3.95.42 \(\int \frac {(-6+2 e^{3+e^{3+x}+x}) \log (-6+e^{e^{3+x}}-3 x)}{-6+e^{e^{3+x}}-3 x} \, dx\)

Optimal. Leaf size=15 \[ \log ^2\left (-6+e^{e^{3+x}}-3 x\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6684, 6686} \begin {gather*} \log ^2\left (-3 x+e^{e^{x+3}}-6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-6 + 2*E^(3 + E^(3 + x) + x))*Log[-6 + E^E^(3 + x) - 3*x])/(-6 + E^E^(3 + x) - 3*x),x]

[Out]

Log[-6 + E^E^(3 + x) - 3*x]^2

Rule 6684

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log ^2\left (-6+e^{e^{3+x}}-3 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 15, normalized size = 1.00 \begin {gather*} \log ^2\left (-6+e^{e^{3+x}}-3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-6 + 2*E^(3 + E^(3 + x) + x))*Log[-6 + E^E^(3 + x) - 3*x])/(-6 + E^E^(3 + x) - 3*x),x]

[Out]

Log[-6 + E^E^(3 + x) - 3*x]^2

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fricas [B]  time = 0.46, size = 31, normalized size = 2.07 \begin {gather*} \log \left (-{\left (3 \, {\left (x + 2\right )} e^{\left (x + 3\right )} - e^{\left (x + e^{\left (x + 3\right )} + 3\right )}\right )} e^{\left (-x - 3\right )}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(3+x)*exp(exp(3+x))-6)*log(exp(exp(3+x))-3*x-6)/(exp(exp(3+x))-3*x-6),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (e^{\left (x + e^{\left (x + 3\right )} + 3\right )} - 3\right )} \log \left (-3 \, x + e^{\left (e^{\left (x + 3\right )}\right )} - 6\right )}{3 \, x - e^{\left (e^{\left (x + 3\right )}\right )} + 6}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(3+x)*exp(exp(3+x))-6)*log(exp(exp(3+x))-3*x-6)/(exp(exp(3+x))-3*x-6),x, algorithm="giac")

[Out]

integrate(-2*(e^(x + e^(x + 3) + 3) - 3)*log(-3*x + e^(e^(x + 3)) - 6)/(3*x - e^(e^(x + 3)) + 6), x)

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maple [A]  time = 0.05, size = 14, normalized size = 0.93

method	result	size
default	\(\ln \left ({\mathrm e}^{{\mathrm e}^{3+x}}-3 x -6\right )^{2}\)	\(14\)
norman	\(\ln \left ({\mathrm e}^{{\mathrm e}^{3+x}}-3 x -6\right )^{2}\)	\(14\)
risch	\(\ln \left ({\mathrm e}^{{\mathrm e}^{3+x}}-3 x -6\right )^{2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(3+x)*exp(exp(3+x))-6)*ln(exp(exp(3+x))-3*x-6)/(exp(exp(3+x))-3*x-6),x,method=_RETURNVERBOSE)

[Out]

ln(exp(exp(3+x))-3*x-6)^2

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maxima [B]  time = 0.37, size = 44, normalized size = 2.93 \begin {gather*} -\log \left (3 \, x - e^{\left (e^{\left (x + 3\right )}\right )} + 6\right )^{2} + 2 \, \log \left (3 \, x - e^{\left (e^{\left (x + 3\right )}\right )} + 6\right ) \log \left (-3 \, x + e^{\left (e^{\left (x + 3\right )}\right )} - 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(3+x)*exp(exp(3+x))-6)*log(exp(exp(3+x))-3*x-6)/(exp(exp(3+x))-3*x-6),x, algorithm="maxima")

[Out]

-log(3*x - e^(e^(x + 3)) + 6)^2 + 2*log(3*x - e^(e^(x + 3)) + 6)*log(-3*x + e^(e^(x + 3)) - 6)

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mupad [B]  time = 5.48, size = 14, normalized size = 0.93 \begin {gather*} {\ln \left ({\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}-3\,x-6\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(exp(exp(x + 3)) - 3*x - 6)*(2*exp(x + 3)*exp(exp(x + 3)) - 6))/(3*x - exp(exp(x + 3)) + 6),x)

[Out]

log(exp(exp(3)*exp(x)) - 3*x - 6)^2

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sympy [A]  time = 0.48, size = 14, normalized size = 0.93 \begin {gather*} \log {\left (- 3 x + e^{e^{x + 3}} - 6 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(3+x)*exp(exp(3+x))-6)*ln(exp(exp(3+x))-3*x-6)/(exp(exp(3+x))-3*x-6),x)

[Out]

log(-3*x + exp(exp(x + 3)) - 6)**2

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