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

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

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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 {gather*} \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {-265 x-512 x^2-e^{-2+x} \left (-256 x-256 x^2\right )-\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )-\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{256 \left (x-\log \left (4 e^{2 x}\right )\right )} \, dx\\ &=\frac {1}{256} \int \frac {-265 x-512 x^2-e^{-2+x} \left (-256 x-256 x^2\right )-\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )-\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{x-\log \left (4 e^{2 x}\right )} \, dx\\ &=\frac {1}{256} \int \left (256 e^{-2+x} (1+x)+\frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )+256 x \log \left (-x+\log \left (4 e^{2 x}\right )\right )-256 \log \left (4 e^{2 x}\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{x-\log \left (4 e^{2 x}\right )}\right ) \, dx\\ &=\frac {1}{256} \int \frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )+256 x \log \left (-x+\log \left (4 e^{2 x}\right )\right )-256 \log \left (4 e^{2 x}\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{x-\log \left (4 e^{2 x}\right )} \, dx+\int e^{-2+x} (1+x) \, dx\\ &=e^{-2+x} (1+x)+\frac {1}{256} \int \left (\frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )}{x-\log \left (4 e^{2 x}\right )}+256 \log \left (-x+\log \left (4 e^{2 x}\right )\right )\right ) \, dx-\int e^{-2+x} \, dx\\ &=-e^{-2+x}+e^{-2+x} (1+x)+\frac {1}{256} \int \frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )}{x-\log \left (4 e^{2 x}\right )} \, dx+\int \log \left (-x+\log \left (4 e^{2 x}\right )\right ) \, dx\\ &=-e^{-2+x}+e^{-2+x} (1+x)+\frac {1}{256} \int \left (-9-512 x-\frac {256 x}{x-\log \left (4 e^{2 x}\right )}\right ) \, dx+\int \log \left (-x+\log \left (4 e^{2 x}\right )\right ) \, dx\\ &=-e^{-2+x}-\frac {9 x}{256}-x^2+e^{-2+x} (1+x)-\int \frac {x}{x-\log \left (4 e^{2 x}\right )} \, dx+\int \log \left (-x+\log \left (4 e^{2 x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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

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 {gather*} -x^{2} + x e^{\left (x - 2\right )} + x \log \left (x + 2 \, \log \relax (2)\right ) - \frac {9}{256} \, x \end {gather*}

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 {gather*} -\frac {1}{256} \, {\left (256 \, x^{2} e^{2} - 256 \, x e^{2} \log \left (x + 2 \, \log \relax (2)\right ) + 9 \, x e^{2} - 256 \, x e^{x}\right )} e^{\left (-2\right )} \end {gather*}

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	\(x \ln \left (2 \ln \relax (2)+2 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )^{2}}{2}-x \right )-x^{2}-\frac {9 x}{256}+x \,{\mathrm e}^{x -2}\)	\(63\)
default	\(-x^{2}+\frac {503 x}{256}-2 \ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )+\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right )-\ln \left (4 \,{\mathrm e}^{2 x}\right )-6 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+2 \,{\mathrm e}^{x -2}-5 \left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+{\mathrm e}^{x -2} \left (x -2\right )-2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{x -2}-\left (\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )^{2}+4 \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+4 \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{2}+4 \ln \left (4 \,{\mathrm e}^{2 x}\right )-8 x +4\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-6 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+2 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )-2 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) x \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )+{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )^{2}+2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left ({\mathrm e}^{x -2}-\left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegralEi \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )\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 {gather*} 8 \, \log \relax (2)^{2} \log \left (x + 2 \, \log \relax (2)\right ) + x^{2} - \frac {1}{128} \, {\left (256 \, x^{2} e^{2} - x {\left (512 \, \log \relax (2) - 137\right )} e^{2} - 128 \, x e^{x} + {\left ({\left (1024 \, \log \relax (2)^{2} - 265 \, \log \relax (2)\right )} e^{2} - 128 \, x e^{2}\right )} \log \left (x + 2 \, \log \relax (2)\right )\right )} e^{\left (-2\right )} - 4 \, x \log \relax (2) - \frac {265}{128} \, \log \relax (2) \log \left (x + 2 \, \log \relax (2)\right ) + \frac {265}{256} \, x \end {gather*}

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 {gather*} x\,{\mathrm {e}}^{x-2}-\frac {9\,x}{256}-x^2+x\,\ln \left (x+2\,\ln \relax (2)\right ) \end {gather*}

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 {gather*} \text {Timed out} \end {gather*}

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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