3.377 \(\int \frac{1}{\left (1+x^2\right ) \left (3+\frac{10 x}{1+x^2}\right )} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]

[Out]

-Log[3 + x]/8 + Log[1 + 3*x]/8

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Rubi [A]  time = 0.109778, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x^2)*(3 + (10*x)/(1 + x^2))),x]

[Out]

-Log[3 + x]/8 + Log[1 + 3*x]/8

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Rubi in Sympy [A]  time = 14.7092, size = 14, normalized size = 0.74 \[ - \frac{\log{\left (x + 3 \right )}}{8} + \frac{\log{\left (3 x + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+1)/(3+10*x/(x**2+1)),x)

[Out]

-log(x + 3)/8 + log(3*x + 1)/8

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Mathematica [A]  time = 0.00511877, size = 19, normalized size = 1. \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 + x^2)*(3 + (10*x)/(1 + x^2))),x]

[Out]

-Log[3 + x]/8 + Log[1 + 3*x]/8

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Maple [A]  time = 0.008, size = 16, normalized size = 0.8 \[ -{\frac{\ln \left ( 3+x \right ) }{8}}+{\frac{\ln \left ( 1+3\,x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)/(3+10*x/(x^2+1)),x)

[Out]

-1/8*ln(3+x)+1/8*ln(1+3*x)

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Maxima [A]  time = 0.804295, size = 20, normalized size = 1.05 \[ \frac{1}{8} \, \log \left (3 \, x + 1\right ) - \frac{1}{8} \, \log \left (x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*(10*x/(x^2 + 1) + 3)),x, algorithm="maxima")

[Out]

1/8*log(3*x + 1) - 1/8*log(x + 3)

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Fricas [A]  time = 0.275956, size = 20, normalized size = 1.05 \[ \frac{1}{8} \, \log \left (3 \, x + 1\right ) - \frac{1}{8} \, \log \left (x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*(10*x/(x^2 + 1) + 3)),x, algorithm="fricas")

[Out]

1/8*log(3*x + 1) - 1/8*log(x + 3)

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Sympy [A]  time = 0.209658, size = 14, normalized size = 0.74 \[ \frac{\log{\left (x + \frac{1}{3} \right )}}{8} - \frac{\log{\left (x + 3 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+1)/(3+10*x/(x**2+1)),x)

[Out]

log(x + 1/3)/8 - log(x + 3)/8

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GIAC/XCAS [A]  time = 0.261683, size = 23, normalized size = 1.21 \[ \frac{1}{8} \,{\rm ln}\left ({\left | 3 \, x + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*(10*x/(x^2 + 1) + 3)),x, algorithm="giac")

[Out]

1/8*ln(abs(3*x + 1)) - 1/8*ln(abs(x + 3))