3.63.100 \(\int \frac {e^x (x^4-x^5)+e^{\frac {e^{4 x}}{x^4}} (e^{5 x} (-4+4 x)+e^{4 x} (12 x-12 x^2))}{e^x x^5-3 x^6} \, dx\) [6300]

Optimal. Leaf size=26 \[ 3+e^{\frac {e^{4 x}}{x^4}}-\log \left (3-\frac {e^x}{x}\right ) \]

[Out]

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

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Rubi [F]
time = 0.85, 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 {e^x \left (x^4-x^5\right )+e^{\frac {e^{4 x}}{x^4}} \left (e^{5 x} (-4+4 x)+e^{4 x} \left (12 x-12 x^2\right )\right )}{e^x x^5-3 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(x^4 - x^5) + E^(E^(4*x)/x^4)*(E^(5*x)*(-4 + 4*x) + E^(4*x)*(12*x - 12*x^2)))/(E^x*x^5 - 3*x^6),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x (1-x) \left (-4 e^{\frac {e^{4 x}}{x^4}+4 x}+12 e^{\frac {e^{4 x}}{x^4}+3 x} x+x^4\right )}{e^x x^5-3 x^6} \, dx\\ &=\int \left (\frac {4 e^{\frac {e^{4 x}}{x^4}+4 x} (-1+x)}{x^5}-\frac {e^x (-1+x)}{\left (e^x-3 x\right ) x}\right ) \, dx\\ &=4 \int \frac {e^{\frac {e^{4 x}}{x^4}+4 x} (-1+x)}{x^5} \, dx-\int \frac {e^x (-1+x)}{\left (e^x-3 x\right ) x} \, dx\\ &=4 \int \left (-\frac {e^{\frac {e^{4 x}}{x^4}+4 x}}{x^5}+\frac {e^{\frac {e^{4 x}}{x^4}+4 x}}{x^4}\right ) \, dx-\text {Subst}\left (\int \frac {1}{-3+x} \, dx,x,\frac {e^x}{x}\right )\\ &=-\log \left (3-\frac {e^x}{x}\right )-4 \int \frac {e^{\frac {e^{4 x}}{x^4}+4 x}}{x^5} \, dx+4 \int \frac {e^{\frac {e^{4 x}}{x^4}+4 x}}{x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.34, size = 24, normalized size = 0.92 \begin {gather*} e^{\frac {e^{4 x}}{x^4}}-\log \left (e^x-3 x\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(x^4 - x^5) + E^(E^(4*x)/x^4)*(E^(5*x)*(-4 + 4*x) + E^(4*x)*(12*x - 12*x^2)))/(E^x*x^5 - 3*x^6)
,x]

[Out]

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

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Maple [A]
time = 0.22, size = 22, normalized size = 0.85

method	result	size
risch	\(\ln \left (x \right )-\ln \left ({\mathrm e}^{x}-3 x \right )+{\mathrm e}^{\frac {{\mathrm e}^{4 x}}{x^{4}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x-4)*exp(x)^5+(-12*x^2+12*x)*exp(x)^4)*exp(exp(x)^4/x^4)+(-x^5+x^4)*exp(x))/(x^5*exp(x)-3*x^6),x,meth
od=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 21, normalized size = 0.81 \begin {gather*} e^{\left (\frac {e^{\left (4 \, x\right )}}{x^{4}}\right )} + \log \left (x\right ) - \log \left (-3 \, x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4+4*x)*exp(x)^5+(-12*x^2+12*x)*exp(x)^4)*exp(exp(x)^4/x^4)+(-x^5+x^4)*exp(x))/(x^5*exp(x)-3*x^6)
,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 21, normalized size = 0.81 \begin {gather*} e^{\left (\frac {e^{\left (4 \, x\right )}}{x^{4}}\right )} + \log \left (x\right ) - \log \left (-3 \, x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4+4*x)*exp(x)^5+(-12*x^2+12*x)*exp(x)^4)*exp(exp(x)^4/x^4)+(-x^5+x^4)*exp(x))/(x^5*exp(x)-3*x^6)
,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.17, size = 20, normalized size = 0.77 \begin {gather*} e^{\frac {e^{4 x}}{x^{4}}} + \log {\left (x \right )} - \log {\left (- 3 x + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4+4*x)*exp(x)**5+(-12*x**2+12*x)*exp(x)**4)*exp(exp(x)**4/x**4)+(-x**5+x**4)*exp(x))/(x**5*exp(x
)-3*x**6),x)

[Out]

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

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Giac [A]
time = 0.39, size = 46, normalized size = 1.77 \begin {gather*} -{\left (e^{\left (4 \, x\right )} \log \left (3 \, x - e^{x}\right ) - e^{\left (4 \, x\right )} \log \left (x\right ) - e^{\left (\frac {4 \, x^{5} + e^{\left (4 \, x\right )}}{x^{4}}\right )}\right )} e^{\left (-4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4+4*x)*exp(x)^5+(-12*x^2+12*x)*exp(x)^4)*exp(exp(x)^4/x^4)+(-x^5+x^4)*exp(x))/(x^5*exp(x)-3*x^6)
,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.24, size = 21, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{4\,x}}{x^4}}-\ln \left ({\mathrm {e}}^x-3\,x\right )+\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(4*x)/x^4)*(exp(4*x)*(12*x - 12*x^2) + exp(5*x)*(4*x - 4)) + exp(x)*(x^4 - x^5))/(x^5*exp(x) - 3*x
^6),x)

[Out]

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

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