3.92.34 \(\int \frac {1}{e^6 x} \, dx\)

Optimal. Leaf size=10 \[ 8+\frac {4+\log (x)}{e^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.60, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 29} \begin {gather*} \frac {\log (x)}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^6*x),x]

[Out]

Log[x]/E^6

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 {\int \frac {1}{x} \, dx}{e^6}\\ &=\frac {\log (x)}{e^6}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 6, normalized size = 0.60 \begin {gather*} \frac {\log (x)}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^6*x),x]

[Out]

Log[x]/E^6

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 5, normalized size = 0.50 \begin {gather*} e^{\left (-6\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/exp(6),x, algorithm="fricas")

[Out]

e^(-6)*log(x)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 6, normalized size = 0.60 \begin {gather*} e^{\left (-6\right )} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/exp(6),x, algorithm="giac")

[Out]

e^(-6)*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 6, normalized size = 0.60

method	result	size
risch	\({\mathrm e}^{-6} \ln \relax (x )\)	\(6\)
default	\({\mathrm e}^{-6} \ln \relax (x )\)	\(8\)
norman	\({\mathrm e}^{-6} \ln \relax (x )\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/exp(6),x,method=_RETURNVERBOSE)

[Out]

exp(-6)*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 5, normalized size = 0.50 \begin {gather*} e^{\left (-6\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/exp(6),x, algorithm="maxima")

[Out]

e^(-6)*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 5, normalized size = 0.50 \begin {gather*} {\mathrm {e}}^{-6}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-6)/x,x)

[Out]

exp(-6)*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 5, normalized size = 0.50 \begin {gather*} \frac {\log {\relax (x )}}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/exp(6),x)

[Out]

exp(-6)*log(x)

________________________________________________________________________________________