3.39.25 \(\int \frac {20 e^{\frac {-400+60 e^2+20 e^2 \log (x)}{e^2}}}{x} \, dx\)

Optimal. Leaf size=13 \[ e^{20 \left (3-\frac {20}{e^2}+\log (x)\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 1586, 30} \begin {gather*} e^{60-\frac {400}{e^2}} x^{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20*E^((-400 + 60*E^2 + 20*E^2*Log[x])/E^2))/x,x]

[Out]

E^(60 - 400/E^2)*x^20

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=20 \int \frac {e^{\frac {-400+60 e^2+20 e^2 \log (x)}{e^2}}}{x} \, dx\\ &=20 \int e^{60-\frac {400}{e^2}} x^{19} \, dx\\ &=\left (20 e^{60-\frac {400}{e^2}}\right ) \int x^{19} \, dx\\ &=e^{60-\frac {400}{e^2}} x^{20}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} e^{60-\frac {400}{e^2}} x^{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*E^((-400 + 60*E^2 + 20*E^2*Log[x])/E^2))/x,x]

[Out]

E^(60 - 400/E^2)*x^20

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fricas [A]  time = 0.48, size = 15, normalized size = 1.15 \begin {gather*} x^{20} e^{\left (20 \, {\left (3 \, e^{2} - 20\right )} e^{\left (-2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*exp((20*exp(2)*log(x)+60*exp(2)-400)/exp(2))/x,x, algorithm="fricas")

[Out]

x^20*e^(20*(3*e^2 - 20)*e^(-2))

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giac [A]  time = 0.14, size = 11, normalized size = 0.85 \begin {gather*} x^{20} e^{\left (-400 \, e^{\left (-2\right )} + 60\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*exp((20*exp(2)*log(x)+60*exp(2)-400)/exp(2))/x,x, algorithm="giac")

[Out]

x^20*e^(-400*e^(-2) + 60)

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maple [A]  time = 0.09, size = 12, normalized size = 0.92

method	result	size
risch	\(x^{20} {\mathrm e}^{60-400 \,{\mathrm e}^{-2}}\)	\(12\)
norman	\({\mathrm e}^{60} {\mathrm e}^{-400 \,{\mathrm e}^{-2}} x^{20}\)	\(14\)
gosper	\({\mathrm e}^{20 \left ({\mathrm e}^{2} \ln \relax (x )+3 \,{\mathrm e}^{2}-20\right ) {\mathrm e}^{-2}}\)	\(19\)
derivativedivides	\({\mathrm e}^{\left (20 \,{\mathrm e}^{2} \ln \relax (x )+60 \,{\mathrm e}^{2}-400\right ) {\mathrm e}^{-2}}\)	\(19\)
default	\({\mathrm e}^{\left (20 \,{\mathrm e}^{2} \ln \relax (x )+60 \,{\mathrm e}^{2}-400\right ) {\mathrm e}^{-2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(20*exp((20*exp(2)*ln(x)+60*exp(2)-400)/exp(2))/x,x,method=_RETURNVERBOSE)

[Out]

x^20*exp(60-400*exp(-2))

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maxima [A]  time = 0.34, size = 16, normalized size = 1.23 \begin {gather*} e^{\left (20 \, {\left (e^{2} \log \relax (x) + 3 \, e^{2} - 20\right )} e^{\left (-2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*exp((20*exp(2)*log(x)+60*exp(2)-400)/exp(2))/x,x, algorithm="maxima")

[Out]

e^(20*(e^2*log(x) + 3*e^2 - 20)*e^(-2))

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mupad [B]  time = 0.06, size = 11, normalized size = 0.85 \begin {gather*} x^{20}\,{\mathrm {e}}^{60-400\,{\mathrm {e}}^{-2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*exp(exp(-2)*(60*exp(2) + 20*exp(2)*log(x) - 400)))/x,x)

[Out]

x^20*exp(60 - 400*exp(-2))

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sympy [A]  time = 0.06, size = 12, normalized size = 0.92 \begin {gather*} \frac {x^{20} e^{60}}{e^{\frac {400}{e^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20*exp((20*exp(2)*ln(x)+60*exp(2)-400)/exp(2))/x,x)

[Out]

x**20*exp(60)*exp(-400*exp(-2))

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