3.18.15 $\int \frac{e^{\frac{36+12\log (\frac{4}{x})}{x\log (e^{-e^{4}x}x)}}(-36+36e^{4}x+(-12+12e^{4}x)\log (\frac{4}{x})+(-48-12\log (\frac{4}{x}))\log (e^{-e^{4}x}x))}{x^2{\log }^2(e^{-e^{4}x}x)}dx$

Optimal. Leaf size=28 $$e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}$$

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Rubi [F]  time = 8.11, 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 = {} $$\begin{array}{c}\int \frac{e^{\frac{36+12\log \left(\frac{4}{x}\right)}{x\log \left(e^{-e^{4}x}x\right)}}(-36+36e^{4}x+(-12+12e^{4}x)\log \left(\frac{4}{x}\right)+(-48-12\log \left(\frac{4}{x}\right))\log \left(e^{-e^{4}x}x\right))}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^((36 + 12*Log[4/x])/(x*Log[x/E^(E^4*x)]))*(-36 + 36*E^4*x + (-12 + 12*E^4*x)*Log[4/x] + (-48 - 12*Log[4
/x])*Log[x/E^(E^4*x)]))/(x^2*Log[x/E^(E^4*x)]^2),x]

[Out]

-36*Defer[Int][E^((12*(3 + Log[4/x]))/(x*Log[x/E^(E^4*x)]))/(x^2*Log[x/E^(E^4*x)]^2), x] + 36*Defer[Int][E^(4
+ (12*(3 + Log[4/x]))/(x*Log[x/E^(E^4*x)]))/(x*Log[x/E^(E^4*x)]^2), x] - 12*Defer[Int][(E^((12*(3 + Log[4/x]))
/(x*Log[x/E^(E^4*x)]))*Log[4/x])/(x^2*Log[x/E^(E^4*x)]^2), x] + 12*Defer[Int][(E^(4 + (12*(3 + Log[4/x]))/(x*L
og[x/E^(E^4*x)]))*Log[4/x])/(x*Log[x/E^(E^4*x)]^2), x] - 48*Defer[Int][E^((12*(3 + Log[4/x]))/(x*Log[x/E^(E^4*
x)]))/(x^2*Log[x/E^(E^4*x)]), x] - 12*Defer[Int][(E^((12*(3 + Log[4/x]))/(x*Log[x/E^(E^4*x)]))*Log[4/x])/(x^2*
Log[x/E^(E^4*x)]), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(-36+36e^{4}x+(-12+12e^{4}x)\log \left(\frac{4}{x}\right)+(-48-12\log \left(\frac{4}{x}\right))\log \left(e^{-e^{4}x}x\right))}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}dx\\ & =\int (\frac{12e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(-1+e^{4}x)(3+\log \left(\frac{4}{x}\right))}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}-\frac{12e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(4+\log \left(\frac{4}{x}\right))}{x^2\log \left(e^{-e^{4}x}x\right)})dx\\ & =12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(-1+e^{4}x)(3+\log \left(\frac{4}{x}\right))}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}dx-12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(4+\log \left(\frac{4}{x}\right))}{x^2\log \left(e^{-e^{4}x}x\right)}dx\\ & =12\int (\frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(-3-\log \left(\frac{4}{x}\right))}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}+\frac{e^{4+\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(3+\log \left(\frac{4}{x}\right))}{x{\log }^2\left(e^{-e^{4}x}x\right)})dx-12\int (\frac{4e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x^2\log \left(e^{-e^{4}x}x\right)}+\frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x^2\log \left(e^{-e^{4}x}x\right)})dx\\ & =12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(-3-\log \left(\frac{4}{x}\right))}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}dx+12\int \frac{e^{4+\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}(3+\log \left(\frac{4}{x}\right))}{x{\log }^2\left(e^{-e^{4}x}x\right)}dx-12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x^2\log \left(e^{-e^{4}x}x\right)}dx-48\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x^2\log \left(e^{-e^{4}x}x\right)}dx\\ & =12\int (-\frac{3e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}-\frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x^2{\log }^2\left(e^{-e^{4}x}x\right)})dx+12\int (\frac{3e^{4+\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x{\log }^2\left(e^{-e^{4}x}x\right)}+\frac{e^{4+\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x{\log }^2\left(e^{-e^{4}x}x\right)})dx-12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x^2\log \left(e^{-e^{4}x}x\right)}dx-48\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x^2\log \left(e^{-e^{4}x}x\right)}dx\\ & =-(12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}dx)+12\int \frac{e^{4+\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x{\log }^2\left(e^{-e^{4}x}x\right)}dx-12\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\log \left(\frac{4}{x}\right)}{x^2\log \left(e^{-e^{4}x}x\right)}dx-36\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x^2{\log }^2\left(e^{-e^{4}x}x\right)}dx+36\int \frac{e^{4+\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x{\log }^2\left(e^{-e^{4}x}x\right)}dx-48\int \frac{e^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}}{x^2\log \left(e^{-e^{4}x}x\right)}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.12, size = 28, normalized size = 1.00 $$\begin{array}{c}{e}^{\frac{12(3+\log \left(\frac{4}{x}\right))}{x\log \left(e^{-e^{4}x}x\right)}}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^((36 + 12*Log[4/x])/(x*Log[x/E^(E^4*x)]))*(-36 + 36*E^4*x + (-12 + 12*E^4*x)*Log[4/x] + (-48 - 12
*Log[4/x])*Log[x/E^(E^4*x)]))/(x^2*Log[x/E^(E^4*x)]^2),x]

[Out]

E^((12*(3 + Log[4/x]))/(x*Log[x/E^(E^4*x)]))

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fricas [A]  time = 0.69, size = 33, normalized size = 1.18 $$\begin{array}{c}{e}^{(-\frac{12(\log \left(\frac{4}{x}\right)+3)}{x^2{e}^4-2x\log (2)+x\log \left(\frac{4}{x}\right)})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*log(4/x)-48)*log(x/exp(x*exp(4)))+(12*x*exp(4)-12)*log(4/x)+36*x*exp(4)-36)*exp((12*log(4/x)+3
6)/x/log(x/exp(x*exp(4))))/x^2/log(x/exp(x*exp(4)))^2,x, algorithm="fricas")

[Out]

e^(-12*(log(4/x) + 3)/(x^2*e^4 - 2*x*log(2) + x*log(4/x)))

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giac [A]  time = 1.29, size = 40, normalized size = 1.43 $$\begin{array}{c}{e}^{(\frac{12\log \left(\frac{4}{x}\right)}{x\log \left(x{e}^{(-x{e}^4)}\right)}+\frac{36}{x\log \left(x{e}^{(-x{e}^4)}\right)})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*log(4/x)-48)*log(x/exp(x*exp(4)))+(12*x*exp(4)-12)*log(4/x)+36*x*exp(4)-36)*exp((12*log(4/x)+3
6)/x/log(x/exp(x*exp(4))))/x^2/log(x/exp(x*exp(4)))^2,x, algorithm="giac")

[Out]

e^(12*log(4/x)/(x*log(x*e^(-x*e^4))) + 36/(x*log(x*e^(-x*e^4))))

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maple [C]  time = 0.77, size = 128, normalized size = 4.57

method	result	size
risch	${\mathrm{e}}^{\frac{48\ln (2)-24\ln (x)+72}{x(-i\mathrm{csgn}{\left(ix{\mathrm{e}}^{-x{\mathrm{e}}^4}\right)}^3\pi +i\mathrm{csgn}{\left(ix{\mathrm{e}}^{-x{\mathrm{e}}^4}\right)}^2\mathrm{csgn}\left(ix\right)\pi +i\mathrm{csgn}{\left(ix{\mathrm{e}}^{-x{\mathrm{e}}^4}\right)}^2\mathrm{csgn}\left(i{\mathrm{e}}^{-x{\mathrm{e}}^4}\right)\pi -i\mathrm{csgn}\left(ix{\mathrm{e}}^{-x{\mathrm{e}}^4}\right)\mathrm{csgn}\left(ix\right)\mathrm{csgn}\left(i{\mathrm{e}}^{-x{\mathrm{e}}^4}\right)\pi +2\ln (x)-2\ln \left({\mathrm{e}}^{x{\mathrm{e}}^4}\right))}}$	$128$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*ln(4/x)-48)*ln(x/exp(x*exp(4)))+(12*x*exp(4)-12)*ln(4/x)+36*x*exp(4)-36)*exp((12*ln(4/x)+36)/x/ln(x/
exp(x*exp(4))))/x^2/ln(x/exp(x*exp(4)))^2,x,method=_RETURNVERBOSE)

[Out]

exp(24*(2*ln(2)-ln(x)+3)/x/(-I*csgn(I*x*exp(-x*exp(4)))^3*Pi+I*csgn(I*x*exp(-x*exp(4)))^2*csgn(I*x)*Pi+I*csgn(
I*x*exp(-x*exp(4)))^2*csgn(I*exp(-x*exp(4)))*Pi-I*csgn(I*x*exp(-x*exp(4)))*csgn(I*x)*csgn(I*exp(-x*exp(4)))*Pi
+2*ln(x)-2*ln(exp(x*exp(4)))))

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maxima [B]  time = 0.86, size = 82, normalized size = 2.93 $$\begin{array}{c}{e}^{(-\frac{24e^4\log (2)}{x{e}^4\log (x)-\log (x{)}^2}-\frac{36e^4}{x{e}^4\log (x)-\log (x{)}^2}+\frac{12e^4}{x{e}^4-\log (x)}-\frac{12}{x}+\frac{24\log (2)}{x\log (x)}+\frac{36}{x\log (x)})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*log(4/x)-48)*log(x/exp(x*exp(4)))+(12*x*exp(4)-12)*log(4/x)+36*x*exp(4)-36)*exp((12*log(4/x)+3
6)/x/log(x/exp(x*exp(4))))/x^2/log(x/exp(x*exp(4)))^2,x, algorithm="maxima")

[Out]

e^(-24*e^4*log(2)/(x*e^4*log(x) - log(x)^2) - 36*e^4/(x*e^4*log(x) - log(x)^2) + 12*e^4/(x*e^4 - log(x)) - 12/
x + 24*log(2)/(x*log(x)) + 36/(x*log(x)))

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mupad [B]  time = 1.66, size = 60, normalized size = 2.14 $$\begin{array}{c}\frac{{\mathrm{e}}^{-\frac{36}{x^2{\mathrm{e}}^4-x\ln (x)}}}{2^{\frac{24}{x^2{\mathrm{e}}^4-x\ln (x)}}{\left(\frac{1}{x}\right)}^{\frac{12}{x^2{\mathrm{e}}^4-x\ln (x)}}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((12*log(4/x) + 36)/(x*log(x*exp(-x*exp(4)))))*(36*x*exp(4) + log(4/x)*(12*x*exp(4) - 12) - log(x*exp(
-x*exp(4)))*(12*log(4/x) + 48) - 36))/(x^2*log(x*exp(-x*exp(4)))^2),x)

[Out]

exp(-36/(x^2*exp(4) - x*log(x)))/(2^(24/(x^2*exp(4) - x*log(x)))*(1/x)^(12/(x^2*exp(4) - x*log(x))))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*ln(4/x)-48)*ln(x/exp(x*exp(4)))+(12*x*exp(4)-12)*ln(4/x)+36*x*exp(4)-36)*exp((12*ln(4/x)+36)/x
/ln(x/exp(x*exp(4))))/x**2/ln(x/exp(x*exp(4)))**2,x)

[Out]

Timed out

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