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

Optimal. Leaf size=10 \[ \frac {1}{2} \log \left (1+x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {266} \begin {gather*} \frac {1}{2} \log \left (x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \log \left (1+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (1+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^2),x]

[Out]

Log[1 + x^2]/2

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Maple [A]
time = 0.01, size = 9, normalized size = 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	\(\frac {\ln \left (x^{2}+1\right )}{2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 + 1)

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Fricas [A]
time = 0.57, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 + 1)

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Sympy [A]
time = 0.02, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (x^{2} + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + 1)/2

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Giac [A]
time = 0.42, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 + 1)

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Mupad [B]
time = 0.03, size = 8, normalized size = 0.80 \begin {gather*} \frac {\ln \left (x^2+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 + 1),x)

[Out]

log(x^2 + 1)/2

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