3.59.73 \(\int -\frac {1350 e^{\frac {13 x^2+675 \log (3)}{x^2}} \log (3)}{x^3+e^{\frac {13 x^2+675 \log (3)}{x^2}} x^3} \, dx\)

Optimal. Leaf size=14 \[ \log \left (1+e^{13+\frac {675 \log (3)}{x^2}}\right ) \]

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Rubi [A]  time = 0.20, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 6688, 6684} \begin {gather*} \log \left (e^{13} 3^{\frac {675}{x^2}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1350*E^((13*x^2 + 675*Log[3])/x^2)*Log[3])/(x^3 + E^((13*x^2 + 675*Log[3])/x^2)*x^3),x]

[Out]

Log[1 + 3^(675/x^2)*E^13]

Rule 12

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

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left ((1350 \log (3)) \int \frac {e^{\frac {13 x^2+675 \log (3)}{x^2}}}{x^3+e^{\frac {13 x^2+675 \log (3)}{x^2}} x^3} \, dx\right )\\ &=-\left ((1350 \log (3)) \int \frac {3^{\frac {675}{x^2}} e^{13}}{\left (1+3^{\frac {675}{x^2}} e^{13}\right ) x^3} \, dx\right )\\ &=-\left (\left (1350 e^{13} \log (3)\right ) \int \frac {3^{\frac {675}{x^2}}}{\left (1+3^{\frac {675}{x^2}} e^{13}\right ) x^3} \, dx\right )\\ &=\log \left (1+3^{\frac {675}{x^2}} e^{13}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \log \left (1+3^{\frac {675}{x^2}} e^{13}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1350*E^((13*x^2 + 675*Log[3])/x^2)*Log[3])/(x^3 + E^((13*x^2 + 675*Log[3])/x^2)*x^3),x]

[Out]

Log[1 + 3^(675/x^2)*E^13]

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fricas [A]  time = 0.64, size = 18, normalized size = 1.29 \begin {gather*} \log \left (e^{\left (\frac {13 \, x^{2} + 675 \, \log \relax (3)}{x^{2}}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1350*log(3)*exp((675*log(3)+13*x^2)/x^2)/(x^3*exp((675*log(3)+13*x^2)/x^2)+x^3),x, algorithm="frica
s")

[Out]

log(e^((13*x^2 + 675*log(3))/x^2) + 1)

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giac [A]  time = 0.19, size = 18, normalized size = 1.29 \begin {gather*} \log \left (e^{\left (\frac {13 \, x^{2} + 675 \, \log \relax (3)}{x^{2}}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1350*log(3)*exp((675*log(3)+13*x^2)/x^2)/(x^3*exp((675*log(3)+13*x^2)/x^2)+x^3),x, algorithm="giac"
)

[Out]

log(e^((13*x^2 + 675*log(3))/x^2) + 1)

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maple [A]  time = 0.12, size = 19, normalized size = 1.36

method	result	size
norman	\(\ln \left ({\mathrm e}^{\frac {675 \ln \relax (3)+13 x^{2}}{x^{2}}}+1\right )\)	\(19\)
risch	\(\frac {675 \ln \relax (3)}{x^{2}}-\frac {675 \ln \relax (3)+13 x^{2}}{x^{2}}+\ln \left ({\mathrm e}^{\frac {675 \ln \relax (3)+13 x^{2}}{x^{2}}}+1\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1350*ln(3)*exp((675*ln(3)+13*x^2)/x^2)/(x^3*exp((675*ln(3)+13*x^2)/x^2)+x^3),x,method=_RETURNVERBOSE)

[Out]

ln(exp((675*ln(3)+13*x^2)/x^2)+1)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1350*log(3)*exp((675*log(3)+13*x^2)/x^2)/(x^3*exp((675*log(3)+13*x^2)/x^2)+x^3),x, algorithm="maxim
a")

[Out]

Timed out

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mupad [B]  time = 4.03, size = 13, normalized size = 0.93 \begin {gather*} \ln \left (3^{\frac {675}{x^2}}\,{\mathrm {e}}^{13}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1350*exp((675*log(3) + 13*x^2)/x^2)*log(3))/(x^3*exp((675*log(3) + 13*x^2)/x^2) + x^3),x)

[Out]

log(3^(675/x^2)*exp(13) + 1)

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sympy [A]  time = 0.12, size = 17, normalized size = 1.21 \begin {gather*} \log {\left (e^{\frac {13 x^{2} + 675 \log {\relax (3 )}}{x^{2}}} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1350*ln(3)*exp((675*ln(3)+13*x**2)/x**2)/(x**3*exp((675*ln(3)+13*x**2)/x**2)+x**3),x)

[Out]

log(exp((13*x**2 + 675*log(3))/x**2) + 1)

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