3.33.49 $\int \frac{265x+512x^2+e^{-2+x}(-256x-256x^2)+(-9-512x+e^{-2+x}(256+256x))\log (4e^{2x})+(-256x+256\log (4e^{2x}))\log (-x+\log (4e^{2x}))}{-256x+256\log (4e^{2x})}dx$

Optimal. Leaf size=27 $$x(-\frac{9}{256}+e^{-2+x}-x+\log (-x+\log \left(4e^{2x}\right)))$$

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Rubi [F]  time = 1.03, 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 = {} $$\begin{array}{c}\int \frac{265x+512x^2+e^{-2+x}(-256x-256x^2)+(-9-512x+e^{-2+x}(256+256x))\log \left(4e^{2x}\right)+(-256x+256\log \left(4e^{2x}\right))\log (-x+\log \left(4e^{2x}\right))}{-256x+256\log \left(4e^{2x}\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(265*x + 512*x^2 + E^(-2 + x)*(-256*x - 256*x^2) + (-9 - 512*x + E^(-2 + x)*(256 + 256*x))*Log[4*E^(2*x)]
+ (-256*x + 256*Log[4*E^(2*x)])*Log[-x + Log[4*E^(2*x)]])/(-256*x + 256*Log[4*E^(2*x)]),x]

[Out]

-E^(-2 + x) - (9*x)/256 - x^2 + E^(-2 + x)*(1 + x) - Defer[Int][x/(x - Log[4*E^(2*x)]), x] + Defer[Int][Log[-x
+ Log[4*E^(2*x)]], x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-265x-512x^2-e^{-2+x}(-256x-256x^2)-(-9-512x+e^{-2+x}(256+256x))\log \left(4e^{2x}\right)-(-256x+256\log \left(4e^{2x}\right))\log (-x+\log \left(4e^{2x}\right))}{256(x-\log \left(4e^{2x}\right))}dx\\ & =\frac{1}{256}\int \frac{-265x-512x^2-e^{-2+x}(-256x-256x^2)-(-9-512x+e^{-2+x}(256+256x))\log \left(4e^{2x}\right)-(-256x+256\log \left(4e^{2x}\right))\log (-x+\log \left(4e^{2x}\right))}{x-\log \left(4e^{2x}\right)}dx\\ & =\frac{1}{256}\int (256e^{-2+x}(1+x)+\frac{-265x-512x^2+9\log \left(4e^{2x}\right)+512x\log \left(4e^{2x}\right)+256x\log (-x+\log \left(4e^{2x}\right))-256\log \left(4e^{2x}\right)\log (-x+\log \left(4e^{2x}\right))}{x-\log \left(4e^{2x}\right)})dx\\ & =\frac{1}{256}\int \frac{-265x-512x^2+9\log \left(4e^{2x}\right)+512x\log \left(4e^{2x}\right)+256x\log (-x+\log \left(4e^{2x}\right))-256\log \left(4e^{2x}\right)\log (-x+\log \left(4e^{2x}\right))}{x-\log \left(4e^{2x}\right)}dx+\int e^{-2+x}(1+x)dx\\ & =e^{-2+x}(1+x)+\frac{1}{256}\int (\frac{-265x-512x^2+9\log \left(4e^{2x}\right)+512x\log \left(4e^{2x}\right)}{x-\log \left(4e^{2x}\right)}+256\log (-x+\log \left(4e^{2x}\right)))dx-\int e^{-2+x}dx\\ & =-e^{-2+x}+e^{-2+x}(1+x)+\frac{1}{256}\int \frac{-265x-512x^2+9\log \left(4e^{2x}\right)+512x\log \left(4e^{2x}\right)}{x-\log \left(4e^{2x}\right)}dx+\int \log (-x+\log \left(4e^{2x}\right))dx\\ & =-e^{-2+x}+e^{-2+x}(1+x)+\frac{1}{256}\int (-9-512x-\frac{256x}{x-\log \left(4e^{2x}\right)})dx+\int \log (-x+\log \left(4e^{2x}\right))dx\\ & =-e^{-2+x}-\frac{9x}{256}-x^2+e^{-2+x}(1+x)-\int \frac{x}{x-\log \left(4e^{2x}\right)}dx+\int \log (-x+\log \left(4e^{2x}\right))dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.20, size = 32, normalized size = 1.19 $$\begin{array}{c}-\frac{1}{256}x(9-256e^{-2+x}+256x-256\log (-x+\log \left(4e^{2x}\right)))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(265*x + 512*x^2 + E^(-2 + x)*(-256*x - 256*x^2) + (-9 - 512*x + E^(-2 + x)*(256 + 256*x))*Log[4*E^(
2*x)] + (-256*x + 256*Log[4*E^(2*x)])*Log[-x + Log[4*E^(2*x)]])/(-256*x + 256*Log[4*E^(2*x)]),x]

[Out]

-1/256*(x*(9 - 256*E^(-2 + x) + 256*x - 256*Log[-x + Log[4*E^(2*x)]]))

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fricas [A]  time = 0.52, size = 24, normalized size = 0.89 $$\begin{array}{c}-x^2+x{e}^{(x-2)}+x\log (x+2\log (2))-\frac{9}{256}x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*log(4*exp(x)^2)-256*x)*log(log(4*exp(x)^2)-x)+((256*x+256)*exp(x-2)-512*x-9)*log(4*exp(x)^2)+(
-256*x^2-256*x)*exp(x-2)+512*x^2+265*x)/(256*log(4*exp(x)^2)-256*x),x, algorithm="fricas")

[Out]

-x^2 + x*e^(x - 2) + x*log(x + 2*log(2)) - 9/256*x

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giac [A]  time = 0.20, size = 34, normalized size = 1.26 $$\begin{array}{c}-\frac{1}{256}(256x^2{e}^2-256x{e}^2\log (x+2\log (2))+9x{e}^2-256x{e}^x)e^{(-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*log(4*exp(x)^2)-256*x)*log(log(4*exp(x)^2)-x)+((256*x+256)*exp(x-2)-512*x-9)*log(4*exp(x)^2)+(
-256*x^2-256*x)*exp(x-2)+512*x^2+265*x)/(256*log(4*exp(x)^2)-256*x),x, algorithm="giac")

[Out]

-1/256*(256*x^2*e^2 - 256*x*e^2*log(x + 2*log(2)) + 9*x*e^2 - 256*x*e^x)*e^(-2)

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maple [C]  time = 0.10, size = 63, normalized size = 2.33

method	result	size
risch	$x\ln (2\ln (2)+2\ln \left({\mathrm{e}}^x\right)-\frac{i\pi \mathrm{csgn}\left(i{\mathrm{e}}^{2x}\right){(-\mathrm{csgn}\left(i{\mathrm{e}}^{2x}\right)+\mathrm{csgn}\left(i{\mathrm{e}}^x\right))}^2}{2}-x)-x^2-\frac{9x}{256}+x{\mathrm{e}}^{x-2}$	$63$
default	$-x^2+\frac{503x}{256}-2\ln (\ln \left(4{\mathrm{e}}^{2x}\right)-x)(\ln \left({\mathrm{e}}^x\right)-x)-\ln (\ln \left(4{\mathrm{e}}^{2x}\right)-x)(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right))+\ln (\ln \left(4{\mathrm{e}}^{2x}\right)-x)(\ln \left(4{\mathrm{e}}^{2x}\right)-x)-\ln \left(4{\mathrm{e}}^{2x}\right)-6{\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))+2{\mathrm{e}}^{x-2}-5(-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)){\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))+{\mathrm{e}}^{x-2}(x-2)-2(\ln \left({\mathrm{e}}^x\right)-x){\mathrm{e}}^{x-2}-({(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right))}^2+4(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right))(\ln \left({\mathrm{e}}^x\right)-x)+4{(\ln \left({\mathrm{e}}^x\right)-x)}^2+4\ln \left(4{\mathrm{e}}^{2x}\right)-8x+4){\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))-(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right)){\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))-6(\ln \left({\mathrm{e}}^x\right)-x){\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))+2\ln \left({\mathrm{e}}^x\right){\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right))-2{\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right))x(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right))+{\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right)){(\ln \left(4{\mathrm{e}}^{2x}\right)-2\ln \left({\mathrm{e}}^x\right))}^2+2(\ln \left({\mathrm{e}}^x\right)-x)({\mathrm{e}}^{x-2}-(-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)){\mathrm{e}}^{-2+2x-\ln \left(4{\mathrm{e}}^{2x}\right)}\text{expIntegralEi}(1,x-\ln \left(4{\mathrm{e}}^{2x}\right)))$	$556$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((256*ln(4*exp(x)^2)-256*x)*ln(ln(4*exp(x)^2)-x)+((256*x+256)*exp(x-2)-512*x-9)*ln(4*exp(x)^2)+(-256*x^2-2
56*x)*exp(x-2)+512*x^2+265*x)/(256*ln(4*exp(x)^2)-256*x),x,method=_RETURNVERBOSE)

[Out]

x*ln(2*ln(2)+2*ln(exp(x))-1/2*I*Pi*csgn(I*exp(2*x))*(-csgn(I*exp(2*x))+csgn(I*exp(x)))^2-x)-x^2-9/256*x+x*exp(
x-2)

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maxima [B]  time = 0.54, size = 92, normalized size = 3.41 $$\begin{array}{c}8\log (2{)}^2\log (x+2\log (2))+x^2-\frac{1}{128}(256x^2{e}^2-x(512\log (2)-137)e^2-128x{e}^x+((1024\log (2{)}^2-265\log (2))e^2-128x{e}^2)\log (x+2\log (2)))e^{(-2)}-4x\log (2)-\frac{265}{128}\log (2)\log (x+2\log (2))+\frac{265}{256}x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*log(4*exp(x)^2)-256*x)*log(log(4*exp(x)^2)-x)+((256*x+256)*exp(x-2)-512*x-9)*log(4*exp(x)^2)+(
-256*x^2-256*x)*exp(x-2)+512*x^2+265*x)/(256*log(4*exp(x)^2)-256*x),x, algorithm="maxima")

[Out]

8*log(2)^2*log(x + 2*log(2)) + x^2 - 1/128*(256*x^2*e^2 - x*(512*log(2) - 137)*e^2 - 128*x*e^x + ((1024*log(2)
^2 - 265*log(2))*e^2 - 128*x*e^2)*log(x + 2*log(2)))*e^(-2) - 4*x*log(2) - 265/128*log(2)*log(x + 2*log(2)) +
265/256*x

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mupad [B]  time = 2.09, size = 24, normalized size = 0.89 $$\begin{array}{c}x{\mathrm{e}}^{x-2}-\frac{9x}{256}-x^2+x\ln (x+2\ln (2))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - 2)*(256*x + 256*x^2) - 265*x + log(4*exp(2*x))*(512*x - exp(x - 2)*(256*x + 256) + 9) + log(log(4
*exp(2*x)) - x)*(256*x - 256*log(4*exp(2*x))) - 512*x^2)/(256*x - 256*log(4*exp(2*x))),x)

[Out]

x*exp(x - 2) - (9*x)/256 - x^2 + x*log(x + 2*log(2))

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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(((256*ln(4*exp(x)**2)-256*x)*ln(ln(4*exp(x)**2)-x)+((256*x+256)*exp(x-2)-512*x-9)*ln(4*exp(x)**2)+(-
256*x**2-256*x)*exp(x-2)+512*x**2+265*x)/(256*ln(4*exp(x)**2)-256*x),x)

[Out]

Timed out

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