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

Optimal. Leaf size=115 \[ \frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{27} e \log \left (1-x^2\right )-\frac{1}{27} e \log \left (4-x^2\right )+\frac{e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )} \]

[Out]

(e*(5 - 2*x^2))/(18*(4 - 5*x^2 + x^4)) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(72
*(4 - 5*x^2 + x^4)) + ((19*d + 52*f)*ArcTanh[x/2])/432 - ((d + 7*f)*ArcTanh[x])/
54 + (e*Log[1 - x^2])/27 - (e*Log[4 - x^2])/27

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Rubi [A]  time = 0.285621, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{27} e \log \left (1-x^2\right )-\frac{1}{27} e \log \left (4-x^2\right )+\frac{e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )} \]

Antiderivative was successfully verified.

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

[Out]

(e*(5 - 2*x^2))/(18*(4 - 5*x^2 + x^4)) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(72
*(4 - 5*x^2 + x^4)) + ((19*d + 52*f)*ArcTanh[x/2])/432 - ((d + 7*f)*ArcTanh[x])/
54 + (e*Log[1 - x^2])/27 - (e*Log[4 - x^2])/27

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Rubi in Sympy [A]  time = 40.634, size = 88, normalized size = 0.77 \[ \frac{e \log{\left (- x^{2} + 1 \right )}}{27} - \frac{e \log{\left (- x^{2} + 4 \right )}}{27} + \frac{x \left (17 d - 5 e x^{3} + 17 e x + 20 f - x^{2} \left (5 d + 8 f\right )\right )}{72 \left (x^{4} - 5 x^{2} + 4\right )} - \left (\frac{d}{54} + \frac{7 f}{54}\right ) \operatorname{atanh}{\left (x \right )} + \left (\frac{19 d}{432} + \frac{13 f}{108}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} \]

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)**2,x)

[Out]

e*log(-x**2 + 1)/27 - e*log(-x**2 + 4)/27 + x*(17*d - 5*e*x**3 + 17*e*x + 20*f -
 x**2*(5*d + 8*f))/(72*(x**4 - 5*x**2 + 4)) - (d/54 + 7*f/54)*atanh(x) + (19*d/4
32 + 13*f/108)*atanh(x/2)

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Mathematica [A]  time = 0.219271, size = 112, normalized size = 0.97 \[ \frac{1}{864} \left (\frac{12 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x\right )}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f)-\log (2-x) (19 d+32 e+52 f)-8 \log (x+1) (d-4 e+7 f)+\log (x+2) (19 d-32 e+52 f)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2)/(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 - 5*x^2 + x^4) +
 8*(d + 4*e + 7*f)*Log[1 - x] - (19*d + 32*e + 52*f)*Log[2 - x] - 8*(d - 4*e + 7
*f)*Log[1 + x] + (19*d - 32*e + 52*f)*Log[2 + x])/864

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Maple [A]  time = 0.026, size = 182, normalized size = 1.6 \[ -{\frac{d}{288+144\,x}}+{\frac{e}{144+72\,x}}-{\frac{f}{72+36\,x}}+{\frac{19\,\ln \left ( 2+x \right ) d}{864}}-{\frac{\ln \left ( 2+x \right ) e}{27}}+{\frac{13\,\ln \left ( 2+x \right ) f}{216}}-{\frac{d}{-36+36\,x}}-{\frac{e}{-36+36\,x}}-{\frac{f}{-36+36\,x}}+{\frac{\ln \left ( -1+x \right ) d}{108}}+{\frac{\ln \left ( -1+x \right ) e}{27}}+{\frac{7\,\ln \left ( -1+x \right ) f}{108}}-{\frac{\ln \left ( 1+x \right ) d}{108}}+{\frac{\ln \left ( 1+x \right ) e}{27}}-{\frac{7\,\ln \left ( 1+x \right ) f}{108}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{f}{36+36\,x}}-{\frac{19\,\ln \left ( x-2 \right ) d}{864}}-{\frac{\ln \left ( x-2 \right ) e}{27}}-{\frac{13\,\ln \left ( x-2 \right ) f}{216}}-{\frac{d}{144\,x-288}}-{\frac{e}{72\,x-144}}-{\frac{f}{36\,x-72}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/144/(2+x)*d+1/72/(2+x)*e-1/36/(2+x)*f+19/864*ln(2+x)*d-1/27*ln(2+x)*e+13/216*
ln(2+x)*f-1/36/(-1+x)*d-1/36/(-1+x)*e-1/36/(-1+x)*f+1/108*ln(-1+x)*d+1/27*ln(-1+
x)*e+7/108*ln(-1+x)*f-1/108*ln(1+x)*d+1/27*ln(1+x)*e-7/108*ln(1+x)*f-1/36/(1+x)*
d+1/36/(1+x)*e-1/36/(1+x)*f-19/864*ln(x-2)*d-1/27*ln(x-2)*e-13/216*ln(x-2)*f-1/1
44/(x-2)*d-1/72/(x-2)*e-1/36/(x-2)*f

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Maxima [A]  time = 0.704709, size = 143, normalized size = 1.24 \[ \frac{1}{864} \,{\left (19 \, d - 32 \, e + 52 \, f\right )} \log \left (x + 2\right ) - \frac{1}{108} \,{\left (d - 4 \, e + 7 \, f\right )} \log \left (x + 1\right ) + \frac{1}{108} \,{\left (d + 4 \, e + 7 \, f\right )} \log \left (x - 1\right ) - \frac{1}{864} \,{\left (19 \, d + 32 \, e + 52 \, f\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 8 \, f\right )} x^{3} + 8 \, e x^{2} -{\left (17 \, d + 20 \, f\right )} x - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.330101, size = 293, normalized size = 2.55 \[ -\frac{12 \,{\left (5 \, d + 8 \, f\right )} x^{3} + 96 \, e x^{2} - 12 \,{\left (17 \, d + 20 \, f\right )} x -{\left ({\left (19 \, d - 32 \, e + 52 \, f\right )} x^{4} - 5 \,{\left (19 \, d - 32 \, e + 52 \, f\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f\right )} \log \left (x + 2\right ) + 8 \,{\left ({\left (d - 4 \, e + 7 \, f\right )} x^{4} - 5 \,{\left (d - 4 \, e + 7 \, f\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f\right )} \log \left (x + 1\right ) - 8 \,{\left ({\left (d + 4 \, e + 7 \, f\right )} x^{4} - 5 \,{\left (d + 4 \, e + 7 \, f\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f\right )} \log \left (x - 1\right ) +{\left ({\left (19 \, d + 32 \, e + 52 \, f\right )} x^{4} - 5 \,{\left (19 \, d + 32 \, e + 52 \, f\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f\right )} \log \left (x - 2\right ) - 240 \, e}{864 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/864*(12*(5*d + 8*f)*x^3 + 96*e*x^2 - 12*(17*d + 20*f)*x - ((19*d - 32*e + 52*
f)*x^4 - 5*(19*d - 32*e + 52*f)*x^2 + 76*d - 128*e + 208*f)*log(x + 2) + 8*((d -
 4*e + 7*f)*x^4 - 5*(d - 4*e + 7*f)*x^2 + 4*d - 16*e + 28*f)*log(x + 1) - 8*((d
+ 4*e + 7*f)*x^4 - 5*(d + 4*e + 7*f)*x^2 + 4*d + 16*e + 28*f)*log(x - 1) + ((19*
d + 32*e + 52*f)*x^4 - 5*(19*d + 32*e + 52*f)*x^2 + 76*d + 128*e + 208*f)*log(x
- 2) - 240*e)/(x^4 - 5*x^2 + 4)

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Sympy [A]  time = 109.052, size = 2689, normalized size = 23.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d - 4*e + 7*f)*log(x + (-6006260*d**5*e + 2341251*d**5*(d - 4*e + 7*f) - 24601
6240*d**4*e*f + 31626180*d**4*f*(d - 4*e + 7*f) - 18247680*d**3*e**3 + 24099840*
d**3*e**2*(d - 4*e + 7*f) - 2758371200*d**3*e*f**2 + 7387904*d**3*e*(d - 4*e + 7
*f)**2 + 171122976*d**3*f**2*(d - 4*e + 7*f) - 665280*d**3*(d - 4*e + 7*f)**3 +
298598400*d**2*e**3*f + 369487872*d**2*e**2*f*(d - 4*e + 7*f) - 13192256000*d**2
*e*f**3 + 90885120*d**2*e*f*(d - 4*e + 7*f)**2 + 441486720*d**2*f**3*(d - 4*e +
7*f) - 5536512*d**2*f*(d - 4*e + 7*f)**3 + 587202560*d*e**5 - 12582912*d*e**4*(d
 - 4*e + 7*f) + 1353646080*d*e**3*f**2 - 36700160*d*e**3*(d - 4*e + 7*f)**2 + 14
48755200*d*e**2*f**2*(d - 4*e + 7*f) + 786432*d*e**2*(d - 4*e + 7*f)**3 - 282823
93600*d*e*f**4 + 362729472*d*e*f**2*(d - 4*e + 7*f)**2 + 399575808*d*f**4*(d - 4
*e + 7*f) - 10368000*d*f**2*(d - 4*e + 7*f)**3 + 2751463424*e**5*f + 251658240*e
**4*f*(d - 4*e + 7*f) - 530841600*e**3*f**3 - 171966464*e**3*f*(d - 4*e + 7*f)**
2 + 1935212544*e**2*f**3*(d - 4*e + 7*f) - 15728640*e**2*f*(d - 4*e + 7*f)**3 -
21886889984*e*f**5 + 483737600*e*f**3*(d - 4*e + 7*f)**2 - 212474880*f**5*(d - 4
*e + 7*f) + 4534272*f**3*(d - 4*e + 7*f)**3)/(1675971*d**6 + 28507545*d**5*f - 6
6150400*d**4*e**2 + 168075324*d**4*f**2 - 1091117056*d**3*e**2*f + 384095520*d**
3*f**3 + 318767104*d**2*e**4 - 6528860160*d**2*e**2*f**2 + 162082944*d**2*f**4 +
 3103784960*d*e**4*f - 17414619136*d*e**2*f**3 - 305130240*d*f**5 + 6106906624*e
**4*f**2 - 17414225920*e**2*f**4 + 67931136*f**6))/108 + (d + 4*e + 7*f)*log(x +
 (-6006260*d**5*e - 2341251*d**5*(d + 4*e + 7*f) - 246016240*d**4*e*f - 31626180
*d**4*f*(d + 4*e + 7*f) - 18247680*d**3*e**3 - 24099840*d**3*e**2*(d + 4*e + 7*f
) - 2758371200*d**3*e*f**2 + 7387904*d**3*e*(d + 4*e + 7*f)**2 - 171122976*d**3*
f**2*(d + 4*e + 7*f) + 665280*d**3*(d + 4*e + 7*f)**3 + 298598400*d**2*e**3*f -
369487872*d**2*e**2*f*(d + 4*e + 7*f) - 13192256000*d**2*e*f**3 + 90885120*d**2*
e*f*(d + 4*e + 7*f)**2 - 441486720*d**2*f**3*(d + 4*e + 7*f) + 5536512*d**2*f*(d
 + 4*e + 7*f)**3 + 587202560*d*e**5 + 12582912*d*e**4*(d + 4*e + 7*f) + 13536460
80*d*e**3*f**2 - 36700160*d*e**3*(d + 4*e + 7*f)**2 - 1448755200*d*e**2*f**2*(d
+ 4*e + 7*f) - 786432*d*e**2*(d + 4*e + 7*f)**3 - 28282393600*d*e*f**4 + 3627294
72*d*e*f**2*(d + 4*e + 7*f)**2 - 399575808*d*f**4*(d + 4*e + 7*f) + 10368000*d*f
**2*(d + 4*e + 7*f)**3 + 2751463424*e**5*f - 251658240*e**4*f*(d + 4*e + 7*f) -
530841600*e**3*f**3 - 171966464*e**3*f*(d + 4*e + 7*f)**2 - 1935212544*e**2*f**3
*(d + 4*e + 7*f) + 15728640*e**2*f*(d + 4*e + 7*f)**3 - 21886889984*e*f**5 + 483
737600*e*f**3*(d + 4*e + 7*f)**2 + 212474880*f**5*(d + 4*e + 7*f) - 4534272*f**3
*(d + 4*e + 7*f)**3)/(1675971*d**6 + 28507545*d**5*f - 66150400*d**4*e**2 + 1680
75324*d**4*f**2 - 1091117056*d**3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*
e**4 - 6528860160*d**2*e**2*f**2 + 162082944*d**2*f**4 + 3103784960*d*e**4*f - 1
7414619136*d*e**2*f**3 - 305130240*d*f**5 + 6106906624*e**4*f**2 - 17414225920*e
**2*f**4 + 67931136*f**6))/108 + (19*d - 32*e + 52*f)*log(x + (-6006260*d**5*e -
 2341251*d**5*(19*d - 32*e + 52*f)/8 - 246016240*d**4*e*f - 7906545*d**4*f*(19*d
 - 32*e + 52*f)/2 - 18247680*d**3*e**3 - 3012480*d**3*e**2*(19*d - 32*e + 52*f)
- 2758371200*d**3*e*f**2 + 115436*d**3*e*(19*d - 32*e + 52*f)**2 - 21390372*d**3
*f**2*(19*d - 32*e + 52*f) + 10395*d**3*(19*d - 32*e + 52*f)**3/8 + 298598400*d*
*2*e**3*f - 46185984*d**2*e**2*f*(19*d - 32*e + 52*f) - 13192256000*d**2*e*f**3
+ 1420080*d**2*e*f*(19*d - 32*e + 52*f)**2 - 55185840*d**2*f**3*(19*d - 32*e + 5
2*f) + 21627*d**2*f*(19*d - 32*e + 52*f)**3/2 + 587202560*d*e**5 + 1572864*d*e**
4*(19*d - 32*e + 52*f) + 1353646080*d*e**3*f**2 - 573440*d*e**3*(19*d - 32*e + 5
2*f)**2 - 181094400*d*e**2*f**2*(19*d - 32*e + 52*f) - 1536*d*e**2*(19*d - 32*e
+ 52*f)**3 - 28282393600*d*e*f**4 + 5667648*d*e*f**2*(19*d - 32*e + 52*f)**2 - 4
9946976*d*f**4*(19*d - 32*e + 52*f) + 20250*d*f**2*(19*d - 32*e + 52*f)**3 + 275
1463424*e**5*f - 31457280*e**4*f*(19*d - 32*e + 52*f) - 530841600*e**3*f**3 - 26
86976*e**3*f*(19*d - 32*e + 52*f)**2 - 241901568*e**2*f**3*(19*d - 32*e + 52*f)
+ 30720*e**2*f*(19*d - 32*e + 52*f)**3 - 21886889984*e*f**5 + 7558400*e*f**3*(19
*d - 32*e + 52*f)**2 + 26559360*f**5*(19*d - 32*e + 52*f) - 8856*f**3*(19*d - 32
*e + 52*f)**3)/(1675971*d**6 + 28507545*d**5*f - 66150400*d**4*e**2 + 168075324*
d**4*f**2 - 1091117056*d**3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*e**4 -
 6528860160*d**2*e**2*f**2 + 162082944*d**2*f**4 + 3103784960*d*e**4*f - 1741461
9136*d*e**2*f**3 - 305130240*d*f**5 + 6106906624*e**4*f**2 - 17414225920*e**2*f*
*4 + 67931136*f**6))/864 - (19*d + 32*e + 52*f)*log(x + (-6006260*d**5*e + 23412
51*d**5*(19*d + 32*e + 52*f)/8 - 246016240*d**4*e*f + 7906545*d**4*f*(19*d + 32*
e + 52*f)/2 - 18247680*d**3*e**3 + 3012480*d**3*e**2*(19*d + 32*e + 52*f) - 2758
371200*d**3*e*f**2 + 115436*d**3*e*(19*d + 32*e + 52*f)**2 + 21390372*d**3*f**2*
(19*d + 32*e + 52*f) - 10395*d**3*(19*d + 32*e + 52*f)**3/8 + 298598400*d**2*e**
3*f + 46185984*d**2*e**2*f*(19*d + 32*e + 52*f) - 13192256000*d**2*e*f**3 + 1420
080*d**2*e*f*(19*d + 32*e + 52*f)**2 + 55185840*d**2*f**3*(19*d + 32*e + 52*f) -
 21627*d**2*f*(19*d + 32*e + 52*f)**3/2 + 587202560*d*e**5 - 1572864*d*e**4*(19*
d + 32*e + 52*f) + 1353646080*d*e**3*f**2 - 573440*d*e**3*(19*d + 32*e + 52*f)**
2 + 181094400*d*e**2*f**2*(19*d + 32*e + 52*f) + 1536*d*e**2*(19*d + 32*e + 52*f
)**3 - 28282393600*d*e*f**4 + 5667648*d*e*f**2*(19*d + 32*e + 52*f)**2 + 4994697
6*d*f**4*(19*d + 32*e + 52*f) - 20250*d*f**2*(19*d + 32*e + 52*f)**3 + 275146342
4*e**5*f + 31457280*e**4*f*(19*d + 32*e + 52*f) - 530841600*e**3*f**3 - 2686976*
e**3*f*(19*d + 32*e + 52*f)**2 + 241901568*e**2*f**3*(19*d + 32*e + 52*f) - 3072
0*e**2*f*(19*d + 32*e + 52*f)**3 - 21886889984*e*f**5 + 7558400*e*f**3*(19*d + 3
2*e + 52*f)**2 - 26559360*f**5*(19*d + 32*e + 52*f) + 8856*f**3*(19*d + 32*e + 5
2*f)**3)/(1675971*d**6 + 28507545*d**5*f - 66150400*d**4*e**2 + 168075324*d**4*f
**2 - 1091117056*d**3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*e**4 - 65288
60160*d**2*e**2*f**2 + 162082944*d**2*f**4 + 3103784960*d*e**4*f - 17414619136*d
*e**2*f**3 - 305130240*d*f**5 + 6106906624*e**4*f**2 - 17414225920*e**2*f**4 + 6
7931136*f**6))/864 - (8*e*x**2 - 20*e + x**3*(5*d + 8*f) + x*(-17*d - 20*f))/(72
*x**4 - 360*x**2 + 288)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272902, size = 155, normalized size = 1.35 \[ \frac{1}{864} \,{\left (19 \, d + 52 \, f - 32 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{108} \,{\left (d + 7 \, f - 4 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{108} \,{\left (d + 7 \, f + 4 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{864} \,{\left (19 \, d + 52 \, f + 32 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{5 \, d x^{3} + 8 \, f x^{3} + 8 \, x^{2} e - 17 \, d x - 20 \, f x - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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