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

Optimal. Leaf size=22 \[ \log \left (e^{e^{(x+x \log (x))^4}}+3 x-x^3\right ) \]

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Rubi [B]  time = 0.53, antiderivative size = 52, normalized size of antiderivative = 2.36, number of steps used = 1, number of rules used = 1, integrand size = 179, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {6684} \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[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 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} &=\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 {aligned} \end {gather*}

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Mathematica [B]  time = 0.61, size = 51, normalized size = 2.32 \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[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]

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fricas [B]  time = 1.29, size = 165, normalized size = 7.50 \begin {gather*} -x^{4} \log \relax (x)^{4} - 4 \, x^{4} \log \relax (x)^{3} - 6 \, x^{4} \log \relax (x)^{2} - 4 \, x^{4} \log \relax (x) - x^{4} + \log \left (-{\left (x^{3} - 3 \, x\right )} e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4}\right )} + e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4} + e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4}\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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giac [B]  time = 4.78, size = 202, normalized size = 9.18 \begin {gather*} -x^{4} \log \relax (x)^{4} - 4 \, x^{4} \log \relax (x)^{3} - 6 \, x^{4} \log \relax (x)^{2} - 4 \, x^{4} \log \relax (x) - x^{4} + \log \left (-x^{3} e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4}\right )} + 3 \, x e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4}\right )} + e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4} + e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4}\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [B]  time = 0.03, size = 43, normalized size = 1.95

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

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [B]  time = 0.74, size = 49, normalized size = 2.23 \begin {gather*} \log \left (-x^{3} + 3 \, x + e^{\left (e^{\left (x^{4} \log \relax (x)^{4} + 4 \, x^{4} \log \relax (x)^{3} + 6 \, x^{4} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x) + x^{4}\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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mupad [B]  time = 1.95, size = 52, normalized size = 2.36 \begin {gather*} \ln \left (3\,x+{\mathrm {e}}^{x^{4\,x^4}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{x^4\,{\ln \relax (x)}^4}\,{\mathrm {e}}^{4\,x^4\,{\ln \relax (x)}^3}\,{\mathrm {e}}^{6\,x^4\,{\ln \relax (x)}^2}}-x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.92, size = 51, normalized size = 2.32 \begin {gather*} \log {\left (- x^{3} + 3 x + e^{e^{x^{4} \log {\relax (x )}^{4} + 4 x^{4} \log {\relax (x )}^{3} + 6 x^{4} \log {\relax (x )}^{2} + 4 x^{4} \log {\relax (x )} + x^{4}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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