3.117 \(\int \frac{x}{\left (1+x^2\right ) \left (2+x^2\right ) \left (3+x^2\right ) \left (4+x^2\right )} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]

[Out]

Log[1 + x^2]/12 - Log[2 + x^2]/4 + Log[3 + x^2]/4 - Log[4 + x^2]/12

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Rubi [A]  time = 0.529343, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]

[Out]

Log[1 + x^2]/12 - Log[2 + x^2]/4 + Log[3 + x^2]/4 - Log[4 + x^2]/12

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Rubi in Sympy [A]  time = 50.5609, size = 32, normalized size = 0.78 \[ \frac{\log{\left (x^{2} + 1 \right )}}{12} - \frac{\log{\left (x^{2} + 2 \right )}}{4} + \frac{\log{\left (x^{2} + 3 \right )}}{4} - \frac{\log{\left (x^{2} + 4 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**2+1)/(x**2+2)/(x**2+3)/(x**2+4),x)

[Out]

log(x**2 + 1)/12 - log(x**2 + 2)/4 + log(x**2 + 3)/4 - log(x**2 + 4)/12

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Mathematica [A]  time = 0.0132908, size = 41, normalized size = 1. \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]

[Out]

Log[1 + x^2]/12 - Log[2 + x^2]/4 + Log[3 + x^2]/4 - Log[4 + x^2]/12

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Maple [A]  time = 0.016, size = 34, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+1 \right ) }{12}}-{\frac{\ln \left ({x}^{2}+2 \right ) }{4}}+{\frac{\ln \left ({x}^{2}+3 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+4 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x)

[Out]

1/12*ln(x^2+1)-1/4*ln(x^2+2)+1/4*ln(x^2+3)-1/12*ln(x^2+4)

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Maxima [A]  time = 1.43785, size = 45, normalized size = 1.1 \[ -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^2 + 4)*(x^2 + 3)*(x^2 + 2)*(x^2 + 1)),x, algorithm="maxima")

[Out]

-1/12*log(x^2 + 4) + 1/4*log(x^2 + 3) - 1/4*log(x^2 + 2) + 1/12*log(x^2 + 1)

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Fricas [A]  time = 0.233671, size = 45, normalized size = 1.1 \[ -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^2 + 4)*(x^2 + 3)*(x^2 + 2)*(x^2 + 1)),x, algorithm="fricas")

[Out]

-1/12*log(x^2 + 4) + 1/4*log(x^2 + 3) - 1/4*log(x^2 + 2) + 1/12*log(x^2 + 1)

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Sympy [A]  time = 0.257805, size = 32, normalized size = 0.78 \[ \frac{\log{\left (x^{2} + 1 \right )}}{12} - \frac{\log{\left (x^{2} + 2 \right )}}{4} + \frac{\log{\left (x^{2} + 3 \right )}}{4} - \frac{\log{\left (x^{2} + 4 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**2+1)/(x**2+2)/(x**2+3)/(x**2+4),x)

[Out]

log(x**2 + 1)/12 - log(x**2 + 2)/4 + log(x**2 + 3)/4 - log(x**2 + 4)/12

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GIAC/XCAS [A]  time = 0.217703, size = 45, normalized size = 1.1 \[ -\frac{1}{12} \,{\rm ln}\left (x^{2} + 4\right ) + \frac{1}{4} \,{\rm ln}\left (x^{2} + 3\right ) - \frac{1}{4} \,{\rm ln}\left (x^{2} + 2\right ) + \frac{1}{12} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^2 + 4)*(x^2 + 3)*(x^2 + 2)*(x^2 + 1)),x, algorithm="giac")

[Out]

-1/12*ln(x^2 + 4) + 1/4*ln(x^2 + 3) - 1/4*ln(x^2 + 2) + 1/12*ln(x^2 + 1)