3.1.95 \(\int \frac {e^{\frac {-x+(5+x^3) \log (e^{e^{6 x}} x)}{\log (e^{e^{6 x}} x)}} (1+6 e^{6 x} x-\log (e^{e^{6 x}} x)+3 x^2 \log ^2(e^{e^{6 x}} x))}{\log ^2(e^{e^{6 x}} x)} \, dx\)

Optimal. Leaf size=22 \[ e^{5+x^3-\frac {x}{\log \left (e^{e^{6 x}} x\right )}} \]

________________________________________________________________________________________

Rubi [A]  time = 1.36, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6688, 6706} \begin {gather*} e^{x^3-\frac {x}{\log \left (e^{e^{6 x}} x\right )}+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-x + (5 + x^3)*Log[E^E^(6*x)*x])/Log[E^E^(6*x)*x])*(1 + 6*E^(6*x)*x - Log[E^E^(6*x)*x] + 3*x^2*Log[E^
E^(6*x)*x]^2))/Log[E^E^(6*x)*x]^2,x]

[Out]

E^(5 + x^3 - x/Log[E^E^(6*x)*x])

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{5+x^3-\frac {x}{\log \left (e^{e^{6 x}} x\right )}} \left (1+6 e^{6 x} x-\log \left (e^{e^{6 x}} x\right )+3 x^2 \log ^2\left (e^{e^{6 x}} x\right )\right )}{\log ^2\left (e^{e^{6 x}} x\right )} \, dx\\ &=e^{5+x^3-\frac {x}{\log \left (e^{e^{6 x}} x\right )}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.01, size = 22, normalized size = 1.00 \begin {gather*} e^{5+x^3-\frac {x}{\log \left (e^{e^{6 x}} x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-x + (5 + x^3)*Log[E^E^(6*x)*x])/Log[E^E^(6*x)*x])*(1 + 6*E^(6*x)*x - Log[E^E^(6*x)*x] + 3*x^2*
Log[E^E^(6*x)*x]^2))/Log[E^E^(6*x)*x]^2,x]

[Out]

E^(5 + x^3 - x/Log[E^E^(6*x)*x])

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 30, normalized size = 1.36 \begin {gather*} e^{\left (\frac {{\left (x^{3} + 5\right )} \log \left (x e^{\left (e^{\left (6 \, x\right )}\right )}\right ) - x}{\log \left (x e^{\left (e^{\left (6 \, x\right )}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x*exp(exp(3*x)^2))^2-log(x*exp(exp(3*x)^2))+6*x*exp(3*x)^2+1)*exp(((x^3+5)*log(x*exp(exp(
3*x)^2))-x)/log(x*exp(exp(3*x)^2)))/log(x*exp(exp(3*x)^2))^2,x, algorithm="fricas")

[Out]

e^(((x^3 + 5)*log(x*e^(e^(6*x))) - x)/log(x*e^(e^(6*x))))

________________________________________________________________________________________

giac [A]  time = 0.55, size = 19, normalized size = 0.86 \begin {gather*} e^{\left (x^{3} - \frac {x}{\log \left (x e^{\left (e^{\left (6 \, x\right )}\right )}\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x*exp(exp(3*x)^2))^2-log(x*exp(exp(3*x)^2))+6*x*exp(3*x)^2+1)*exp(((x^3+5)*log(x*exp(exp(
3*x)^2))-x)/log(x*exp(exp(3*x)^2)))/log(x*exp(exp(3*x)^2))^2,x, algorithm="giac")

[Out]

e^(x^3 - x/log(x*e^(e^(6*x))) + 5)

________________________________________________________________________________________

maple [C]  time = 1.57, size = 334, normalized size = 15.18

method	result	size
risch	\({\mathrm e}^{\frac {-i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{3} x^{3}+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{2} \mathrm {csgn}\left (i x \right ) x^{3}+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{6 x}}\right ) x^{3}-i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{6 x}}\right ) x^{3}-5 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{3}+5 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{2} \mathrm {csgn}\left (i x \right )+5 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{6 x}}\right )-5 i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{6 x}}\right )+2 x^{3} \ln \relax (x )+2 \ln \left ({\mathrm e}^{{\mathrm e}^{6 x}}\right ) x^{3}+10 \ln \relax (x )+10 \ln \left ({\mathrm e}^{{\mathrm e}^{6 x}}\right )-2 x}{-i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{3}+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{6 x}}\right )-i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{{\mathrm e}^{6 x}}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{6 x}}\right )+2 \ln \relax (x )+2 \ln \left ({\mathrm e}^{{\mathrm e}^{6 x}}\right )}}\)	\(334\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*ln(x*exp(exp(3*x)^2))^2-ln(x*exp(exp(3*x)^2))+6*x*exp(3*x)^2+1)*exp(((x^3+5)*ln(x*exp(exp(3*x)^2))-
x)/ln(x*exp(exp(3*x)^2)))/ln(x*exp(exp(3*x)^2))^2,x,method=_RETURNVERBOSE)

[Out]

exp((-I*Pi*csgn(I*x*exp(exp(6*x)))^3*x^3+I*Pi*csgn(I*x*exp(exp(6*x)))^2*csgn(I*x)*x^3+I*Pi*csgn(I*x*exp(exp(6*
x)))^2*csgn(I*exp(exp(6*x)))*x^3-I*Pi*csgn(I*x*exp(exp(6*x)))*csgn(I*x)*csgn(I*exp(exp(6*x)))*x^3-5*I*Pi*csgn(
I*x*exp(exp(6*x)))^3+5*I*Pi*csgn(I*x*exp(exp(6*x)))^2*csgn(I*x)+5*I*Pi*csgn(I*x*exp(exp(6*x)))^2*csgn(I*exp(ex
p(6*x)))-5*I*Pi*csgn(I*x*exp(exp(6*x)))*csgn(I*x)*csgn(I*exp(exp(6*x)))+2*x^3*ln(x)+2*ln(exp(exp(6*x)))*x^3+10
*ln(x)+10*ln(exp(exp(6*x)))-2*x)/(-I*Pi*csgn(I*x*exp(exp(6*x)))^3+I*Pi*csgn(I*x*exp(exp(6*x)))^2*csgn(I*x)+I*P
i*csgn(I*x*exp(exp(6*x)))^2*csgn(I*exp(exp(6*x)))-I*Pi*csgn(I*x*exp(exp(6*x)))*csgn(I*x)*csgn(I*exp(exp(6*x)))
+2*ln(x)+2*ln(exp(exp(6*x)))))

________________________________________________________________________________________

maxima [A]  time = 0.85, size = 18, normalized size = 0.82 \begin {gather*} e^{\left (x^{3} - \frac {x}{e^{\left (6 \, x\right )} + \log \relax (x)} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x*exp(exp(3*x)^2))^2-log(x*exp(exp(3*x)^2))+6*x*exp(3*x)^2+1)*exp(((x^3+5)*log(x*exp(exp(
3*x)^2))-x)/log(x*exp(exp(3*x)^2)))/log(x*exp(exp(3*x)^2))^2,x, algorithm="maxima")

[Out]

e^(x^3 - x/(e^(6*x) + log(x)) + 5)

________________________________________________________________________________________

mupad [B]  time = 0.38, size = 65, normalized size = 2.95 \begin {gather*} x^{\frac {x^3+5}{{\mathrm {e}}^{6\,x}+\ln \relax (x)}}\,{\mathrm {e}}^{-\frac {x}{{\mathrm {e}}^{6\,x}+\ln \relax (x)}}\,{\mathrm {e}}^{\frac {x^3\,{\mathrm {e}}^{6\,x}}{{\mathrm {e}}^{6\,x}+\ln \relax (x)}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{6\,x}}{{\mathrm {e}}^{6\,x}+\ln \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x - log(x*exp(exp(6*x)))*(x^3 + 5))/log(x*exp(exp(6*x))))*(6*x*exp(6*x) - log(x*exp(exp(6*x))) + 3*
x^2*log(x*exp(exp(6*x)))^2 + 1))/log(x*exp(exp(6*x)))^2,x)

[Out]

x^((x^3 + 5)/(exp(6*x) + log(x)))*exp(-x/(exp(6*x) + log(x)))*exp((x^3*exp(6*x))/(exp(6*x) + log(x)))*exp((5*e
xp(6*x))/(exp(6*x) + log(x)))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2*ln(x*exp(exp(3*x)**2))**2-ln(x*exp(exp(3*x)**2))+6*x*exp(3*x)**2+1)*exp(((x**3+5)*ln(x*exp(e
xp(3*x)**2))-x)/ln(x*exp(exp(3*x)**2)))/ln(x*exp(exp(3*x)**2))**2,x)

[Out]

Timed out

________________________________________________________________________________________