3.90.75 \(\int \frac {e^{4-x} (4+(-4-2 x) \log (\frac {4}{x^2}))}{5 x^3 \log ^2(\frac {4}{x^2})} \, dx\)

Optimal. Leaf size=22 \[ \frac {2 e^{4-x}}{5 x^2 \log \left (\frac {4}{x^2}\right )} \]

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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^{4-x} \left (4+(-4-2 x) \log \left (\frac {4}{x^2}\right )\right )}{5 x^3 \log ^2\left (\frac {4}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 e^{4-x}}{5 x^2 \log \left (\frac {4}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*E^(4 - x))/(5*x^2*Log[4/x^2])

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fricas [A]  time = 0.49, size = 19, normalized size = 0.86 \begin {gather*} \frac {2 \, e^{\left (-x + 4\right )}}{5 \, x^{2} \log \left (\frac {4}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-2*x-4)*log(4/x^2)+4)/x^3/exp(x-4)/log(4/x^2)^2,x, algorithm="fricas")

[Out]

2/5*e^(-x + 4)/(x^2*log(4/x^2))

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giac [A]  time = 0.21, size = 19, normalized size = 0.86 \begin {gather*} \frac {2 \, e^{\left (-x + 4\right )}}{5 \, x^{2} \log \left (\frac {4}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-2*x-4)*log(4/x^2)+4)/x^3/exp(x-4)/log(4/x^2)^2,x, algorithm="giac")

[Out]

2/5*e^(-x + 4)/(x^2*log(4/x^2))

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maple [C]  time = 0.12, size = 72, normalized size = 3.27

method	result	size
risch	\(\frac {4 i {\mathrm e}^{-x +4}}{5 x^{2} \left (-4 i \ln \relax (x )-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (2)\right )}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((-2*x-4)*ln(4/x^2)+4)/x^3/exp(x-4)/ln(4/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

4/5*I/x^2*exp(-x+4)/(-4*I*ln(x)-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+4*I*l
n(2))

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maxima [A]  time = 0.48, size = 24, normalized size = 1.09 \begin {gather*} \frac {e^{\left (-x + 4\right )}}{5 \, {\left (x^{2} \log \relax (2) - x^{2} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-2*x-4)*log(4/x^2)+4)/x^3/exp(x-4)/log(4/x^2)^2,x, algorithm="maxima")

[Out]

1/5*e^(-x + 4)/(x^2*log(2) - x^2*log(x))

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mupad [B]  time = 7.82, size = 19, normalized size = 0.86 \begin {gather*} \frac {2\,{\mathrm {e}}^{4-x}}{5\,x^2\,\ln \left (\frac {4}{x^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*exp(4 - x))/(5*x^2*log(4/x^2))

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sympy [A]  time = 0.29, size = 17, normalized size = 0.77 \begin {gather*} \frac {2 e^{4 - x}}{5 x^{2} \log {\left (\frac {4}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((-2*x-4)*ln(4/x**2)+4)/x**3/exp(x-4)/ln(4/x**2)**2,x)

[Out]

2*exp(4 - x)/(5*x**2*log(4/x**2))

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