3.41.48 $\int \frac{2\log (2)-4e^{3+x}\log (2)-2e^{3+x}x\log (2)\log (x^2)+(\log (2)-2e^{3+x}\log (2))\log (x^2)\log ((1-2e^{3+x})\log (x^2))}{(-x^2+2e^{3+x}{x}^2)\log (x^2){\log }^2((1-2e^{3+x})\log (x^2))}dx$

Optimal. Leaf size=24 $$\frac{\log (2)}{x\log (-((-1+2e^{3+x})\log \left(x^2\right)))}$$

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Rubi [F]  time = 1.74, 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 = {} $$\begin{array}{c}\int \frac{2\log (2)-4e^{3+x}\log (2)-2e^{3+x}x\log (2)\log \left(x^2\right)+(\log (2)-2e^{3+x}\log (2))\log \left(x^2\right)\log ((1-2e^{3+x})\log \left(x^2\right))}{(-x^2+2e^{3+x}{x}^2)\log \left(x^2\right){\log }^2((1-2e^{3+x})\log \left(x^2\right))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(2*Log[2] - 4*E^(3 + x)*Log[2] - 2*E^(3 + x)*x*Log[2]*Log[x^2] + (Log[2] - 2*E^(3 + x)*Log[2])*Log[x^2]*Lo
g[(1 - 2*E^(3 + x))*Log[x^2]])/((-x^2 + 2*E^(3 + x)*x^2)*Log[x^2]*Log[(1 - 2*E^(3 + x))*Log[x^2]]^2),x]

[Out]

-(Log[2]*Defer[Int][1/(x*Log[(1 - 2*E^(3 + x))*Log[x^2]]^2), x]) - Log[2]*Defer[Int][1/((-1 + 2*E^(3 + x))*x*L
og[(1 - 2*E^(3 + x))*Log[x^2]]^2), x] - 2*Log[2]*Defer[Int][1/(x^2*Log[x^2]*Log[(1 - 2*E^(3 + x))*Log[x^2]]^2)
, x] - Log[2]*Defer[Int][1/(x^2*Log[(1 - 2*E^(3 + x))*Log[x^2]]), x]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 23, normalized size = 0.96 $$\begin{array}{c}\frac{\log (2)}{x\log ((1-2e^{3+x})\log \left(x^2\right))}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(2*Log[2] - 4*E^(3 + x)*Log[2] - 2*E^(3 + x)*x*Log[2]*Log[x^2] + (Log[2] - 2*E^(3 + x)*Log[2])*Log[x
^2]*Log[(1 - 2*E^(3 + x))*Log[x^2]])/((-x^2 + 2*E^(3 + x)*x^2)*Log[x^2]*Log[(1 - 2*E^(3 + x))*Log[x^2]]^2),x]

[Out]

Log[2]/(x*Log[(1 - 2*E^(3 + x))*Log[x^2]])

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fricas [A]  time = 0.64, size = 23, normalized size = 0.96 $$\begin{array}{c}\frac{\log (2)}{x\log (-(2e^{(x+3)}-1)\log \left(x^2\right))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2)/(x*log(-(2*e^(x + 3) - 1)*log(x^2)))

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giac [A]  time = 0.60, size = 24, normalized size = 1.00 $$\begin{array}{c}\frac{\log (2)}{x\log (-2e^{(x+3)}\log \left(x^2\right)+\log \left(x^2\right))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.26, size = 831, normalized size = 34.62

method	result	size
risch	$\frac{2i\ln (2)}{x(\pi \mathrm{csgn}\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)\mathrm{csgn}\left(({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)+\pi \mathrm{csgn}{\left(({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)}^2+\pi \mathrm{csgn}\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})\right)\mathrm{csgn}\left(i(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)\mathrm{csgn}\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)-\pi \mathrm{csgn}\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})\right)\mathrm{csgn}{\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)}^2-\pi \mathrm{csgn}\left(i(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)\mathrm{csgn}{\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)}^2+\pi \mathrm{csgn}{\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)}^3-\pi \mathrm{csgn}\left(i({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)\mathrm{csgn}{\left(({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)}^2-\pi \mathrm{csgn}{\left(({\mathrm{e}}^{3+x}-\frac{1}{2})(\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2)\right)}^3-\pi +2i\ln ({\mathrm{e}}^{3+x}-\frac{1}{2})+2i\ln (\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+4i\ln (x)+\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2))}$	$831$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*ln(2)/x/(Pi*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*cs
gn(I*x^2)^2))*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+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((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*
x^2)^2))^2+Pi*csgn(I*(exp(3+x)-1/2))*csgn(I*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I
*x)*csgn(I*x^2)^2))*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+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*(exp(3+x)-1/2))*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*c
sgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2-Pi*csgn(I*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2
*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*P
i*csgn(I*x)*csgn(I*x^2)^2))^2+Pi*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-
2*Pi*csgn(I*x)*csgn(I*x^2)^2))^3-Pi*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^
2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2
*Pi*csgn(I*x)*csgn(I*x^2)^2))^2-Pi*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-
2*Pi*csgn(I*x)*csgn(I*x^2)^2))^3-Pi+2*I*ln(exp(3+x)-1/2)+2*I*ln(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn
(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))

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maxima [C]  time = 0.51, size = 30, normalized size = 1.25 $$\begin{array}{c}\frac{\log (2)}{(i\pi +\log (2))x+x\log (2e^{(x+3)}-1)+x\log (\log (x))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2)/((I*pi + log(2))*x + x*log(2*e^(x + 3) - 1) + x*log(log(x)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 $$\begin{array}{c}\int \frac{2\ln (2)-4{\mathrm{e}}^{x+3}\ln (2)+\ln (-\ln \left(x^2\right)(2{\mathrm{e}}^{x+3}-1))\ln \left(x^2\right)(\ln (2)-2{\mathrm{e}}^{x+3}\ln (2))-2x\ln \left(x^2\right){\mathrm{e}}^{x+3}\ln (2)}{{\ln (-\ln \left(x^2\right)(2{\mathrm{e}}^{x+3}-1))}^2\ln \left(x^2\right)(2x^2{\mathrm{e}}^{x+3}-x^2)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2) - 4*exp(x + 3)*log(2) + log(-log(x^2)*(2*exp(x + 3) - 1))*log(x^2)*(log(2) - 2*exp(x + 3)*log(2)
) - 2*x*log(x^2)*exp(x + 3)*log(2))/(log(-log(x^2)*(2*exp(x + 3) - 1))^2*log(x^2)*(2*x^2*exp(x + 3) - x^2)),x)

[Out]

int((2*log(2) - 4*exp(x + 3)*log(2) + log(-log(x^2)*(2*exp(x + 3) - 1))*log(x^2)*(log(2) - 2*exp(x + 3)*log(2)
) - 2*x*log(x^2)*exp(x + 3)*log(2))/(log(-log(x^2)*(2*exp(x + 3) - 1))^2*log(x^2)*(2*x^2*exp(x + 3) - x^2)), x
)

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sympy [A]  time = 0.56, size = 19, normalized size = 0.79 $$\begin{array}{c}\frac{\log (2)}{x\log ((1-2e^{x+3})\log \left(x^2\right))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2)/(x*log((1 - 2*exp(x + 3))*log(x**2)))

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