3.75.59 \(\int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} (2 x+e^{-4 x^2+2 x^3} (-8 x+6 x^2)-2 \log (5)) \, dx\)

Optimal. Leaf size=24 \[ e^{e^{-4+e^{2 (-2+x) x^2}+(x-\log (5))^2}} \]

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Rubi [F]  time = 7.25, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \exp \left (-4+e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)+\log ^2(5)\right ) \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*Log[5]
+ Log[5]^2)*(2*x + E^(-4*x^2 + 2*x^3)*(-8*x + 6*x^2) - 2*Log[5]),x]

[Out]

-2*Log[5]*Defer[Int][E^(E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 -
2*x*Log[5] - 4*(1 - Log[5]^2/4)), x] + 2*Defer[Int][E^(E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2
- 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*Log[5] - 4*(1 - Log[5]^2/4))*x, x] - 8*Defer[Int][E^(E^(-4*x^2 + 2*x^3) +
E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 + 2*(-2 + x)*x^2 - 2*x*Log[5] - 4*(1 - Log[5]
^2/4))*x, x] + 6*Defer[Int][E^(E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2)
+ x^2 + 2*(-2 + x)*x^2 - 2*x*Log[5] - 4*(1 - Log[5]^2/4))*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx\\ &=\int \left (2 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x+2 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x (-4+3 x)-2 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \log (5)\right ) \, dx\\ &=2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx+2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x (-4+3 x) \, dx-(2 \log (5)) \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \, dx\\ &=2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx+2 \int \left (-4 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x+3 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x^2\right ) \, dx-(2 \log (5)) \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \, dx\\ &=2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx+6 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x^2 \, dx-8 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx-(2 \log (5)) \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.56, size = 0, normalized size = 0.00 \begin {gather*} \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*L
og[5] + Log[5]^2)*(2*x + E^(-4*x^2 + 2*x^3)*(-8*x + 6*x^2) - 2*Log[5]),x]

[Out]

Integrate[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*L
og[5] + Log[5]^2)*(2*x + E^(-4*x^2 + 2*x^3)*(-8*x + 6*x^2) - 2*Log[5]), x]

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fricas [A]  time = 0.65, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (e^{\left (x^{2} - 2 \, x \log \relax (5) + \log \relax (5)^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*log(5)+2*x)*exp(exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)*exp(exp(
exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)),x, algorithm="fricas")

[Out]

e^(e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 2 \, {\left ({\left (3 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} + x - \log \relax (5)\right )} e^{\left (x^{2} - 2 \, x \log \relax (5) + \log \relax (5)^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} + e^{\left (x^{2} - 2 \, x \log \relax (5) + \log \relax (5)^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )} - 4\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*log(5)+2*x)*exp(exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)*exp(exp(
exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)),x, algorithm="giac")

[Out]

integrate(2*((3*x^2 - 4*x)*e^(2*x^3 - 4*x^2) + x - log(5))*e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2)
+ e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4) - 4), x)

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maple [A]  time = 0.07, size = 25, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{\left (\frac {1}{25}\right )^{x} {\mathrm e}^{{\mathrm e}^{2 \left (x -2\right ) x^{2}}+\ln \relax (5)^{2}-4+x^{2}}}\)	\(25\)
derivativedivides	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \relax (5)^{2}-2 x \ln \relax (5)+x^{2}-4}}\)	\(29\)
default	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \relax (5)^{2}-2 x \ln \relax (5)+x^{2}-4}}\)	\(29\)
norman	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \relax (5)^{2}-2 x \ln \relax (5)+x^{2}-4}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*ln(5)+2*x)*exp(exp(2*x^3-4*x^2)+ln(5)^2-2*x*ln(5)+x^2-4)*exp(exp(exp(2*x^3
-4*x^2)+ln(5)^2-2*x*ln(5)+x^2-4)),x,method=_RETURNVERBOSE)

[Out]

exp((1/25)^x*exp(exp(2*(x-2)*x^2)+ln(5)^2-4+x^2))

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maxima [A]  time = 1.05, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (e^{\left (x^{2} - 2 \, x \log \relax (5) + \log \relax (5)^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*log(5)+2*x)*exp(exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)*exp(exp(
exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)),x, algorithm="maxima")

[Out]

e^(e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4))

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mupad [B]  time = 5.20, size = 34, normalized size = 1.42 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{{\ln \relax (5)}^2}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{-4\,x^2}}}{5^{2\,x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(exp(exp(2*x^3 - 4*x^2) - 2*x*log(5) + log(5)^2 + x^2 - 4))*exp(exp(2*x^3 - 4*x^2) - 2*x*log(5) + log(
5)^2 + x^2 - 4)*(2*log(5) - 2*x + exp(2*x^3 - 4*x^2)*(8*x - 6*x^2)),x)

[Out]

exp((exp(log(5)^2)*exp(x^2)*exp(-4)*exp(exp(2*x^3)*exp(-4*x^2)))/5^(2*x))

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sympy [A]  time = 1.39, size = 31, normalized size = 1.29 \begin {gather*} e^{e^{x^{2} - 2 x \log {\relax (5 )} + e^{2 x^{3} - 4 x^{2}} - 4 + \log {\relax (5 )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**2-8*x)*exp(2*x**3-4*x**2)-2*ln(5)+2*x)*exp(exp(2*x**3-4*x**2)+ln(5)**2-2*x*ln(5)+x**2-4)*exp(
exp(exp(2*x**3-4*x**2)+ln(5)**2-2*x*ln(5)+x**2-4)),x)

[Out]

exp(exp(x**2 - 2*x*log(5) + exp(2*x**3 - 4*x**2) - 4 + log(5)**2))

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