3.97.15 \(\int \frac {-4+4 x^2+e^{\frac {1}{2} (-12+3 e^{e^x}+6 x)} (6 x+3 e^{e^x+x} x)}{2 e^{\frac {1}{2} (-12+3 e^{e^x}+6 x)} x+2 x^3-2 x \log (\frac {4 x^2}{25})} \, dx\)

Optimal. Leaf size=31 \[ \log \left (e^{3 \left (-2+\frac {e^{e^x}}{2}+x\right )}+x^2-\log \left (\frac {4 x^2}{25}\right )\right ) \]

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Rubi [F]  time = 3.76, 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 \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 + 4*x^2 + E^((-12 + 3*E^E^x + 6*x)/2)*(6*x + 3*E^(E^x + x)*x))/(2*E^((-12 + 3*E^E^x + 6*x)/2)*x + 2*x^
3 - 2*x*Log[(4*x^2)/25]),x]

[Out]

(3*E^E^x)/2 + 3*x - 2*E^6*Defer[Int][1/(x*(E^((3*E^E^x)/2 + 3*x) + E^6*x^2 - E^6*Log[(4*x^2)/25])), x] + 2*E^6
*Defer[Int][x/(E^((3*E^E^x)/2 + 3*x) + E^6*x^2 - E^6*Log[(4*x^2)/25]), x] - 3*E^6*Defer[Int][x^2/(E^((3*E^E^x)
/2 + 3*x) + E^6*x^2 - E^6*Log[(4*x^2)/25]), x] - (3*E^6*Defer[Int][(E^(E^x + x)*x^2)/(E^((3*E^E^x)/2 + 3*x) +
E^6*x^2 - E^6*Log[(4*x^2)/25]), x])/2 + (3*E^6*Defer[Int][(E^(E^x + x)*Log[(4*x^2)/25])/(E^((3*E^E^x)/2 + 3*x)
+ E^6*x^2 - E^6*Log[(4*x^2)/25]), x])/2 - 3*E^6*Defer[Int][Log[(4*x^2)/25]/(-E^((3*E^E^x)/2 + 3*x) - E^6*x^2
+ E^6*Log[(4*x^2)/25]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3+\frac {3 e^{e^x+x}}{2}-\frac {e^6 \left (4-4 x^2+6 x^3+3 e^{e^x+x} x^3-6 x \log \left (\frac {4 x^2}{25}\right )-3 e^{e^x+x} x \log \left (\frac {4 x^2}{25}\right )\right )}{2 x \left (e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )\right )}\right ) \, dx\\ &=3 x+\frac {3}{2} \int e^{e^x+x} \, dx-\frac {1}{2} e^6 \int \frac {4-4 x^2+6 x^3+3 e^{e^x+x} x^3-6 x \log \left (\frac {4 x^2}{25}\right )-3 e^{e^x+x} x \log \left (\frac {4 x^2}{25}\right )}{x \left (e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )\right )} \, dx\\ &=3 x+\frac {3}{2} \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )-\frac {1}{2} e^6 \int \left (\frac {4}{x \left (e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )\right )}-\frac {4 x}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )}+\frac {6 x^2}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )}+\frac {3 e^{e^x+x} x^2}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )}-\frac {3 e^{e^x+x} \log \left (\frac {4 x^2}{25}\right )}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )}+\frac {6 \log \left (\frac {4 x^2}{25}\right )}{-e^{\frac {3 e^{e^x}}{2}+3 x}-e^6 x^2+e^6 \log \left (\frac {4 x^2}{25}\right )}\right ) \, dx\\ &=\frac {3 e^{e^x}}{2}+3 x-\frac {1}{2} \left (3 e^6\right ) \int \frac {e^{e^x+x} x^2}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )} \, dx+\frac {1}{2} \left (3 e^6\right ) \int \frac {e^{e^x+x} \log \left (\frac {4 x^2}{25}\right )}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )} \, dx-\left (2 e^6\right ) \int \frac {1}{x \left (e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )\right )} \, dx+\left (2 e^6\right ) \int \frac {x}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )} \, dx-\left (3 e^6\right ) \int \frac {x^2}{e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )} \, dx-\left (3 e^6\right ) \int \frac {\log \left (\frac {4 x^2}{25}\right )}{-e^{\frac {3 e^{e^x}}{2}+3 x}-e^6 x^2+e^6 \log \left (\frac {4 x^2}{25}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.86, size = 37, normalized size = 1.19 \begin {gather*} \log \left (e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 4*x^2 + E^((-12 + 3*E^E^x + 6*x)/2)*(6*x + 3*E^(E^x + x)*x))/(2*E^((-12 + 3*E^E^x + 6*x)/2)*x
+ 2*x^3 - 2*x*Log[(4*x^2)/25]),x]

[Out]

Log[E^((3*E^E^x)/2 + 3*x) + E^6*x^2 - E^6*Log[(4*x^2)/25]]

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fricas [A]  time = 0.65, size = 33, normalized size = 1.06 \begin {gather*} \log \left (x^{2} + e^{\left (\frac {3}{2} \, {\left (2 \, {\left (x - 2\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - \log \left (\frac {4}{25} \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4)/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*
x*log(4/25*x^2)+2*x^3),x, algorithm="fricas")

[Out]

log(x^2 + e^(3/2*(2*(x - 2)*e^x + e^(x + e^x))*e^(-x)) - log(4/25*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, x^{2} + 3 \, {\left (x e^{\left (x + e^{x}\right )} + 2 \, x\right )} e^{\left (3 \, x + \frac {3}{2} \, e^{\left (e^{x}\right )} - 6\right )} - 4}{2 \, {\left (x^{3} + x e^{\left (3 \, x + \frac {3}{2} \, e^{\left (e^{x}\right )} - 6\right )} - x \log \left (\frac {4}{25} \, x^{2}\right )\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4)/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*
x*log(4/25*x^2)+2*x^3),x, algorithm="giac")

[Out]

integrate(1/2*(4*x^2 + 3*(x*e^(x + e^x) + 2*x)*e^(3*x + 3/2*e^(e^x) - 6) - 4)/(x^3 + x*e^(3*x + 3/2*e^(e^x) -
6) - x*log(4/25*x^2)), x)

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maple [C]  time = 0.29, size = 85, normalized size = 2.74

method	result	size
risch	\(6+\ln \left ({\mathrm e}^{\frac {3 \,{\mathrm e}^{{\mathrm e}^{x}}}{2}+3 x -6}+\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i x^{2}+4 i \ln \relax (2)-4 i \ln \relax (5)+4 i \ln \relax (x )\right )}{2}\right )\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4)/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*x*ln(4
/25*x^2)+2*x^3),x,method=_RETURNVERBOSE)

[Out]

6+ln(exp(3/2*exp(exp(x))+3*x-6)+1/2*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^
3-2*I*x^2+4*I*ln(2)-4*I*ln(5)+4*I*ln(x)))

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maxima [A]  time = 0.51, size = 44, normalized size = 1.42 \begin {gather*} 3 \, x + \log \left ({\left (x^{2} e^{6} + 2 \, {\left (\log \relax (5) - \log \relax (2)\right )} e^{6} - 2 \, e^{6} \log \relax (x) + e^{\left (3 \, x + \frac {3}{2} \, e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-3 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4)/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*
x*log(4/25*x^2)+2*x^3),x, algorithm="maxima")

[Out]

3*x + log((x^2*e^6 + 2*(log(5) - log(2))*e^6 - 2*e^6*log(x) + e^(3*x + 3/2*e^(e^x)))*e^(-3*x))

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mupad [B]  time = 5.84, size = 26, normalized size = 0.84 \begin {gather*} \ln \left (x^2-\ln \left (\frac {4\,x^2}{25}\right )+{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-6}\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}\right )}^{3/2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x + (3*exp(exp(x)))/2 - 6)*(6*x + 3*x*exp(exp(x))*exp(x)) + 4*x^2 - 4)/(2*x*exp(3*x + (3*exp(exp(x)
))/2 - 6) - 2*x*log((4*x^2)/25) + 2*x^3),x)

[Out]

log(x^2 - log((4*x^2)/25) + exp(3*x)*exp(-6)*exp(exp(exp(x)))^(3/2))

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sympy [A]  time = 0.61, size = 27, normalized size = 0.87 \begin {gather*} \log {\left (x^{2} + e^{3 x + \frac {3 e^{e^{x}}}{2} - 6} - \log {\left (\frac {4 x^{2}}{25} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x**2-4)/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2
*x*ln(4/25*x**2)+2*x**3),x)

[Out]

log(x**2 + exp(3*x + 3*exp(exp(x))/2 - 6) - log(4*x**2/25))

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