3.85.78 \(\int \frac {-4+3 x+e^{e^x+x} x}{-3 x+e^{e^x} x+3 x^2-x \log (x^4)} \, dx\)

Optimal. Leaf size=17 \[ \log \left (3-e^{e^x}-3 x+\log \left (x^4\right )\right ) \]

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Rubi [F]  time = 0.74, 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+3 x+e^{e^x+x} x}{-3 x+e^{e^x} x+3 x^2-x \log \left (x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 17, normalized size = 1.00 \begin {gather*} \log \left (3-e^{e^x}-3 x+\log \left (x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[3 - E^E^x - 3*x + Log[x^4]]

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fricas [A]  time = 1.15, size = 26, normalized size = 1.53 \begin {gather*} -x + \log \left (3 \, {\left (x - 1\right )} e^{x} - e^{x} \log \left (x^{4}\right ) + e^{\left (x + e^{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(3*(x - 1)*e^x - e^x*log(x^4) + e^(x + e^x))

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giac [A]  time = 0.16, size = 28, normalized size = 1.65 \begin {gather*} -x + \log \left (3 \, x e^{x} - e^{x} \log \left (x^{4}\right ) + e^{\left (x + e^{x}\right )} - 3 \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.15, size = 201, normalized size = 11.82

method	result	size
risch	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\pi \mathrm {csgn}\left (i x^{3}\right )^{3}-\pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\pi \mathrm {csgn}\left (i x^{4}\right )^{3}-6 i x +8 i \ln \relax (x )+6 i\right )}{2}\right )\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(exp(exp(x))+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)*csgn(I*x^2)*csgn(I*
x^3)-Pi*csgn(I*x)*csgn(I*x^3)^2+Pi*csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)-Pi*csgn(I*x)*csgn(I*x^4)^2+Pi*csgn(I*x^2)
^3-Pi*csgn(I*x^2)*csgn(I*x^3)^2+Pi*csgn(I*x^3)^3-Pi*csgn(I*x^3)*csgn(I*x^4)^2+Pi*csgn(I*x^4)^3-6*I*x+8*I*ln(x)
+6*I))

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maxima [A]  time = 0.38, size = 13, normalized size = 0.76 \begin {gather*} \log \left (3 \, x + e^{\left (e^{x}\right )} - 4 \, \log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x + e^(e^x) - 4*log(x) - 3)

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mupad [B]  time = 5.40, size = 15, normalized size = 0.88 \begin {gather*} \ln \left (3\,x+{\mathrm {e}}^{{\mathrm {e}}^x}-\ln \left (x^4\right )-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x + exp(exp(x)) - log(x^4) - 3)

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sympy [A]  time = 0.40, size = 15, normalized size = 0.88 \begin {gather*} \log {\left (3 x + e^{e^{x}} - \log {\left (x^{4} \right )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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