3.33.49 \(\int \frac {265 x+512 x^2+e^{-2+x} (-256 x-256 x^2)+(-9-512 x+e^{-2+x} (256+256 x)) \log (4 e^{2 x})+(-256 x+256 \log (4 e^{2 x})) \log (-x+\log (4 e^{2 x}))}{-256 x+256 \log (4 e^{2 x})} \, dx\) [3249]

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

[Out]

(ln(ln(4*exp(x)^2)-x)-x-9/256+exp(-2+x))*x

________________________________________________________________________________________

Rubi [F]
time = 0.88, 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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.14, 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)]]))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.77, size = 556, normalized size = 20.59

method	result	size
risch	\(x \ln \left (2 \ln \left (2\right )+2 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )^{2}}{2}-x \right )-x^{2}-\frac {9 x}{256}+x \,{\mathrm e}^{x -2}\)	\(63\)
default	\(-x^{2}+\frac {503 x}{256}-\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 )-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 )-x \right )-\ln \left (4 \,{\mathrm e}^{2 x}\right )-6 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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 )} \expIntegral \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^2+503/256*x-ln(ln(4*exp(x)^2)-x)*(ln(4*exp(x)^2)-2*ln(exp(x)))-2*ln(ln(4*exp(x)^2)-x)*(ln(exp(x))-x)+ln(ln(
4*exp(x)^2)-x)*(ln(4*exp(x)^2)-x)-ln(4*exp(x)^2)-6*exp(-2+2*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))+2*exp(x-2
)-5*(-2+2*x-ln(4*exp(x)^2))*exp(-2+2*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))+exp(x-2)*(x-2)-2*(ln(exp(x))-x)*
exp(x-2)-((ln(4*exp(x)^2)-2*ln(exp(x)))^2+4*(ln(4*exp(x)^2)-2*ln(exp(x)))*(ln(exp(x))-x)+4*(ln(exp(x))-x)^2+4*
ln(4*exp(x)^2)-8*x+4)*exp(-2+2*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))-(ln(4*exp(x)^2)-2*ln(exp(x)))*exp(-2+2
*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))-6*(ln(exp(x))-x)*exp(-2+2*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))+2
*ln(exp(x))*exp(-2+2*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))*(ln(4*exp(x)^2)-2*ln(exp(x)))-2*exp(-2+2*x-ln(4*
exp(x)^2))*Ei(1,x-ln(4*exp(x)^2))*x*(ln(4*exp(x)^2)-2*ln(exp(x)))+exp(-2+2*x-ln(4*exp(x)^2))*Ei(1,x-ln(4*exp(x
)^2))*(ln(4*exp(x)^2)-2*ln(exp(x)))^2+2*(ln(exp(x))-x)*(exp(x-2)-(-2+2*x-ln(4*exp(x)^2))*exp(-2+2*x-ln(4*exp(x
)^2))*Ei(1,x-ln(4*exp(x)^2)))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (28) = 56\).
time = 0.53, size = 92, normalized size = 3.41 \begin {gather*} 8 \, \log \left (2\right )^{2} \log \left (x + 2 \, \log \left (2\right )\right ) + x^{2} - \frac {1}{128} \, {\left (256 \, x^{2} e^{2} - x {\left (512 \, \log \left (2\right ) - 137\right )} e^{2} - 128 \, x e^{x} + {\left ({\left (1024 \, \log \left (2\right )^{2} - 265 \, \log \left (2\right )\right )} e^{2} - 128 \, x e^{2}\right )} \log \left (x + 2 \, \log \left (2\right )\right )\right )} e^{\left (-2\right )} - 4 \, x \log \left (2\right ) - \frac {265}{128} \, \log \left (2\right ) \log \left (x + 2 \, \log \left (2\right )\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(-2+x)-512*x-9)*log(4*exp(x)^2)+
(-256*x^2-256*x)*exp(-2+x)+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

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 24, normalized size = 0.89 \begin {gather*} -x^{2} + x e^{\left (x - 2\right )} + x \log \left (x + 2 \, \log \left (2\right )\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(-2+x)-512*x-9)*log(4*exp(x)^2)+
(-256*x^2-256*x)*exp(-2+x)+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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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(-2+x)-512*x-9)*ln(4*exp(x)**2)+(
-256*x**2-256*x)*exp(-2+x)+512*x**2+265*x)/(256*ln(4*exp(x)**2)-256*x),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.42, 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 \left (2\right )\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(-2+x)-512*x-9)*log(4*exp(x)^2)+
(-256*x^2-256*x)*exp(-2+x)+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)

________________________________________________________________________________________

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 \left (2\right )\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))

________________________________________________________________________________________