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

Optimal. Leaf size=46 \[ -\frac{7}{100} \log \left (x^2+1\right )+\frac{2}{5 (2 x+1)}-\frac{1}{2} \log (x+1)+\frac{16}{25} \log (2 x+1)+\frac{1}{50} \tan ^{-1}(x) \]

[Out]

2/(5*(1 + 2*x)) + ArcTan[x]/50 - Log[1 + x]/2 + (16*Log[1 + 2*x])/25 - (7*Log[1
+ x^2])/100

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Rubi [A]  time = 0.353672, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{7}{100} \log \left (x^2+1\right )+\frac{2}{5 (2 x+1)}-\frac{1}{2} \log (x+1)+\frac{16}{25} \log (2 x+1)+\frac{1}{50} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

2/(5*(1 + 2*x)) + ArcTan[x]/50 - Log[1 + x]/2 + (16*Log[1 + 2*x])/25 - (7*Log[1
+ x^2])/100

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Rubi in Sympy [A]  time = 28.029, size = 37, normalized size = 0.8 \[ - \frac{\log{\left (x + 1 \right )}}{2} + \frac{16 \log{\left (2 x + 1 \right )}}{25} - \frac{7 \log{\left (x^{2} + 1 \right )}}{100} + \frac{\operatorname{atan}{\left (x \right )}}{50} + \frac{2}{5 \left (2 x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 1)/2 + 16*log(2*x + 1)/25 - 7*log(x**2 + 1)/100 + atan(x)/50 + 2/(5*(2*
x + 1))

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Mathematica [A]  time = 0.0314044, size = 40, normalized size = 0.87 \[ \frac{1}{100} \left (-7 \log \left (x^2+1\right )+\frac{40}{2 x+1}-50 \log (x+1)+64 \log (2 x+1)+2 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(40/(1 + 2*x) + 2*ArcTan[x] - 50*Log[1 + x] + 64*Log[1 + 2*x] - 7*Log[1 + x^2])/
100

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Maple [A]  time = 0.012, size = 37, normalized size = 0.8 \[{\frac{2}{5+10\,x}}+{\frac{\arctan \left ( x \right ) }{50}}-{\frac{\ln \left ( 1+x \right ) }{2}}+{\frac{16\,\ln \left ( 1+2\,x \right ) }{25}}-{\frac{7\,\ln \left ({x}^{2}+1 \right ) }{100}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/(1+2*x)+1/50*arctan(x)-1/2*ln(1+x)+16/25*ln(1+2*x)-7/100*ln(x^2+1)

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Maxima [A]  time = 1.52773, size = 49, normalized size = 1.07 \[ \frac{2}{5 \,{\left (2 \, x + 1\right )}} + \frac{1}{50} \, \arctan \left (x\right ) - \frac{7}{100} \, \log \left (x^{2} + 1\right ) + \frac{16}{25} \, \log \left (2 \, x + 1\right ) - \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/(2*x + 1) + 1/50*arctan(x) - 7/100*log(x^2 + 1) + 16/25*log(2*x + 1) - 1/2*l
og(x + 1)

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Fricas [A]  time = 0.215212, size = 77, normalized size = 1.67 \[ \frac{2 \,{\left (2 \, x + 1\right )} \arctan \left (x\right ) - 7 \,{\left (2 \, x + 1\right )} \log \left (x^{2} + 1\right ) + 64 \,{\left (2 \, x + 1\right )} \log \left (2 \, x + 1\right ) - 50 \,{\left (2 \, x + 1\right )} \log \left (x + 1\right ) + 40}{100 \,{\left (2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/100*(2*(2*x + 1)*arctan(x) - 7*(2*x + 1)*log(x^2 + 1) + 64*(2*x + 1)*log(2*x +
1) - 50*(2*x + 1)*log(x + 1) + 40)/(2*x + 1)

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Sympy [A]  time = 0.300442, size = 37, normalized size = 0.8 \[ \frac{16 \log{\left (x + \frac{1}{2} \right )}}{25} - \frac{\log{\left (x + 1 \right )}}{2} - \frac{7 \log{\left (x^{2} + 1 \right )}}{100} + \frac{\operatorname{atan}{\left (x \right )}}{50} + \frac{2}{10 x + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*log(x + 1/2)/25 - log(x + 1)/2 - 7*log(x**2 + 1)/100 + atan(x)/50 + 2/(10*x +
5)

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GIAC/XCAS [A]  time = 0.201311, size = 84, normalized size = 1.83 \[ \frac{2}{5 \,{\left (2 \, x + 1\right )}} + \frac{1}{50} \, \arctan \left (-\frac{5}{2 \,{\left (2 \, x + 1\right )}} + \frac{1}{2}\right ) - \frac{7}{100} \,{\rm ln}\left (-\frac{2}{2 \, x + 1} + \frac{5}{{\left (2 \, x + 1\right )}^{2}} + 1\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | -\frac{1}{2 \, x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/(2*x + 1) + 1/50*arctan(-5/2/(2*x + 1) + 1/2) - 7/100*ln(-2/(2*x + 1) + 5/(2
*x + 1)^2 + 1) - 1/2*ln(abs(-1/(2*x + 1) - 1))