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

Optimal. Leaf size=11 \[ x+\frac {x^2}{2}+\log (x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {x^2}{2}+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + x^(-1) + x,x]

[Out]

x + x^2/2 + Log[x]

Rubi steps

\begin {align*} \int \left (1+\frac {1}{x}+x\right ) \, dx &=x+\frac {x^2}{2}+\log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} x+\frac {x^2}{2}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + x^(-1) + x,x]

[Out]

x + x^2/2 + Log[x]

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Maple [A]
time = 0.01, size = 10, normalized size = 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	\(x +\frac {x^{2}}{2}+\ln \left (x \right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + x + log(x)

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Fricas [A]
time = 0.54, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + x + log(x)

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Sympy [A]
time = 0.02, size = 8, normalized size = 0.73 \begin {gather*} \frac {x^{2}}{2} + x + \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 + x + log(x)

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Giac [A]
time = 1.40, size = 10, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{2} + x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + x + log(abs(x))

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Mupad [B]
time = 0.03, size = 9, normalized size = 0.82 \begin {gather*} x+\ln \left (x\right )+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x + 1/x + 1,x)

[Out]

x + log(x) + x^2/2

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