3.12.14 \(\int \frac {e^{-4 e^5} (-2 e^{4 e^5} x^2-\log (3))}{x^2 \log (5)} \, dx\)

Optimal. Leaf size=22 \[ \frac {-2 x+\frac {e^{-4 e^5} \log (3)}{x}}{\log (5)} \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 14} \begin {gather*} \frac {e^{-4 e^5} \log (3)}{x \log (5)}-\frac {2 x}{\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^(4*E^5)*x^2 - Log[3])/(E^(4*E^5)*x^2*Log[5]),x]

[Out]

(-2*x)/Log[5] + Log[3]/(E^(4*E^5)*x*Log[5])

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{-4 e^5} \int \frac {-2 e^{4 e^5} x^2-\log (3)}{x^2} \, dx}{\log (5)}\\ &=\frac {e^{-4 e^5} \int \left (-2 e^{4 e^5}-\frac {\log (3)}{x^2}\right ) \, dx}{\log (5)}\\ &=-\frac {2 x}{\log (5)}+\frac {e^{-4 e^5} \log (3)}{x \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 1.09 \begin {gather*} -\frac {2 x-\frac {e^{-4 e^5} \log (3)}{x}}{\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^(4*E^5)*x^2 - Log[3])/(E^(4*E^5)*x^2*Log[5]),x]

[Out]

-((2*x - Log[3]/(E^(4*E^5)*x))/Log[5])

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fricas [A]  time = 0.66, size = 29, normalized size = 1.32 \begin {gather*} -\frac {{\left (2 \, x^{2} e^{\left (4 \, e^{5}\right )} - \log \relax (3)\right )} e^{\left (-4 \, e^{5}\right )}}{x \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*exp(4*exp(5))-log(3))/x^2/log(5)/exp(4*exp(5)),x, algorithm="fricas")

[Out]

-(2*x^2*e^(4*e^5) - log(3))*e^(-4*e^5)/(x*log(5))

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giac [A]  time = 0.34, size = 27, normalized size = 1.23 \begin {gather*} -\frac {{\left (2 \, x e^{\left (4 \, e^{5}\right )} - \frac {\log \relax (3)}{x}\right )} e^{\left (-4 \, e^{5}\right )}}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*exp(4*exp(5))-log(3))/x^2/log(5)/exp(4*exp(5)),x, algorithm="giac")

[Out]

-(2*x*e^(4*e^5) - log(3)/x)*e^(-4*e^5)/log(5)

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maple [A]  time = 0.06, size = 24, normalized size = 1.09

method	result	size
risch	\(-\frac {2 x}{\ln \relax (5)}+\frac {{\mathrm e}^{-4 \,{\mathrm e}^{5}} \ln \relax (3)}{\ln \relax (5) x}\)	\(24\)
norman	\(\frac {\frac {{\mathrm e}^{-4 \,{\mathrm e}^{5}} \ln \relax (3)}{\ln \relax (5)}-\frac {2 x^{2}}{\ln \relax (5)}}{x}\)	\(27\)
default	\(\frac {{\mathrm e}^{-4 \,{\mathrm e}^{5}} \left (-2 x \,{\mathrm e}^{4 \,{\mathrm e}^{5}}+\frac {\ln \relax (3)}{x}\right )}{\ln \relax (5)}\)	\(28\)
gosper	\(\frac {\left (-2 x^{2} {\mathrm e}^{4 \,{\mathrm e}^{5}}+\ln \relax (3)\right ) {\mathrm e}^{-4 \,{\mathrm e}^{5}}}{x \ln \relax (5)}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2*exp(4*exp(5))-ln(3))/x^2/ln(5)/exp(4*exp(5)),x,method=_RETURNVERBOSE)

[Out]

-2*x/ln(5)+1/ln(5)*exp(-4*exp(5))*ln(3)/x

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maxima [A]  time = 0.38, size = 27, normalized size = 1.23 \begin {gather*} -\frac {{\left (2 \, x e^{\left (4 \, e^{5}\right )} - \frac {\log \relax (3)}{x}\right )} e^{\left (-4 \, e^{5}\right )}}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2*exp(4*exp(5))-log(3))/x^2/log(5)/exp(4*exp(5)),x, algorithm="maxima")

[Out]

-(2*x*e^(4*e^5) - log(3)/x)*e^(-4*e^5)/log(5)

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mupad [B]  time = 0.06, size = 26, normalized size = 1.18 \begin {gather*} \frac {{\mathrm {e}}^{-4\,{\mathrm {e}}^5}\,\left (\ln \relax (3)-2\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^5}\right )}{x\,\ln \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4*exp(5))*(log(3) + 2*x^2*exp(4*exp(5))))/(x^2*log(5)),x)

[Out]

(exp(-4*exp(5))*(log(3) - 2*x^2*exp(4*exp(5))))/(x*log(5))

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sympy [A]  time = 0.09, size = 24, normalized size = 1.09 \begin {gather*} \frac {- 2 x e^{4 e^{5}} + \frac {\log {\relax (3 )}}{x}}{e^{4 e^{5}} \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2*exp(4*exp(5))-ln(3))/x**2/ln(5)/exp(4*exp(5)),x)

[Out]

(-2*x*exp(4*exp(5)) + log(3)/x)*exp(-4*exp(5))/log(5)

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