\(\int \frac {x}{1+x^2} \, dx\) [97]

Optimal result

Integrand size = 9, antiderivative size = 10 \[ \int \frac {x}{1+x^2} \, dx=\frac {1}{2} \log \left (1+x^2\right ) \]

[Out]

1/2*ln(x^2+1)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {266} \[ \int \frac {x}{1+x^2} \, dx=\frac {1}{2} \log \left (x^2+1\right ) \]

[In]

Int[x/(1 + x^2),x]

[Out]

Log[1 + x^2]/2

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log \left (1+x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x}{1+x^2} \, dx=\frac {1}{2} \log \left (1+x^2\right ) \]

[In]

Integrate[x/(1 + x^2),x]

[Out]

Log[1 + x^2]/2

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)
default	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)
norman	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)
meijerg	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)
risch	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)
parallelrisch	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)

[In]

int(1/(x^2+1)*x,x,method=_RETURNVERBOSE)

[Out]

1/2*ln(x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x}{1+x^2} \, dx=\frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

[In]

integrate(x/(x^2+1),x, algorithm="fricas")

[Out]

1/2*log(x^2 + 1)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {x}{1+x^2} \, dx=\frac {\log {\left (x^{2} + 1 \right )}}{2} \]

[In]

integrate(x/(x**2+1),x)

[Out]

log(x**2 + 1)/2

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x}{1+x^2} \, dx=\frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

[In]

integrate(x/(x^2+1),x, algorithm="maxima")

[Out]

1/2*log(x^2 + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x}{1+x^2} \, dx=\frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

[In]

integrate(x/(x^2+1),x, algorithm="giac")

[Out]

1/2*log(x^2 + 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x}{1+x^2} \, dx=\frac {\ln \left (x^2+1\right )}{2} \]

[In]

int(x/(x^2 + 1),x)

[Out]

log(x^2 + 1)/2