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

Optimal. Leaf size=45 \[ -\frac{1}{6} d \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} d \tanh ^{-1}(x)-\frac{1}{6} e \log \left (1-x^2\right )+\frac{1}{6} e \log \left (4-x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.06644, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ -\frac{1}{6} d \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} d \tanh ^{-1}(x)-\frac{1}{6} e \log \left (1-x^2\right )+\frac{1}{6} e \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.0075, size = 34, normalized size = 0.76 \[ - \frac{d \operatorname{atanh}{\left (\frac{x}{2} \right )}}{6} + \frac{d \operatorname{atanh}{\left (x \right )}}{3} - \frac{e \log{\left (- x^{2} + 1 \right )}}{6} + \frac{e \log{\left (- x^{2} + 4 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0320466, size = 50, normalized size = 1.11 \[ \frac{1}{12} (-2 (d+e) \log (1-x)+(d+2 e) \log (2-x)+2 (d-e) \log (x+1)-(d-2 e) \log (x+2)) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 58, normalized size = 1.3 \[ -{\frac{\ln \left ( 2+x \right ) d}{12}}+{\frac{\ln \left ( 2+x \right ) e}{6}}-{\frac{\ln \left ( -1+x \right ) d}{6}}-{\frac{\ln \left ( -1+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) d}{12}}+{\frac{\ln \left ( x-2 \right ) e}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(x^4-5*x^2+4),x)

[Out]

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

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Maxima [A]  time = 0.70226, size = 58, normalized size = 1.29 \[ -\frac{1}{12} \,{\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.271535, size = 58, normalized size = 1.29 \[ -\frac{1}{12} \,{\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.81026, size = 515, normalized size = 11.44 \[ - \frac{\left (d - 2 e\right ) \log{\left (x + \frac{- 35 d^{4} e + \frac{51 d^{4} \left (d - 2 e\right )}{2} - 180 d^{2} e^{3} - 90 d^{2} e^{2} \left (d - 2 e\right ) + 41 d^{2} e \left (d - 2 e\right )^{2} - \frac{15 d^{2} \left (d - 2 e\right )^{3}}{2} + 320 e^{5} - 96 e^{4} \left (d - 2 e\right ) - 80 e^{3} \left (d - 2 e\right )^{2} + 24 e^{2} \left (d - 2 e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{12} + \frac{\left (d - e\right ) \log{\left (x + \frac{- 35 d^{4} e - 51 d^{4} \left (d - e\right ) - 180 d^{2} e^{3} + 180 d^{2} e^{2} \left (d - e\right ) + 164 d^{2} e \left (d - e\right )^{2} + 60 d^{2} \left (d - e\right )^{3} + 320 e^{5} + 192 e^{4} \left (d - e\right ) - 320 e^{3} \left (d - e\right )^{2} - 192 e^{2} \left (d - e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{6} - \frac{\left (d + e\right ) \log{\left (x + \frac{- 35 d^{4} e + 51 d^{4} \left (d + e\right ) - 180 d^{2} e^{3} - 180 d^{2} e^{2} \left (d + e\right ) + 164 d^{2} e \left (d + e\right )^{2} - 60 d^{2} \left (d + e\right )^{3} + 320 e^{5} - 192 e^{4} \left (d + e\right ) - 320 e^{3} \left (d + e\right )^{2} + 192 e^{2} \left (d + e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{6} + \frac{\left (d + 2 e\right ) \log{\left (x + \frac{- 35 d^{4} e - \frac{51 d^{4} \left (d + 2 e\right )}{2} - 180 d^{2} e^{3} + 90 d^{2} e^{2} \left (d + 2 e\right ) + 41 d^{2} e \left (d + 2 e\right )^{2} + \frac{15 d^{2} \left (d + 2 e\right )^{3}}{2} + 320 e^{5} + 96 e^{4} \left (d + 2 e\right ) - 80 e^{3} \left (d + 2 e\right )^{2} - 24 e^{2} \left (d + 2 e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d - 2*e)*log(x + (-35*d**4*e + 51*d**4*(d - 2*e)/2 - 180*d**2*e**3 - 90*d**2*e
**2*(d - 2*e) + 41*d**2*e*(d - 2*e)**2 - 15*d**2*(d - 2*e)**3/2 + 320*e**5 - 96*
e**4*(d - 2*e) - 80*e**3*(d - 2*e)**2 + 24*e**2*(d - 2*e)**3)/(9*d**5 - 160*d**3
*e**2 + 256*d*e**4))/12 + (d - e)*log(x + (-35*d**4*e - 51*d**4*(d - e) - 180*d*
*2*e**3 + 180*d**2*e**2*(d - e) + 164*d**2*e*(d - e)**2 + 60*d**2*(d - e)**3 + 3
20*e**5 + 192*e**4*(d - e) - 320*e**3*(d - e)**2 - 192*e**2*(d - e)**3)/(9*d**5
- 160*d**3*e**2 + 256*d*e**4))/6 - (d + e)*log(x + (-35*d**4*e + 51*d**4*(d + e)
- 180*d**2*e**3 - 180*d**2*e**2*(d + e) + 164*d**2*e*(d + e)**2 - 60*d**2*(d +
e)**3 + 320*e**5 - 192*e**4*(d + e) - 320*e**3*(d + e)**2 + 192*e**2*(d + e)**3)
/(9*d**5 - 160*d**3*e**2 + 256*d*e**4))/6 + (d + 2*e)*log(x + (-35*d**4*e - 51*d
**4*(d + 2*e)/2 - 180*d**2*e**3 + 90*d**2*e**2*(d + 2*e) + 41*d**2*e*(d + 2*e)**
2 + 15*d**2*(d + 2*e)**3/2 + 320*e**5 + 96*e**4*(d + 2*e) - 80*e**3*(d + 2*e)**2
- 24*e**2*(d + 2*e)**3)/(9*d**5 - 160*d**3*e**2 + 256*d*e**4))/12

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GIAC/XCAS [A]  time = 0.302092, size = 69, normalized size = 1.53 \[ -\frac{1}{12} \,{\left (d - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\left (d + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{12} \,{\left (d + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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