3.99.92 \(\int \frac {e^2}{3 x} \, dx\)

Optimal. Leaf size=11 \[ -14+\frac {1}{3} e^2 \log (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 29} \begin {gather*} \frac {1}{3} e^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^2/(3*x),x]

[Out]

(E^2*Log[x])/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} e^2 \int \frac {1}{x} \, dx\\ &=\frac {1}{3} e^2 \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{3} e^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^2/(3*x),x]

[Out]

(E^2*Log[x])/3

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 6, normalized size = 0.55 \begin {gather*} \frac {1}{3} \, e^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*exp(2)/x,x, algorithm="fricas")

[Out]

1/3*e^2*log(x)

________________________________________________________________________________________

giac [A]  time = 0.28, size = 7, normalized size = 0.64 \begin {gather*} \frac {1}{3} \, e^{2} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*exp(2)/x,x, algorithm="giac")

[Out]

1/3*e^2*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 7, normalized size = 0.64

method	result	size
default	\(\frac {{\mathrm e}^{2} \ln \relax (x )}{3}\)	\(7\)
norman	\(\frac {{\mathrm e}^{2} \ln \relax (x )}{3}\)	\(7\)
risch	\(\frac {{\mathrm e}^{2} \ln \relax (x )}{3}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*exp(2)/x,x,method=_RETURNVERBOSE)

[Out]

1/3*exp(2)*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 6, normalized size = 0.55 \begin {gather*} \frac {1}{3} \, e^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*exp(2)/x,x, algorithm="maxima")

[Out]

1/3*e^2*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 6, normalized size = 0.55 \begin {gather*} \frac {{\mathrm {e}}^2\,\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2)/(3*x),x)

[Out]

(exp(2)*log(x))/3

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 7, normalized size = 0.64 \begin {gather*} \frac {e^{2} \log {\relax (x )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*exp(2)/x,x)

[Out]

exp(2)*log(x)/3

________________________________________________________________________________________