*e*f**3*(d - 2*e + 4*
f)**2 - 480*f**5*(d - 2*e + 4*f) + 48*f**3*(d - 2*e + 4*f)**3)/(9*d**6 + 45*d**5
*f - 160*d**4*e**2 - 36*d**4*f**2 - 1312*d**3*e**2*f - 360*d**3*f**3 + 256*d**2*
e**4 - 3840*d**2*e**2*f**2 - 144*d**2*f**4 + 1280*d*e**4*f - 5248*d*e**2*f**3 +
720*d*f**5 + 1024*e**4*f**2 - 2560*e**2*f**4 + 576*f**6))/12 + (d - e + f)*log(x
+ (-35*d**5*e - 51*d**5*(d - e + f) - 820*d**4*e*f - 180*d**4*f*(d - e + f) - 1
80*d**3*e**3 + 180*d**3*e**2*(d - e + f) - 4100*d**3*e*f**2 + 164*d**3*e*(d - e
+ f)**2 - 84*d**3*f**2*(d - e + f) + 60*d**3*(d - e + f)**3 + 864*d**2*e**2*f*(d
- e + f) - 8000*d**2*e*f**3 + 960*d**2*e*f*(d - e + f)**2 + 480*d**2*f**3*(d -
e + f) + 96*d**2*f*(d - e + f)**3 + 320*d*e**5 + 192*d*e**4*(d - e + f) + 720*d*
e**3*f**2 - 320*d*e**3*(d - e + f)**2 + 2160*d*e**2*f**2*(d - e + f) - 192*d*e**
2*(d - e + f)**3 - 6400*d*e*f**4 + 1968*d*e*f**2*(d - e + f)**2 + 1152*d*f**4*(d
- e + f) - 240*d*f**2*(d - e + f)**3 + 512*e**5*f - 512*e**3*f*(d - e + f)**2 +
1152*e**2*f**3*(d - e + f) - 1472*e*f**5 + 1280*e*f**3*(d - e + f)**2 + 960*f**
5*(d - e + f) - 384*f**3*(d - e + f)**3)/(9*d**6 + 45*d**5*f - 160*d**4*e**2 - 3
6*d**4*f**2 - 1312*d**3*e**2*f - 360*d**3*f**3 + 256*d**2*e**4 - 3840*d**2*e**2*
f**2 - 144*d**2*f**4 + 1280*d*e**4*f - 5248*d*e**2*f**3 + 720*d*f**5 + 1024*e**4
*f**2 - 2560*e**2*f**4 + 576*f**6))/6 - (d + e + f)*log(x + (-35*d**5*e + 51*d**
5*(d + e + f) - 820*d**4*e*f + 180*d**4*f*(d + e + f) - 180*d**3*e**3 - 180*d**3
*e**2*(d + e + f) - 4100*d**3*e*f**2 + 164*d**3*e*(d + e + f)**2 + 84*d**3*f**2*
(d + e + f) - 60*d**3*(d + e + f)**3 - 864*d**2*e**2*f*(d + e + f) - 8000*d**2*e
*f**3 + 960*d**2*e*f*(d + e + f)**2 - 480*d**2*f**3*(d + e + f) - 96*d**2*f*(d +
e + f)**3 + 320*d*e**5 - 192*d*e**4*(d + e + f) + 720*d*e**3*f**2 - 320*d*e**3*
(d + e + f)**2 - 2160*d*e**2*f**2*(d + e + f) + 192*d*e**2*(d + e + f)**3 - 6400
*d*e*f**4 + 1968*d*e*f**2*(d + e + f)**2 - 1152*d*f**4*(d + e + f) + 240*d*f**2*
(d + e + f)**3 + 512*e**5*f - 512*e**3*f*(d + e + f)**2 - 1152*e**2*f**3*(d + e
+ f) - 1472*e*f**5 + 1280*e*f**3*(d + e + f)**2 - 960*f**5*(d + e + f) + 384*f**
3*(d + e + f)**3)/(9*d**6 + 45*d**5*f - 160*d**4*e**2 - 36*d**4*f**2 - 1312*d**3
*e**2*f - 360*d**3*f**3 + 256*d**2*e**4 - 3840*d**2*e**2*f**2 - 144*d**2*f**4 +
1280*d*e**4*f - 5248*d*e**2*f**3 + 720*d*f**5 + 1024*e**4*f**2 - 2560*e**2*f**4
+ 576*f**6))/6 + (d + 2*e + 4*f)*log(x + (-35*d**5*e - 51*d**5*(d + 2*e + 4*f)/2
- 820*d**4*e*f - 90*d**4*f*(d + 2*e + 4*f) - 180*d**3*e**3 + 90*d**3*e**2*(d +
2*e + 4*f) - 4100*d**3*e*f**2 + 41*d**3*e*(d + 2*e + 4*f)**2 - 42*d**3*f**2*(d +
2*e + 4*f) + 15*d**3*(d + 2*e + 4*f)**3/2 + 432*d**2*e**2*f*(d + 2*e + 4*f) - 8
000*d**2*e*f**3 + 240*d**2*e*f*(d + 2*e + 4*f)**2 + 240*d**2*f**3*(d + 2*e + 4*f
) + 12*d**2*f*(d + 2*e + 4*f)**3 + 320*d*e**5 + 96*d*e**4*(d + 2*e + 4*f) + 720*
d*e**3*f**2 - 80*d*e**3*(d + 2*e + 4*f)**2 + 1080*d*e**2*f**2*(d + 2*e + 4*f) -
24*d*e**2*(d + 2*e + 4*f)**3 - 6400*d*e*f**4 + 492*d*e*f**2*(d + 2*e + 4*f)**2 +
576*d*f**4*(d + 2*e + 4*f) - 30*d*f**2*(d + 2*e + 4*f)**3 + 512*e**5*f - 128*e*
*3*f*(d + 2*e + 4*f)**2 + 576*e**2*f**3*(d + 2*e + 4*f) - 1472*e*f**5 + 320*e*f*
*3*(d + 2*e + 4*f)**2 + 480*f**5*(d + 2*e + 4*f) - 48*f**3*(d + 2*e + 4*f)**3)/(
9*d**6 + 45*d**5*f - 160*d**4*e**2 - 36*d**4*f**2 - 1312*d**3*e**2*f - 360*d**3*
f**3 + 256*d**2*e**4 - 3840*d**2*e**2*f**2 - 144*d**2*f**4 + 1280*d*e**4*f - 524
8*d*e**2*f**3 + 720*d*f**5 + 1024*e**4*f**2 - 2560*e**2*f**4 + 576*f**6))/12

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GIAC/XCAS [A]  time = 0.290333, size = 80, normalized size = 1.57 \[ -\frac{1}{12} \,{\left (d + 4 \, f - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\left (d + f + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{12} \,{\left (d + 4 \, f + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="giac")

[Out]

-1/12*(d + 4*f - 2*e)*ln(abs(x + 2)) + 1/6*(d + f - e)*ln(abs(x + 1)) - 1/6*(d +
f + e)*ln(abs(x - 1)) + 1/12*(d + 4*f + 2*e)*ln(abs(x - 2))