3.18.34 \(\int \frac {36+9 e^{2+e^{2+x}+x}}{-24+e^{e^{2+x}}+4 x} \, dx\)

Optimal. Leaf size=17 \[ 9 \log \left (-6+\frac {e^{e^{2+x}}}{4}+x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6684} \begin {gather*} 9 \log \left (-4 x-e^{e^{x+2}}+24\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 + 9*E^(2 + E^(2 + x) + x))/(-24 + E^E^(2 + x) + 4*x),x]

[Out]

9*Log[24 - E^E^(2 + x) - 4*x]

Rule 6684

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=9 \log \left (24-e^{e^{2+x}}-4 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 16, normalized size = 0.94 \begin {gather*} 9 \log \left (e^{e^{2+x}}+4 (-6+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 + 9*E^(2 + E^(2 + x) + x))/(-24 + E^E^(2 + x) + 4*x),x]

[Out]

9*Log[E^E^(2 + x) + 4*(-6 + x)]

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fricas [A]  time = 0.70, size = 25, normalized size = 1.47 \begin {gather*} -9 \, x + 9 \, \log \left (4 \, {\left (x - 6\right )} e^{\left (x + 2\right )} + e^{\left (x + e^{\left (x + 2\right )} + 2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(2+x)*exp(exp(2+x))+36)/(exp(exp(2+x))+4*x-24),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9 \, {\left (e^{\left (x + e^{\left (x + 2\right )} + 2\right )} + 4\right )}}{4 \, x + e^{\left (e^{\left (x + 2\right )}\right )} - 24}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(2+x)*exp(exp(2+x))+36)/(exp(exp(2+x))+4*x-24),x, algorithm="giac")

[Out]

integrate(9*(e^(x + e^(x + 2) + 2) + 4)/(4*x + e^(e^(x + 2)) - 24), x)

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maple [A]  time = 0.02, size = 14, normalized size = 0.82

method	result	size
default	\(9 \ln \left ({\mathrm e}^{{\mathrm e}^{2+x}}+4 x -24\right )\)	\(14\)
norman	\(9 \ln \left ({\mathrm e}^{{\mathrm e}^{2+x}}+4 x -24\right )\)	\(14\)
risch	\(9 \ln \left ({\mathrm e}^{{\mathrm e}^{2+x}}+4 x -24\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*exp(2+x)*exp(exp(2+x))+36)/(exp(exp(2+x))+4*x-24),x,method=_RETURNVERBOSE)

[Out]

9*ln(exp(exp(2+x))+4*x-24)

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maxima [A]  time = 0.47, size = 13, normalized size = 0.76 \begin {gather*} 9 \, \log \left (4 \, x + e^{\left (e^{\left (x + 2\right )}\right )} - 24\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(2+x)*exp(exp(2+x))+36)/(exp(exp(2+x))+4*x-24),x, algorithm="maxima")

[Out]

9*log(4*x + e^(e^(x + 2)) - 24)

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mupad [B]  time = 1.11, size = 13, normalized size = 0.76 \begin {gather*} 9\,\ln \left (x+\frac {{\mathrm {e}}^{{\mathrm {e}}^{x+2}}}{4}-6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*exp(x + 2)*exp(exp(x + 2)) + 36)/(4*x + exp(exp(x + 2)) - 24),x)

[Out]

9*log(x + exp(exp(x + 2))/4 - 6)

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sympy [A]  time = 0.14, size = 14, normalized size = 0.82 \begin {gather*} 9 \log {\left (4 x + e^{e^{x + 2}} - 24 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(2+x)*exp(exp(2+x))+36)/(exp(exp(2+x))+4*x-24),x)

[Out]

9*log(4*x + exp(exp(x + 2)) - 24)

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