3.88.91 \(\int \frac {-243-783 x-621 x^2-189 x^3-20 x^4+e^{\frac {4}{9+6 x+x^2}} (-243-171 x-81 x^2-9 x^3)}{243 x+513 x^2+351 x^3+99 x^4+10 x^5+e^{\frac {4}{9+6 x+x^2}} (243 x+243 x^2+81 x^3+9 x^4)} \, dx\)

Optimal. Leaf size=23 \[ 6-\log \left (x \left (x+9 \left (1+e^{\frac {4}{(3+x)^2}}+x\right )\right )\right ) \]

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Rubi [A]  time = 0.50, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, integrand size = 111, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6688, 6685} \begin {gather*} -\log \left (x \left (10 x+9 e^{\frac {4}{(x+3)^2}}+9\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-243 - 783*x - 621*x^2 - 189*x^3 - 20*x^4 + E^(4/(9 + 6*x + x^2))*(-243 - 171*x - 81*x^2 - 9*x^3))/(243*x
+ 513*x^2 + 351*x^3 + 99*x^4 + 10*x^5 + E^(4/(9 + 6*x + x^2))*(243*x + 243*x^2 + 81*x^3 + 9*x^4)),x]

[Out]

-Log[x*(9 + 9*E^(4/(3 + x)^2) + 10*x)]

Rule 6685

Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]],
x] /;  !FalseQ[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} &=\int \frac {-(3+x)^3 (9+20 x)-9 e^{\frac {4}{(3+x)^2}} \left (27+19 x+9 x^2+x^3\right )}{x (3+x)^3 \left (9+9 e^{\frac {4}{(3+x)^2}}+10 x\right )} \, dx\\ &=-\log \left (x \left (9+9 e^{\frac {4}{(3+x)^2}}+10 x\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 24, normalized size = 1.04 \begin {gather*} -\log (x)-\log \left (9+9 e^{\frac {4}{(3+x)^2}}+10 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-243 - 783*x - 621*x^2 - 189*x^3 - 20*x^4 + E^(4/(9 + 6*x + x^2))*(-243 - 171*x - 81*x^2 - 9*x^3))/
(243*x + 513*x^2 + 351*x^3 + 99*x^4 + 10*x^5 + E^(4/(9 + 6*x + x^2))*(243*x + 243*x^2 + 81*x^3 + 9*x^4)),x]

[Out]

-Log[x] - Log[9 + 9*E^(4/(3 + x)^2) + 10*x]

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fricas [A]  time = 0.59, size = 28, normalized size = 1.22 \begin {gather*} -\log \left (10 \, x + 9 \, e^{\left (\frac {4}{x^{2} + 6 \, x + 9}\right )} + 9\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^3-81*x^2-171*x-243)*exp(2/(x^2+6*x+9))^2-20*x^4-189*x^3-621*x^2-783*x-243)/((9*x^4+81*x^3+243
*x^2+243*x)*exp(2/(x^2+6*x+9))^2+10*x^5+99*x^4+351*x^3+513*x^2+243*x),x, algorithm="fricas")

[Out]

-log(10*x + 9*e^(4/(x^2 + 6*x + 9)) + 9) - log(x)

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giac [A]  time = 0.15, size = 37, normalized size = 1.61 \begin {gather*} -\log \left (10 \, x + 9 \, e^{\left (-\frac {4 \, {\left (x^{2} + 6 \, x\right )}}{9 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {4}{9}\right )} + 9\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^3-81*x^2-171*x-243)*exp(2/(x^2+6*x+9))^2-20*x^4-189*x^3-621*x^2-783*x-243)/((9*x^4+81*x^3+243
*x^2+243*x)*exp(2/(x^2+6*x+9))^2+10*x^5+99*x^4+351*x^3+513*x^2+243*x),x, algorithm="giac")

[Out]

-log(10*x + 9*e^(-4/9*(x^2 + 6*x)/(x^2 + 6*x + 9) + 4/9) + 9) - log(x)

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maple [A]  time = 0.19, size = 22, normalized size = 0.96

method	result	size
risch	\(-\ln \relax (x )-\ln \left ({\mathrm e}^{\frac {4}{\left (3+x \right )^{2}}}+\frac {10 x}{9}+1\right )\)	\(22\)
norman	\(-\ln \relax (x )-\ln \left (9 \,{\mathrm e}^{\frac {4}{x^{2}+6 x +9}}+10 x +9\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x^3-81*x^2-171*x-243)*exp(2/(x^2+6*x+9))^2-20*x^4-189*x^3-621*x^2-783*x-243)/((9*x^4+81*x^3+243*x^2+2
43*x)*exp(2/(x^2+6*x+9))^2+10*x^5+99*x^4+351*x^3+513*x^2+243*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-ln(exp(4/(3+x)^2)+10/9*x+1)

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maxima [A]  time = 0.39, size = 26, normalized size = 1.13 \begin {gather*} -\log \left (\frac {10}{9} \, x + e^{\left (\frac {4}{x^{2} + 6 \, x + 9}\right )} + 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^3-81*x^2-171*x-243)*exp(2/(x^2+6*x+9))^2-20*x^4-189*x^3-621*x^2-783*x-243)/((9*x^4+81*x^3+243
*x^2+243*x)*exp(2/(x^2+6*x+9))^2+10*x^5+99*x^4+351*x^3+513*x^2+243*x),x, algorithm="maxima")

[Out]

-log(10/9*x + e^(4/(x^2 + 6*x + 9)) + 1) - log(x)

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mupad [B]  time = 5.72, size = 21, normalized size = 0.91 \begin {gather*} -\ln \left (x+\frac {9\,{\mathrm {e}}^{\frac {4}{{\left (x+3\right )}^2}}}{10}+\frac {9}{10}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(783*x + exp(4/(6*x + x^2 + 9))*(171*x + 81*x^2 + 9*x^3 + 243) + 621*x^2 + 189*x^3 + 20*x^4 + 243)/(243*x
+ exp(4/(6*x + x^2 + 9))*(243*x + 243*x^2 + 81*x^3 + 9*x^4) + 513*x^2 + 351*x^3 + 99*x^4 + 10*x^5),x)

[Out]

- log(x + (9*exp(4/(x + 3)^2))/10 + 9/10) - log(x)

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sympy [A]  time = 0.41, size = 24, normalized size = 1.04 \begin {gather*} - \log {\relax (x )} - \log {\left (\frac {10 x}{9} + e^{\frac {4}{x^{2} + 6 x + 9}} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x**3-81*x**2-171*x-243)*exp(2/(x**2+6*x+9))**2-20*x**4-189*x**3-621*x**2-783*x-243)/((9*x**4+81
*x**3+243*x**2+243*x)*exp(2/(x**2+6*x+9))**2+10*x**5+99*x**4+351*x**3+513*x**2+243*x),x)

[Out]

-log(x) - log(10*x/9 + exp(4/(x**2 + 6*x + 9)) + 1)

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