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

Optimal. Leaf size=204 \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right )-\frac{\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2} \]

[Out]

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (5*e + 8*g - (2*
e + 5*g)*x^2)/(36*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x
^2 + x^4)) - (x*(59*d + 380*f - 35*(d + 4*f)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((
313*d + 820*f)*ArcTanh[x/2])/20736 + ((13*d + 25*f)*ArcTanh[x])/648 - ((2*e + 5*
g)*Log[1 - x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

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Rubi [A]  time = 0.5138, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right )-\frac{\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (5*e + 8*g - (2*
e + 5*g)*x^2)/(36*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x
^2 + x^4)) - (x*(59*d + 380*f - 35*(d + 4*f)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((
313*d + 820*f)*ArcTanh[x/2])/20736 + ((13*d + 25*f)*ArcTanh[x])/648 - ((2*e + 5*
g)*Log[1 - x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

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Rubi in Sympy [A]  time = 75.3119, size = 158, normalized size = 0.77 \[ - \frac{x \left (177 d + 1140 f - x^{3} \left (150 e + 456 g\right ) - x^{2} \left (105 d + 420 f\right ) + x \left (366 e + 1320 g\right )\right )}{10368 \left (x^{4} - 5 x^{2} + 4\right )} + \frac{x \left (17 d + 20 f - x^{3} \left (5 e + 8 g\right ) - x^{2} \left (5 d + 8 f\right ) + x \left (17 e + 20 g\right )\right )}{144 \left (x^{4} - 5 x^{2} + 4\right )^{2}} - \left (\frac{313 d}{20736} + \frac{205 f}{5184}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{13 d}{648} + \frac{25 f}{648}\right ) \operatorname{atanh}{\left (x \right )} - \left (\frac{e}{81} + \frac{5 g}{162}\right ) \log{\left (- x^{2} + 1 \right )} + \left (\frac{e}{81} + \frac{5 g}{162}\right ) \log{\left (- x^{2} + 4 \right )} \]

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)**3,x)

[Out]

-x*(177*d + 1140*f - x**3*(150*e + 456*g) - x**2*(105*d + 420*f) + x*(366*e + 13
20*g))/(10368*(x**4 - 5*x**2 + 4)) + x*(17*d + 20*f - x**3*(5*e + 8*g) - x**2*(5
*d + 8*f) + x*(17*e + 20*g))/(144*(x**4 - 5*x**2 + 4)**2) - (313*d/20736 + 205*f
/5184)*atanh(x/2) + (13*d/648 + 25*f/648)*atanh(x) - (e/81 + 5*g/162)*log(-x**2
+ 1) + (e/81 + 5*g/162)*log(-x**2 + 4)

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Mathematica [A]  time = 0.165335, size = 193, normalized size = 0.95 \[ \frac{\frac{12 \left (d x \left (35 x^2-59\right )+64 e \left (2 x^2-5\right )+20 f x \left (7 x^2-19\right )+160 g \left (2 x^2-5\right )\right )}{x^4-5 x^2+4}+\frac{288 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x-4 g \left (5 x^2-8\right )\right )}{\left (x^4-5 x^2+4\right )^2}-32 \log (1-x) (13 d+16 e+25 f+40 g)+\log (2-x) (313 d+512 e+820 f+1280 g)+32 \log (x+1) (13 d-16 e+25 f-40 g)+\log (x+2) (-313 d+512 e-820 f+1280 g)}{41472} \]

Antiderivative was successfully verified.

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

[Out]

((288*(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)^2 + (12*(64*e*(-5 + 2*x^2) + 160*g*(-5 + 2*x^2) + 20*f*x*(-19
+ 7*x^2) + d*x*(-59 + 35*x^2)))/(4 - 5*x^2 + x^4) - 32*(13*d + 16*e + 25*f + 40
*g)*Log[1 - x] + (313*d + 512*e + 820*f + 1280*g)*Log[2 - x] + 32*(13*d - 16*e +
25*f - 40*g)*Log[1 + x] + (-313*d + 512*e - 820*f + 1280*g)*Log[2 + x])/41472

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Maple [A]  time = 0.031, size = 370, normalized size = 1.8 \[ -{\frac{g}{432\, \left ( x-2 \right ) ^{2}}}+{\frac{g}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{g}{432\, \left ( -1+x \right ) ^{2}}}-{\frac{g}{432\, \left ( 2+x \right ) ^{2}}}+{\frac{d}{432\, \left ( -1+x \right ) ^{2}}}+{\frac{e}{432\, \left ( -1+x \right ) ^{2}}}-{\frac{d}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{e}{432\, \left ( 1+x \right ) ^{2}}}-{\frac{d}{3456\, \left ( x-2 \right ) ^{2}}}-{\frac{e}{1728\, \left ( x-2 \right ) ^{2}}}+{\frac{d}{3456\, \left ( 2+x \right ) ^{2}}}-{\frac{e}{1728\, \left ( 2+x \right ) ^{2}}}-{\frac{f}{432\, \left ( 1+x \right ) ^{2}}}-{\frac{f}{864\, \left ( x-2 \right ) ^{2}}}+{\frac{f}{864\, \left ( 2+x \right ) ^{2}}}+{\frac{f}{432\, \left ( -1+x \right ) ^{2}}}-{\frac{7\,g}{432+432\,x}}+{\frac{7\,g}{-432+432\,x}}-{\frac{13\,g}{1728+864\,x}}+{\frac{13\,g}{864\,x-1728}}+{\frac{5\,f}{432+432\,x}}+{\frac{d}{432+432\,x}}-{\frac{e}{144+144\,x}}+{\frac{19\,d}{6912\,x-13824}}+{\frac{17\,e}{3456\,x-6912}}+{\frac{5\,f}{576\,x-1152}}+{\frac{5\,f}{-432+432\,x}}+{\frac{19\,d}{13824+6912\,x}}-{\frac{17\,e}{6912+3456\,x}}+{\frac{d}{-432+432\,x}}+{\frac{e}{-144+144\,x}}+{\frac{5\,f}{1152+576\,x}}+{\frac{13\,\ln \left ( 1+x \right ) d}{1296}}-{\frac{\ln \left ( 1+x \right ) e}{81}}-{\frac{13\,\ln \left ( -1+x \right ) d}{1296}}-{\frac{\ln \left ( -1+x \right ) e}{81}}-{\frac{5\,\ln \left ( 1+x \right ) g}{162}}+{\frac{5\,\ln \left ( x-2 \right ) g}{162}}-{\frac{5\,\ln \left ( -1+x \right ) g}{162}}+{\frac{5\,\ln \left ( 2+x \right ) g}{162}}+{\frac{313\,\ln \left ( x-2 \right ) d}{41472}}+{\frac{\ln \left ( x-2 \right ) e}{81}}+{\frac{\ln \left ( 2+x \right ) e}{81}}+{\frac{205\,\ln \left ( x-2 \right ) f}{10368}}-{\frac{313\,\ln \left ( 2+x \right ) d}{41472}}+{\frac{25\,\ln \left ( 1+x \right ) f}{1296}}-{\frac{25\,\ln \left ( -1+x \right ) f}{1296}}-{\frac{205\,\ln \left ( 2+x \right ) f}{10368}} \]

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)^3,x)

[Out]

-1/432/(x-2)^2*g+1/432/(1+x)^2*g+1/432/(-1+x)^2*g-1/432/(2+x)^2*g+1/432/(-1+x)^2
*d+1/432/(-1+x)^2*e-1/432/(1+x)^2*d+1/432/(1+x)^2*e-1/3456/(x-2)^2*d-1/1728/(x-2
)^2*e+1/3456/(2+x)^2*d-1/1728/(2+x)^2*e-1/432/(1+x)^2*f-1/864/(x-2)^2*f+1/864/(2
+x)^2*f+1/432/(-1+x)^2*f-7/432/(1+x)*g+7/432/(-1+x)*g-13/864/(2+x)*g+13/864/(x-2
)*g+5/432/(1+x)*f+1/432/(1+x)*d-1/144/(1+x)*e+19/6912/(x-2)*d+17/3456/(x-2)*e+5/
576/(x-2)*f+5/432/(-1+x)*f+19/6912/(2+x)*d-17/3456/(2+x)*e+1/432/(-1+x)*d+1/144/
(-1+x)*e+5/576/(2+x)*f+13/1296*ln(1+x)*d-1/81*ln(1+x)*e-13/1296*ln(-1+x)*d-1/81*
ln(-1+x)*e-5/162*ln(1+x)*g+5/162*ln(x-2)*g-5/162*ln(-1+x)*g+5/162*ln(2+x)*g+313/
41472*ln(x-2)*d+1/81*ln(x-2)*e+1/81*ln(2+x)*e+205/10368*ln(x-2)*f-313/41472*ln(2
+x)*d+25/1296*ln(1+x)*f-25/1296*ln(-1+x)*f-205/10368*ln(2+x)*f

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Maxima [A]  time = 0.710456, size = 254, normalized size = 1.25 \[ -\frac{1}{41472} \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} \log \left (x + 2\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} \log \left (x + 1\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} \log \left (x - 1\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} \log \left (x - 2\right ) + \frac{35 \,{\left (d + 4 \, f\right )} x^{7} + 64 \,{\left (2 \, e + 5 \, g\right )} x^{6} - 18 \,{\left (13 \, d + 60 \, f\right )} x^{5} - 480 \,{\left (2 \, e + 5 \, g\right )} x^{4} + 63 \,{\left (5 \, d + 36 \, f\right )} x^{3} + 960 \,{\left (2 \, e + 5 \, g\right )} x^{2} + 4 \,{\left (43 \, d - 260 \, f\right )} x - 800 \, e - 2432 \, g}{3456 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/41472*(313*d - 512*e + 820*f - 1280*g)*log(x + 2) + 1/1296*(13*d - 16*e + 25*
f - 40*g)*log(x + 1) - 1/1296*(13*d + 16*e + 25*f + 40*g)*log(x - 1) + 1/41472*(
313*d + 512*e + 820*f + 1280*g)*log(x - 2) + 1/3456*(35*(d + 4*f)*x^7 + 64*(2*e
+ 5*g)*x^6 - 18*(13*d + 60*f)*x^5 - 480*(2*e + 5*g)*x^4 + 63*(5*d + 36*f)*x^3 +
960*(2*e + 5*g)*x^2 + 4*(43*d - 260*f)*x - 800*e - 2432*g)/(x^8 - 10*x^6 + 33*x^
4 - 40*x^2 + 16)

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Fricas [A]  time = 0.627975, size = 635, normalized size = 3.11 \[ \frac{420 \,{\left (d + 4 \, f\right )} x^{7} + 768 \,{\left (2 \, e + 5 \, g\right )} x^{6} - 216 \,{\left (13 \, d + 60 \, f\right )} x^{5} - 5760 \,{\left (2 \, e + 5 \, g\right )} x^{4} + 756 \,{\left (5 \, d + 36 \, f\right )} x^{3} + 11520 \,{\left (2 \, e + 5 \, g\right )} x^{2} + 48 \,{\left (43 \, d - 260 \, f\right )} x -{\left ({\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{8} - 10 \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{6} + 33 \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{4} - 40 \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{2} + 5008 \, d - 8192 \, e + 13120 \, f - 20480 \, g\right )} \log \left (x + 2\right ) + 32 \,{\left ({\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{8} - 10 \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{6} + 33 \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{4} - 40 \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{2} + 208 \, d - 256 \, e + 400 \, f - 640 \, g\right )} \log \left (x + 1\right ) - 32 \,{\left ({\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{8} - 10 \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{6} + 33 \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{4} - 40 \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{2} + 208 \, d + 256 \, e + 400 \, f + 640 \, g\right )} \log \left (x - 1\right ) +{\left ({\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{8} - 10 \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{6} + 33 \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{4} - 40 \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{2} + 5008 \, d + 8192 \, e + 13120 \, f + 20480 \, g\right )} \log \left (x - 2\right ) - 9600 \, e - 29184 \, g}{41472 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/41472*(420*(d + 4*f)*x^7 + 768*(2*e + 5*g)*x^6 - 216*(13*d + 60*f)*x^5 - 5760*
(2*e + 5*g)*x^4 + 756*(5*d + 36*f)*x^3 + 11520*(2*e + 5*g)*x^2 + 48*(43*d - 260*
f)*x - ((313*d - 512*e + 820*f - 1280*g)*x^8 - 10*(313*d - 512*e + 820*f - 1280*
g)*x^6 + 33*(313*d - 512*e + 820*f - 1280*g)*x^4 - 40*(313*d - 512*e + 820*f - 1
280*g)*x^2 + 5008*d - 8192*e + 13120*f - 20480*g)*log(x + 2) + 32*((13*d - 16*e
+ 25*f - 40*g)*x^8 - 10*(13*d - 16*e + 25*f - 40*g)*x^6 + 33*(13*d - 16*e + 25*f
- 40*g)*x^4 - 40*(13*d - 16*e + 25*f - 40*g)*x^2 + 208*d - 256*e + 400*f - 640*
g)*log(x + 1) - 32*((13*d + 16*e + 25*f + 40*g)*x^8 - 10*(13*d + 16*e + 25*f + 4
0*g)*x^6 + 33*(13*d + 16*e + 25*f + 40*g)*x^4 - 40*(13*d + 16*e + 25*f + 40*g)*x
^2 + 208*d + 256*e + 400*f + 640*g)*log(x - 1) + ((313*d + 512*e + 820*f + 1280*
g)*x^8 - 10*(313*d + 512*e + 820*f + 1280*g)*x^6 + 33*(313*d + 512*e + 820*f + 1
280*g)*x^4 - 40*(313*d + 512*e + 820*f + 1280*g)*x^2 + 5008*d + 8192*e + 13120*f
+ 20480*g)*log(x - 2) - 9600*e - 29184*g)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265825, size = 257, normalized size = 1.26 \[ -\frac{1}{41472} \,{\left (313 \, d + 820 \, f - 1280 \, g - 512 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d + 25 \, f - 40 \, g - 16 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 25 \, f + 40 \, g + 16 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 820 \, f + 1280 \, g + 512 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) + \frac{35 \, d x^{7} + 140 \, f x^{7} + 320 \, g x^{6} + 128 \, x^{6} e - 234 \, d x^{5} - 1080 \, f x^{5} - 2400 \, g x^{4} - 960 \, x^{4} e + 315 \, d x^{3} + 2268 \, f x^{3} + 4800 \, g x^{2} + 1920 \, x^{2} e + 172 \, d x - 1040 \, f x - 2432 \, g - 800 \, e}{3456 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-1/41472*(313*d + 820*f - 1280*g - 512*e)*ln(abs(x + 2)) + 1/1296*(13*d + 25*f -
40*g - 16*e)*ln(abs(x + 1)) - 1/1296*(13*d + 25*f + 40*g + 16*e)*ln(abs(x - 1))
+ 1/41472*(313*d + 820*f + 1280*g + 512*e)*ln(abs(x - 2)) + 1/3456*(35*d*x^7 +
140*f*x^7 + 320*g*x^6 + 128*x^6*e - 234*d*x^5 - 1080*f*x^5 - 2400*g*x^4 - 960*x^
4*e + 315*d*x^3 + 2268*f*x^3 + 4800*g*x^2 + 1920*x^2*e + 172*d*x - 1040*f*x - 24
32*g - 800*e)/(x^4 - 5*x^2 + 4)^2