\(\int (1+\frac {1}{x}+x) \, dx\) [1907]

Optimal result

Integrand size = 6, antiderivative size = 11 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=x+\frac {x^2}{2}+\log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=\frac {x^2}{2}+x+\log (x) \]

[In]

Int[1 + x^(-1) + x,x]

[Out]

x + x^2/2 + Log[x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=x+\frac {x^2}{2}+\log (x) \]

[In]

Integrate[1 + x^(-1) + x,x]

[Out]

x + x^2/2 + Log[x]

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91

method	result	size
default	\(x +\frac {x^{2}}{2}+\ln \left (x \right )\)	\(10\)
norman	\(x +\frac {x^{2}}{2}+\ln \left (x \right )\)	\(10\)
risch	\(x +\frac {x^{2}}{2}+\ln \left (x \right )\)	\(10\)
parallelrisch	\(x +\frac {x^{2}}{2}+\ln \left (x \right )\)	\(10\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=\frac {1}{2} \, x^{2} + x + \log \left (x\right ) \]

[In]

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

[Out]

1/2*x^2 + x + log(x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=\frac {x^{2}}{2} + x + \log {\left (x \right )} \]

[In]

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

[Out]

x**2/2 + x + log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=\frac {1}{2} \, x^{2} + x + \log \left (x\right ) \]

[In]

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

[Out]

1/2*x^2 + x + log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=\frac {1}{2} \, x^{2} + x + \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

1/2*x^2 + x + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \left (1+\frac {1}{x}+x\right ) \, dx=x+\ln \left (x\right )+\frac {x^2}{2} \]

[In]

int(x + 1/x + 1,x)

[Out]

x + log(x) + x^2/2