3.3.35 \(\int \frac {10 x^5-e^x x^5+e^{e^x} (\frac {1}{x})^{\frac {e^{e^x}}{x^4}} (1+(4-e^x x) \log (\frac {1}{x}))}{x^5} \, dx\)

Optimal. Leaf size=25 \[ -1-e^x-\left (\frac {1}{x}\right )^{\frac {e^{e^x}}{x^4}}+10 x \]

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Rubi [F]  time = 0.61, 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 {10 x^5-e^x x^5+e^{e^x} \left (\frac {1}{x}\right )^{\frac {e^{e^x}}{x^4}} \left (1+\left (4-e^x x\right ) \log \left (\frac {1}{x}\right )\right )}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x^5 - E^x*x^5 + E^E^x*(x^(-1))^(E^E^x/x^4)*(1 + (4 - E^x*x)*Log[x^(-1)]))/x^5,x]

[Out]

-E^x + 10*x - Log[x^(-1)]*Defer[Int][E^(E^x + x)*(x^(-1))^(4 + E^E^x/x^4), x] + Defer[Int][E^E^x*(x^(-1))^(5 +
E^E^x/x^4), x] + 4*Log[x^(-1)]*Defer[Int][E^E^x*(x^(-1))^(5 + E^E^x/x^4), x] - Defer[Int][Defer[Int][E^(E^x +
x)*(x^(-1))^(4 + E^E^x/x^4), x]/x, x] + 4*Defer[Int][Defer[Int][E^E^x*(x^(-1))^(5 + E^E^x/x^4), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (10-e^x-e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \left (-1-4 \log \left (\frac {1}{x}\right )+e^x x \log \left (\frac {1}{x}\right )\right )\right ) \, dx\\ &=10 x-\int e^x \, dx-\int e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \left (-1-4 \log \left (\frac {1}{x}\right )+e^x x \log \left (\frac {1}{x}\right )\right ) \, dx\\ &=-e^x+10 x-\int \left (-e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}}+e^{e^x+x} \left (\frac {1}{x}\right )^{4+\frac {e^{e^x}}{x^4}} \log \left (\frac {1}{x}\right )-4 e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \log \left (\frac {1}{x}\right )\right ) \, dx\\ &=-e^x+10 x+4 \int e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \log \left (\frac {1}{x}\right ) \, dx+\int e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \, dx-\int e^{e^x+x} \left (\frac {1}{x}\right )^{4+\frac {e^{e^x}}{x^4}} \log \left (\frac {1}{x}\right ) \, dx\\ &=-e^x+10 x+4 \int \frac {\int e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \, dx}{x} \, dx-\log \left (\frac {1}{x}\right ) \int e^{e^x+x} \left (\frac {1}{x}\right )^{4+\frac {e^{e^x}}{x^4}} \, dx+\left (4 \log \left (\frac {1}{x}\right )\right ) \int e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \, dx+\int e^{e^x} \left (\frac {1}{x}\right )^{5+\frac {e^{e^x}}{x^4}} \, dx-\int \frac {\int e^{e^x+x} \left (\frac {1}{x}\right )^{4+\frac {e^{e^x}}{x^4}} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 24, normalized size = 0.96 \begin {gather*} -e^x-\left (\frac {1}{x}\right )^{\frac {e^{e^x}}{x^4}}+10 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x^5 - E^x*x^5 + E^E^x*(x^(-1))^(E^E^x/x^4)*(1 + (4 - E^x*x)*Log[x^(-1)]))/x^5,x]

[Out]

-E^x - (x^(-1))^(E^E^x/x^4) + 10*x

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fricas [A]  time = 0.67, size = 21, normalized size = 0.84 \begin {gather*} 10 \, x - \frac {1}{x}^{\frac {e^{\left (e^{x}\right )}}{x^{4}}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x+4)*log(1/x)+1)*exp(exp(x))*exp(log(1/x)*exp(exp(x))/x^4)-x^5*exp(x)+10*x^5)/x^5,x, algo
rithm="fricas")

[Out]

10*x - (1/x)^(e^(e^x)/x^4) - e^x

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giac [A]  time = 0.84, size = 21, normalized size = 0.84 \begin {gather*} 10 \, x - \frac {1}{x^{\frac {e^{\left (e^{x}\right )}}{x^{4}}}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x+4)*log(1/x)+1)*exp(exp(x))*exp(log(1/x)*exp(exp(x))/x^4)-x^5*exp(x)+10*x^5)/x^5,x, algo
rithm="giac")

[Out]

10*x - 1/x^(e^(e^x)/x^4) - e^x

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-{\mathrm e}^{x} x +4\right ) \ln \left (\frac {1}{x}\right )+1\right ) {\mathrm e}^{{\mathrm e}^{x}} {\mathrm e}^{\frac {\ln \left (\frac {1}{x}\right ) {\mathrm e}^{{\mathrm e}^{x}}}{x^{4}}}-x^{5} {\mathrm e}^{x}+10 x^{5}}{x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(x)*x+4)*ln(1/x)+1)*exp(exp(x))*exp(ln(1/x)*exp(exp(x))/x^4)-x^5*exp(x)+10*x^5)/x^5,x)

[Out]

int((((-exp(x)*x+4)*ln(1/x)+1)*exp(exp(x))*exp(ln(1/x)*exp(exp(x))/x^4)-x^5*exp(x)+10*x^5)/x^5,x)

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maxima [A]  time = 0.66, size = 21, normalized size = 0.84 \begin {gather*} 10 \, x - \frac {1}{x^{\frac {e^{\left (e^{x}\right )}}{x^{4}}}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x+4)*log(1/x)+1)*exp(exp(x))*exp(log(1/x)*exp(exp(x))/x^4)-x^5*exp(x)+10*x^5)/x^5,x, algo
rithm="maxima")

[Out]

10*x - 1/x^(e^(e^x)/x^4) - e^x

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mupad [B]  time = 0.39, size = 21, normalized size = 0.84 \begin {gather*} 10\,x-{\mathrm {e}}^x-{\left (\frac {1}{x}\right )}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^5*exp(x) - 10*x^5 + exp((log(1/x)*exp(exp(x)))/x^4)*exp(exp(x))*(log(1/x)*(x*exp(x) - 4) - 1))/x^5,x)

[Out]

10*x - exp(x) - (1/x)^(exp(exp(x))/x^4)

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sympy [A]  time = 0.95, size = 20, normalized size = 0.80 \begin {gather*} 10 x - e^{x} - e^{\frac {e^{e^{x}} \log {\left (\frac {1}{x} \right )}}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x+4)*ln(1/x)+1)*exp(exp(x))*exp(ln(1/x)*exp(exp(x))/x**4)-x**5*exp(x)+10*x**5)/x**5,x)

[Out]

10*x - exp(x) - exp(exp(exp(x))*log(1/x)/x**4)

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