3.98.8 \(\int \frac {e^5 (-e^{21}-3 x^2) \log (3)}{(e^{21} x+x^3)^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {e^5 \log (3)}{x \left (e^{21}+x^2\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 1588} \begin {gather*} \frac {e^5 \log (3)}{x^3+e^{21} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(-E^21 - 3*x^2)*Log[3])/(E^21*x + x^3)^2,x]

[Out]

(E^5*Log[3])/(E^21*x + x^3)

Rule 12

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

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (e^5 \log (3)\right ) \int \frac {-e^{21}-3 x^2}{\left (e^{21} x+x^3\right )^2} \, dx\\ &=\frac {e^5 \log (3)}{e^{21} x+x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {e^5 \log (3)}{e^{21} x+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-E^21 - 3*x^2)*Log[3])/(E^21*x + x^3)^2,x]

[Out]

(E^5*Log[3])/(E^21*x + x^3)

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fricas [A]  time = 0.80, size = 15, normalized size = 0.83 \begin {gather*} \frac {e^{5} \log \relax (3)}{x^{3} + x e^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(21)-3*x^2)*exp(log(log(3)/(x*exp(21)+x^3))+5)/(x*exp(21)+x^3),x, algorithm="fricas")

[Out]

e^5*log(3)/(x^3 + x*e^21)

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giac [A]  time = 0.16, size = 15, normalized size = 0.83 \begin {gather*} \frac {e^{5} \log \relax (3)}{x^{3} + x e^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(21)-3*x^2)*exp(log(log(3)/(x*exp(21)+x^3))+5)/(x*exp(21)+x^3),x, algorithm="giac")

[Out]

e^5*log(3)/(x^3 + x*e^21)

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maple [A]  time = 0.09, size = 17, normalized size = 0.94

method	result	size
norman	\(\frac {{\mathrm e}^{5} \ln \relax (3)}{x \left ({\mathrm e}^{21}+x^{2}\right )}\)	\(17\)
risch	\(\frac {{\mathrm e}^{5} \ln \relax (3)}{x \left ({\mathrm e}^{21}+x^{2}\right )}\)	\(17\)
gosper	\({\mathrm e}^{5+\ln \left (\frac {\ln \relax (3)}{\left ({\mathrm e}^{21}+x^{2}\right ) x}\right )}\)	\(19\)
default	\(\frac {{\mathrm e}^{5+\ln \left (\frac {\ln \relax (3)}{x \,{\mathrm e}^{21}+x^{3}}\right )+\ln \left (x \,{\mathrm e}^{21}+x^{3}\right )} {\mathrm e}^{-21}}{x}-\frac {{\mathrm e}^{5+\ln \left (\frac {\ln \relax (3)}{x \,{\mathrm e}^{21}+x^{3}}\right )+\ln \left (x \,{\mathrm e}^{21}+x^{3}\right )} x \,{\mathrm e}^{-21}}{{\mathrm e}^{21}+x^{2}}\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(21)-3*x^2)*exp(ln(ln(3)/(x*exp(21)+x^3))+5)/(x*exp(21)+x^3),x,method=_RETURNVERBOSE)

[Out]

exp(5)*ln(3)/x/(exp(21)+x^2)

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maxima [A]  time = 0.34, size = 15, normalized size = 0.83 \begin {gather*} \frac {e^{5} \log \relax (3)}{x^{3} + x e^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(21)-3*x^2)*exp(log(log(3)/(x*exp(21)+x^3))+5)/(x*exp(21)+x^3),x, algorithm="maxima")

[Out]

e^5*log(3)/(x^3 + x*e^21)

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mupad [B]  time = 0.14, size = 16, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^5\,\ln \relax (3)}{x\,\left (x^2+{\mathrm {e}}^{21}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(log(3)/(x*exp(21) + x^3)) + 5)*(exp(21) + 3*x^2))/(x*exp(21) + x^3),x)

[Out]

(exp(5)*log(3))/(x*(exp(21) + x^2))

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sympy [A]  time = 0.20, size = 14, normalized size = 0.78 \begin {gather*} \frac {e^{5} \log {\relax (3 )}}{x^{3} + x e^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(21)-3*x**2)*exp(ln(ln(3)/(x*exp(21)+x**3))+5)/(x*exp(21)+x**3),x)

[Out]

exp(5)*log(3)/(x**3 + x*exp(21))

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