\(\int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x))}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx\) [2898]

Optimal result

Integrand size = 179, antiderivative size = 22 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=\log \left (e^{e^{(x+x \log (x))^4}}+3 x-x^3\right ) \]

[Out]

ln(exp(exp((x*ln(x)+x)^4))-x^3+3*x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).

Time = 0.32 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {6816} \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=\log \left (\exp \left (x^{4 x^4} \exp \left (x^4+x^4 \log ^4(x)+4 x^4 \log ^3(x)+6 x^4 \log ^2(x)\right )\right )-x^3+3 x\right ) \]

[In]

Int[(3 - 3*x^2 + E^(E^(x^4 + 4*x^4*Log[x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4) + x^4 + 4*x^4*Log[
x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4)*(8*x^3 + 28*x^3*Log[x] + 36*x^3*Log[x]^2 + 20*x^3*Log[x]^
3 + 4*x^3*Log[x]^4))/(E^E^(x^4 + 4*x^4*Log[x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4) + 3*x - x^3),x
]

[Out]

Log[E^(E^(x^4 + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4)*x^(4*x^4)) + 3*x - x^3]

Rule 6816

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

Rubi steps \begin{align*} \text {integral}& = \log \left (\exp \left (\exp \left (x^4+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)\right ) x^{4 x^4}\right )+3 x-x^3\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).

Time = 0.57 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=\log \left (e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3\right ) \]

[In]

Integrate[(3 - 3*x^2 + E^(E^(x^4 + 4*x^4*Log[x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4) + x^4 + 4*x^
4*Log[x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4)*(8*x^3 + 28*x^3*Log[x] + 36*x^3*Log[x]^2 + 20*x^3*L
og[x]^3 + 4*x^3*Log[x]^4))/(E^E^(x^4 + 4*x^4*Log[x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4) + 3*x -
x^3),x]

[Out]

Log[E^E^(x^4 + 4*x^4*Log[x] + 6*x^4*Log[x]^2 + 4*x^4*Log[x]^3 + x^4*Log[x]^4) + 3*x - x^3]

Maple [A] (verified)

Time = 7.69 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77

method	result	size
parallelrisch	\(\ln \left (x^{3}-3 x -{\mathrm e}^{{\mathrm e}^{x^{4} \left (\ln \left (x \right )^{4}+4 \ln \left (x \right )^{3}+6 \ln \left (x \right )^{2}+4 \ln \left (x \right )+1\right )}}\right )\)	\(39\)
risch	\(\ln \left ({\mathrm e}^{x^{4 x^{4}} {\mathrm e}^{x^{4} \left (\ln \left (x \right )^{4}+4 \ln \left (x \right )^{3}+6 \ln \left (x \right )^{2}+1\right )}}-x^{3}+3 x \right )\)	\(43\)
derivativedivides	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x^{4} \ln \left (x \right )^{4}+4 x^{4} \ln \left (x \right )^{3}+6 x^{4} \ln \left (x \right )^{2}+4 x^{4} \ln \left (x \right )+x^{4}}}-x^{3}+3 x \right )\)	\(50\)
default	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x^{4} \ln \left (x \right )^{4}+4 x^{4} \ln \left (x \right )^{3}+6 x^{4} \ln \left (x \right )^{2}+4 x^{4} \ln \left (x \right )+x^{4}}}-x^{3}+3 x \right )\)	\(50\)

[In]

int(((4*x^3*ln(x)^4+20*x^3*ln(x)^3+36*x^3*ln(x)^2+28*x^3*ln(x)+8*x^3)*exp(x^4*ln(x)^4+4*x^4*ln(x)^3+6*x^4*ln(x
)^2+4*x^4*ln(x)+x^4)*exp(exp(x^4*ln(x)^4+4*x^4*ln(x)^3+6*x^4*ln(x)^2+4*x^4*ln(x)+x^4))-3*x^2+3)/(exp(exp(x^4*l
n(x)^4+4*x^4*ln(x)^3+6*x^4*ln(x)^2+4*x^4*ln(x)+x^4))-x^3+3*x),x,method=_RETURNVERBOSE)

[Out]

ln(x^3-3*x-exp(exp(x^4*(ln(x)^4+4*ln(x)^3+6*ln(x)^2+4*ln(x)+1))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (20) = 40\).

Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 7.50 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=-x^{4} \log \left (x\right )^{4} - 4 \, x^{4} \log \left (x\right )^{3} - 6 \, x^{4} \log \left (x\right )^{2} - 4 \, x^{4} \log \left (x\right ) - x^{4} + \log \left (-{\left (x^{3} - 3 \, x\right )} e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4}\right )} + e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4} + e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4}\right )}\right )}\right ) \]

[In]

integrate(((4*x^3*log(x)^4+20*x^3*log(x)^3+36*x^3*log(x)^2+28*x^3*log(x)+8*x^3)*exp(x^4*log(x)^4+4*x^4*log(x)^
3+6*x^4*log(x)^2+4*x^4*log(x)+x^4)*exp(exp(x^4*log(x)^4+4*x^4*log(x)^3+6*x^4*log(x)^2+4*x^4*log(x)+x^4))-3*x^2
+3)/(exp(exp(x^4*log(x)^4+4*x^4*log(x)^3+6*x^4*log(x)^2+4*x^4*log(x)+x^4))-x^3+3*x),x, algorithm="fricas")

[Out]

-x^4*log(x)^4 - 4*x^4*log(x)^3 - 6*x^4*log(x)^2 - 4*x^4*log(x) - x^4 + log(-(x^3 - 3*x)*e^(x^4*log(x)^4 + 4*x^
4*log(x)^3 + 6*x^4*log(x)^2 + 4*x^4*log(x) + x^4) + e^(x^4*log(x)^4 + 4*x^4*log(x)^3 + 6*x^4*log(x)^2 + 4*x^4*
log(x) + x^4 + e^(x^4*log(x)^4 + 4*x^4*log(x)^3 + 6*x^4*log(x)^2 + 4*x^4*log(x) + x^4)))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).

Time = 0.43 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=\log {\left (- x^{3} + 3 x + e^{e^{x^{4} \log {\left (x \right )}^{4} + 4 x^{4} \log {\left (x \right )}^{3} + 6 x^{4} \log {\left (x \right )}^{2} + 4 x^{4} \log {\left (x \right )} + x^{4}}} \right )} \]

[In]

integrate(((4*x**3*ln(x)**4+20*x**3*ln(x)**3+36*x**3*ln(x)**2+28*x**3*ln(x)+8*x**3)*exp(x**4*ln(x)**4+4*x**4*l
n(x)**3+6*x**4*ln(x)**2+4*x**4*ln(x)+x**4)*exp(exp(x**4*ln(x)**4+4*x**4*ln(x)**3+6*x**4*ln(x)**2+4*x**4*ln(x)+
x**4))-3*x**2+3)/(exp(exp(x**4*ln(x)**4+4*x**4*ln(x)**3+6*x**4*ln(x)**2+4*x**4*ln(x)+x**4))-x**3+3*x),x)

[Out]

log(-x**3 + 3*x + exp(exp(x**4*log(x)**4 + 4*x**4*log(x)**3 + 6*x**4*log(x)**2 + 4*x**4*log(x) + x**4)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=\log \left (-x^{3} + 3 \, x + e^{\left (e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4}\right )}\right )}\right ) \]

[In]

integrate(((4*x^3*log(x)^4+20*x^3*log(x)^3+36*x^3*log(x)^2+28*x^3*log(x)+8*x^3)*exp(x^4*log(x)^4+4*x^4*log(x)^
3+6*x^4*log(x)^2+4*x^4*log(x)+x^4)*exp(exp(x^4*log(x)^4+4*x^4*log(x)^3+6*x^4*log(x)^2+4*x^4*log(x)+x^4))-3*x^2
+3)/(exp(exp(x^4*log(x)^4+4*x^4*log(x)^3+6*x^4*log(x)^2+4*x^4*log(x)+x^4))-x^3+3*x),x, algorithm="maxima")

[Out]

log(-x^3 + 3*x + e^(e^(x^4*log(x)^4 + 4*x^4*log(x)^3 + 6*x^4*log(x)^2 + 4*x^4*log(x) + x^4)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (20) = 40\).

Time = 2.34 (sec) , antiderivative size = 202, normalized size of antiderivative = 9.18 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=-x^{4} \log \left (x\right )^{4} - 4 \, x^{4} \log \left (x\right )^{3} - 6 \, x^{4} \log \left (x\right )^{2} - 4 \, x^{4} \log \left (x\right ) - x^{4} + \log \left (-x^{3} e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4}\right )} + 3 \, x e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4}\right )} + e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4} + e^{\left (x^{4} \log \left (x\right )^{4} + 4 \, x^{4} \log \left (x\right )^{3} + 6 \, x^{4} \log \left (x\right )^{2} + 4 \, x^{4} \log \left (x\right ) + x^{4}\right )}\right )}\right ) \]

[In]

integrate(((4*x^3*log(x)^4+20*x^3*log(x)^3+36*x^3*log(x)^2+28*x^3*log(x)+8*x^3)*exp(x^4*log(x)^4+4*x^4*log(x)^
3+6*x^4*log(x)^2+4*x^4*log(x)+x^4)*exp(exp(x^4*log(x)^4+4*x^4*log(x)^3+6*x^4*log(x)^2+4*x^4*log(x)+x^4))-3*x^2
+3)/(exp(exp(x^4*log(x)^4+4*x^4*log(x)^3+6*x^4*log(x)^2+4*x^4*log(x)+x^4))-x^3+3*x),x, algorithm="giac")

[Out]

-x^4*log(x)^4 - 4*x^4*log(x)^3 - 6*x^4*log(x)^2 - 4*x^4*log(x) - x^4 + log(-x^3*e^(x^4*log(x)^4 + 4*x^4*log(x)
^3 + 6*x^4*log(x)^2 + 4*x^4*log(x) + x^4) + 3*x*e^(x^4*log(x)^4 + 4*x^4*log(x)^3 + 6*x^4*log(x)^2 + 4*x^4*log(
x) + x^4) + e^(x^4*log(x)^4 + 4*x^4*log(x)^3 + 6*x^4*log(x)^2 + 4*x^4*log(x) + x^4 + e^(x^4*log(x)^4 + 4*x^4*l
og(x)^3 + 6*x^4*log(x)^2 + 4*x^4*log(x) + x^4)))

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {3-3 x^2+e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}+x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)} \left (8 x^3+28 x^3 \log (x)+36 x^3 \log ^2(x)+20 x^3 \log ^3(x)+4 x^3 \log ^4(x)\right )}{e^{e^{x^4+4 x^4 \log (x)+6 x^4 \log ^2(x)+4 x^4 \log ^3(x)+x^4 \log ^4(x)}}+3 x-x^3} \, dx=\ln \left (3\,x+{\mathrm {e}}^{x^{4\,x^4}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{x^4\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{4\,x^4\,{\ln \left (x\right )}^3}\,{\mathrm {e}}^{6\,x^4\,{\ln \left (x\right )}^2}}-x^3\right ) \]

[In]

int((exp(4*x^4*log(x) + 6*x^4*log(x)^2 + 4*x^4*log(x)^3 + x^4*log(x)^4 + x^4)*exp(exp(4*x^4*log(x) + 6*x^4*log
(x)^2 + 4*x^4*log(x)^3 + x^4*log(x)^4 + x^4))*(28*x^3*log(x) + 36*x^3*log(x)^2 + 20*x^3*log(x)^3 + 4*x^3*log(x
)^4 + 8*x^3) - 3*x^2 + 3)/(3*x + exp(exp(4*x^4*log(x) + 6*x^4*log(x)^2 + 4*x^4*log(x)^3 + x^4*log(x)^4 + x^4))
- x^3),x)

[Out]

log(3*x + exp(x^(4*x^4)*exp(x^4)*exp(x^4*log(x)^4)*exp(4*x^4*log(x)^3)*exp(6*x^4*log(x)^2)) - x^3)