3.49.78 \(\int \frac {-1+e^x x}{x} \, dx\)

Optimal. Leaf size=17 \[ 3+e^x+x-\log \left (e^x x\right )+\log (\log (5)) \]

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 0.47, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {14, 2194} \begin {gather*} e^x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^x*x)/x,x]

[Out]

E^x - Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x-\frac {1}{x}\right ) \, dx\\ &=-\log (x)+\int e^x \, dx\\ &=e^x-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.47 \begin {gather*} e^x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^x*x)/x,x]

[Out]

E^x - Log[x]

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fricas [A]  time = 0.83, size = 7, normalized size = 0.41 \begin {gather*} e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)/x,x, algorithm="fricas")

[Out]

e^x - log(x)

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giac [A]  time = 0.15, size = 7, normalized size = 0.41 \begin {gather*} e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)/x,x, algorithm="giac")

[Out]

e^x - log(x)

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maple [A]  time = 0.02, size = 8, normalized size = 0.47

method	result	size
default	\({\mathrm e}^{x}-\ln \relax (x )\)	\(8\)
norman	\({\mathrm e}^{x}-\ln \relax (x )\)	\(8\)
risch	\({\mathrm e}^{x}-\ln \relax (x )\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

exp(x)-ln(x)

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maxima [A]  time = 0.35, size = 7, normalized size = 0.41 \begin {gather*} e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)/x,x, algorithm="maxima")

[Out]

e^x - log(x)

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mupad [B]  time = 0.03, size = 7, normalized size = 0.41 \begin {gather*} {\mathrm {e}}^x-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x) - 1)/x,x)

[Out]

exp(x) - log(x)

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sympy [A]  time = 0.10, size = 5, normalized size = 0.29 \begin {gather*} e^{x} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)/x,x)

[Out]

exp(x) - log(x)

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