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

Optimal. Leaf size=22 \[ -2 \log \left (x^2+1\right )-\frac{4}{x}+4 \log (x)-4 \tan ^{-1}(x) \]

[Out]

-4/x - 4*ArcTan[x] + 4*Log[x] - 2*Log[1 + x^2]

_______________________________________________________________________________________

Rubi [A]  time = 0.0545094, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -2 \log \left (x^2+1\right )-\frac{4}{x}+4 \log (x)-4 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-4/x - 4*ArcTan[x] + 4*Log[x] - 2*Log[1 + x^2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.06004, size = 20, normalized size = 0.91 \[ 4 \log{\left (x \right )} - 2 \log{\left (x^{2} + 1 \right )} - 4 \operatorname{atan}{\left (x \right )} - \frac{4}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*log(x) - 2*log(x**2 + 1) - 4*atan(x) - 4/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.00812693, size = 24, normalized size = 1.09 \[ 4 \left (-\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{x}+\log (x)-\tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*(-x^(-1) - ArcTan[x] + Log[x] - Log[1 + x^2]/2)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 23, normalized size = 1.1 \[ -4\,{x}^{-1}-4\,\arctan \left ( x \right ) +4\,\ln \left ( x \right ) -2\,\ln \left ({x}^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4+4*x)/x^2/(x^2+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.858785, size = 30, normalized size = 1.36 \[ -\frac{4}{x} - 4 \, \arctan \left (x\right ) - 2 \, \log \left (x^{2} + 1\right ) + 4 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.28056, size = 34, normalized size = 1.55 \[ -\frac{2 \,{\left (2 \, x \arctan \left (x\right ) + x \log \left (x^{2} + 1\right ) - 2 \, x \log \left (x\right ) + 2\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x*arctan(x) + x*log(x^2 + 1) - 2*x*log(x) + 2)/x

_______________________________________________________________________________________

Sympy [A]  time = 0.280935, size = 20, normalized size = 0.91 \[ 4 \log{\left (x \right )} - 2 \log{\left (x^{2} + 1 \right )} - 4 \operatorname{atan}{\left (x \right )} - \frac{4}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*log(x) - 2*log(x**2 + 1) - 4*atan(x) - 4/x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260692, size = 31, normalized size = 1.41 \[ -\frac{4}{x} - 4 \, \arctan \left (x\right ) - 2 \,{\rm ln}\left (x^{2} + 1\right ) + 4 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/x - 4*arctan(x) - 2*ln(x^2 + 1) + 4*ln(abs(x))