\(\int \frac {(-6+2 e^{3+e^{3+x}+x}) \log (-6+e^{e^{3+x}}-3 x)}{-6+e^{e^{3+x}}-3 x} \, dx\) [9442]

Optimal result

Integrand size = 42, antiderivative size = 15 \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=\log ^2\left (-6+e^{e^{3+x}}-3 x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6816, 6818} \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=\log ^2\left (-3 x+e^{e^{x+3}}-6\right ) \]

[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 6816

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

Rule 6818

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=\log ^2\left (-6+e^{e^{3+x}}-3 x\right ) \]

[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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 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	\(\ln \left ({\mathrm e}^{{\mathrm e}^{3+x}}-3 x -6\right )^{2}\)	\(14\)
parallelrisch	\(\ln \left ({\mathrm e}^{{\mathrm e}^{3+x}}-3 x -6\right )^{2}\)	\(14\)

[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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=\log \left (-{\left (3 \, {\left (x + 2\right )} e^{\left (x + 3\right )} - e^{\left (x + e^{\left (x + 3\right )} + 3\right )}\right )} e^{\left (-x - 3\right )}\right )^{2} \]

[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

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=\log {\left (- 3 x + e^{e^{x + 3}} - 6 \right )}^{2} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (13) = 26\).

Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.93 \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=-\log \left (3 \, x - e^{\left (e^{\left (x + 3\right )}\right )} + 6\right )^{2} + 2 \, \log \left (3 \, x - e^{\left (e^{\left (x + 3\right )}\right )} + 6\right ) \log \left (-3 \, x + e^{\left (e^{\left (x + 3\right )}\right )} - 6\right ) \]

[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)

Giac [F]

\[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx=\int { -\frac {2 \, {\left (e^{\left (x + e^{\left (x + 3\right )} + 3\right )} - 3\right )} \log \left (-3 \, x + e^{\left (e^{\left (x + 3\right )}\right )} - 6\right )}{3 \, x - e^{\left (e^{\left (x + 3\right )}\right )} + 6} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 13.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-6+2 e^{3+e^{3+x}+x}\right ) \log \left (-6+e^{e^{3+x}}-3 x\right )}{-6+e^{e^{3+x}}-3 x} \, dx={\ln \left ({\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}-3\,x-6\right )}^2 \]

[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