3.66.16 \(\int \frac {-10000 e^{2 x} x+\log (2 x)}{5000 x} \, dx\)

Optimal. Leaf size=19 \[ 2-e^{2 x}+\frac {\log ^2(2 x)}{10000} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 14, 2194, 2301} \begin {gather*} \frac {\log ^2(2 x)}{10000}-e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10000*E^(2*x)*x + Log[2*x])/(5000*x),x]

[Out]

-E^(2*x) + Log[2*x]^2/10000

Rule 12

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-10000 e^{2 x} x+\log (2 x)}{x} \, dx}{5000}\\ &=\frac {\int \left (-10000 e^{2 x}+\frac {\log (2 x)}{x}\right ) \, dx}{5000}\\ &=\frac {\int \frac {\log (2 x)}{x} \, dx}{5000}-2 \int e^{2 x} \, dx\\ &=-e^{2 x}+\frac {\log ^2(2 x)}{10000}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 18, normalized size = 0.95 \begin {gather*} -e^{2 x}+\frac {\log ^2(2 x)}{10000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10000*E^(2*x)*x + Log[2*x])/(5000*x),x]

[Out]

-E^(2*x) + Log[2*x]^2/10000

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{10000} \, \log \left (2 \, x\right )^{2} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(log(2*x)-10000*x*exp(x)^2)/x,x, algorithm="fricas")

[Out]

1/10000*log(2*x)^2 - e^(2*x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{10000} \, \log \left (2 \, x\right )^{2} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(log(2*x)-10000*x*exp(x)^2)/x,x, algorithm="giac")

[Out]

1/10000*log(2*x)^2 - e^(2*x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 16, normalized size = 0.84

method	result	size
default	\(\frac {\ln \left (2 x \right )^{2}}{10000}-{\mathrm e}^{2 x}\)	\(16\)
norman	\(\frac {\ln \left (2 x \right )^{2}}{10000}-{\mathrm e}^{2 x}\)	\(16\)
risch	\(\frac {\ln \left (2 x \right )^{2}}{10000}-{\mathrm e}^{2 x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5000*(ln(2*x)-10000*x*exp(x)^2)/x,x,method=_RETURNVERBOSE)

[Out]

1/10000*ln(2*x)^2-exp(x)^2

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{10000} \, \log \left (2 \, x\right )^{2} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(log(2*x)-10000*x*exp(x)^2)/x,x, algorithm="maxima")

[Out]

1/10000*log(2*x)^2 - e^(2*x)

________________________________________________________________________________________

mupad [B]  time = 4.14, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\ln \relax (x)}^2}{10000}+\frac {\ln \relax (2)\,\ln \relax (x)}{5000}-{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x)/5000 - 2*x*exp(2*x))/x,x)

[Out]

log(x)^2/10000 - exp(2*x) + (log(2)*log(x))/5000

________________________________________________________________________________________

sympy [A]  time = 0.24, size = 12, normalized size = 0.63 \begin {gather*} - e^{2 x} + \frac {\log {\left (2 x \right )}^{2}}{10000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(ln(2*x)-10000*x*exp(x)**2)/x,x)

[Out]

-exp(2*x) + log(2*x)**2/10000

________________________________________________________________________________________