3.2.22 \(\int \frac {2+x}{x+x^2} \, dx\) [122]

Optimal. Leaf size=11 \[ 2 \log (x)-\log (1+x) \]

[Out]

2*ln(x)-ln(1+x)

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {645} \begin {gather*} 2 \log (x)-\log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(x + x^2),x]

[Out]

2*Log[x] - Log[1 + x]

Rule 645

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} 2 \log (x)-\log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(x + x^2),x]

[Out]

2*Log[x] - Log[1 + x]

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Mathics [A]
time = 1.65, size = 11, normalized size = 1.00 \begin {gather*} -\text {Log}\left [1+x\right ]+2 \text {Log}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(x + 2)/(x^2 + x),x]')

[Out]

-Log[1 + x] + 2 Log[x]

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Maple [A]
time = 0.07, size = 12, normalized size = 1.09

method	result	size
default	\(2 \ln \left (x \right )-\ln \left (1+x \right )\)	\(12\)
norman	\(2 \ln \left (x \right )-\ln \left (1+x \right )\)	\(12\)
meijerg	\(2 \ln \left (x \right )-\ln \left (1+x \right )\)	\(12\)
risch	\(2 \ln \left (x \right )-\ln \left (1+x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(x)-ln(1+x)

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Maxima [A]
time = 0.26, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + 2*log(x)

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Fricas [A]
time = 0.33, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + 2*log(x)

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Sympy [A]
time = 0.05, size = 8, normalized size = 0.73 \begin {gather*} 2 \log {\left (x \right )} - \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x) - log(x + 1)

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Giac [A]
time = 0.00, size = 12, normalized size = 1.09 \begin {gather*} 2 \ln \left |x\right |-\ln \left |x+1\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^2+x),x)

[Out]

-log(abs(x + 1)) + 2*log(abs(x))

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Mupad [B]
time = 0.10, size = 11, normalized size = 1.00 \begin {gather*} 2\,\ln \left (x\right )-\ln \left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2)/(x + x^2),x)

[Out]

2*log(x) - log(x + 1)

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