3.85.50 \(\int \frac {4 e^x+(e^x (-4 x+x^2)+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))) \log (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))})}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx\) [8450]

Optimal. Leaf size=30 \[ e^x \log \left (4+\log ^2(\log (4))+\frac {-4+x}{x (i \pi +\log (\log (4)))}\right ) \]

[Out]

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

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Rubi [A]
time = 0.48, antiderivative size = 39, normalized size of antiderivative = 1.30, number of steps used = 5, number of rules used = 4, integrand size = 146, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6, 1607, 6820, 2326} \begin {gather*} e^x \log \left (-\frac {4}{x (\log (\log (4))+i \pi )}+4+\log ^2(\log (4))+\frac {1}{\log (\log (4))+i \pi }\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^x + (E^x*(-4*x + x^2) + 4*E^x*x^2*(I*Pi + Log[Log[4]]) + E^x*x^2*Log[Log[4]]^2*(I*Pi + Log[Log[4]]))*
Log[(-4 + x + 4*x*(I*Pi + Log[Log[4]]) + x*Log[Log[4]]^2*(I*Pi + Log[Log[4]]))/(x*(I*Pi + Log[Log[4]]))])/(-4*
x + x^2 + 4*x^2*(I*Pi + Log[Log[4]]) + x^2*Log[Log[4]]^2*(I*Pi + Log[Log[4]])),x]

[Out]

E^x*Log[4 + Log[Log[4]]^2 + (I*Pi + Log[Log[4]])^(-1) - 4/(x*(I*Pi + Log[Log[4]]))]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))+x^2 (1+4 (i \pi +\log (\log (4))))} \, dx\\ &=\int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2 \left (1+4 (i \pi +\log (\log (4)))+\log ^2(\log (4)) (i \pi +\log (\log (4)))\right )} \, dx\\ &=\int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{x \left (-4+x \left (1+4 (i \pi +\log (\log (4)))+\log ^2(\log (4)) (i \pi +\log (\log (4)))\right )\right )} \, dx\\ &=\int e^x \left (\frac {4}{x \left (-4+x \left (1+4 \log (\log (4))+\log ^3(\log (4))+i \pi \left (4+\log ^2(\log (4))\right )\right )\right )}+\log \left (4+\log ^2(\log (4))+\frac {1}{i \pi +\log (\log (4))}-\frac {4}{x (i \pi +\log (\log (4)))}\right )\right ) \, dx\\ &=e^x \log \left (4+\log ^2(\log (4))+\frac {1}{i \pi +\log (\log (4))}-\frac {4}{x (i \pi +\log (\log (4)))}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, size = 39, normalized size = 1.30 \begin {gather*} e^x \log \left (4+\log ^2(\log (4))+\frac {1}{i \pi +\log (\log (4))}-\frac {4}{x (i \pi +\log (\log (4)))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^x*Log[4 + Log[Log[4]]^2 + (I*Pi + Log[Log[4]])^(-1) - 4/(x*(I*Pi + Log[Log[4]]))]

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Maple [A]
time = 5.03, size = 41, normalized size = 1.37

method	result	size
default	\({\mathrm e}^{x} \ln \left (\frac {x \ln \left (-2 \ln \left (2\right )\right ) \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \ln \left (-2 \ln \left (2\right )\right )+x -4}{x \ln \left (-2 \ln \left (2\right )\right )}\right )\)	\(41\)
norman	\({\mathrm e}^{x} \ln \left (\frac {x \ln \left (-2 \ln \left (2\right )\right ) \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \ln \left (-2 \ln \left (2\right )\right )+x -4}{x \ln \left (-2 \ln \left (2\right )\right )}\right )\)	\(41\)
risch	\({\mathrm e}^{x} \ln \left (\left (x \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )^{2}+4 x \right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+i \pi \right )+x -4\right )-{\mathrm e}^{x} \ln \left (x \right )+\frac {i \pi \,\mathrm {csgn}\left (i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right )^{2} {\mathrm e}^{x}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) {\mathrm e}^{x}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right ) {\mathrm e}^{x}}{2}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right )^{2} {\mathrm e}^{x}}{2}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right )^{3} {\mathrm e}^{x}}{2}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) {\mathrm e}^{x}}{2}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right )^{2} {\mathrm e}^{x}}{2}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right )^{3} {\mathrm e}^{x}}{2}-\ln \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+i \pi \right ) {\mathrm e}^{x}\)	\(499\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.67, size = 82, normalized size = 2.73 \begin {gather*} -{\left (\log \left (i \, \pi + \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + \log \left (x\right )\right )} e^{x} + e^{x} \log \left ({\left (4 i \, \pi + i \, \pi \log \left (2\right )^{2} + \log \left (2\right )^{3} + {\left (i \, \pi + 3 \, \log \left (2\right )\right )} \log \left (\log \left (2\right )\right )^{2} + \log \left (\log \left (2\right )\right )^{3} + {\left (2 i \, \pi \log \left (2\right ) + 3 \, \log \left (2\right )^{2} + 4\right )} \log \left (\log \left (2\right )\right ) + 4 \, \log \left (2\right ) + 1\right )} x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(I*pi + log(2) + log(log(2))) + log(x))*e^x + e^x*log((4*I*pi + I*pi*log(2)^2 + log(2)^3 + (I*pi + 3*log(
2))*log(log(2))^2 + log(log(2))^3 + (2*I*pi*log(2) + 3*log(2)^2 + 4)*log(log(2)) + 4*log(2) + 1)*x - 4)

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Fricas [C] Result contains complex when optimal does not.
time = 0.43, size = 54, normalized size = 1.80 \begin {gather*} e^{x} \log \left (\frac {2 i \, \pi x \log \left (-2 \, \log \left (2\right )\right )^{2} + x \log \left (-2 \, \log \left (2\right )\right )^{3} - {\left (\pi ^{2} x - 4 \, x\right )} \log \left (-2 \, \log \left (2\right )\right ) + x - 4}{x \log \left (-2 \, \log \left (2\right )\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (29) = 58\).
time = 34.43, size = 313, normalized size = 10.43 \begin {gather*} e^{x} \log {\left (\frac {4 x \log {\left (\log {\left (2 \right )} \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {3 x \log {\left (2 \right )}^{2} \log {\left (\log {\left (2 \right )} \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {x \log {\left (\log {\left (2 \right )} \right )}^{3}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {3 x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}^{2}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {x \log {\left (2 \right )}^{3}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {x}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {4 x \log {\left (2 \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {2 i \pi x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {i \pi x \log {\left (\log {\left (2 \right )} \right )}^{2}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {i \pi x \log {\left (2 \right )}^{2}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {4 i \pi x}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} - \frac {4}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*log(4*x*log(log(2))/(x*log(log(2)) + x*log(2) + I*pi*x) + 3*x*log(2)**2*log(log(2))/(x*log(log(2)) + x*
log(2) + I*pi*x) + x*log(log(2))**3/(x*log(log(2)) + x*log(2) + I*pi*x) + 3*x*log(2)*log(log(2))**2/(x*log(log
(2)) + x*log(2) + I*pi*x) + x*log(2)**3/(x*log(log(2)) + x*log(2) + I*pi*x) + x/(x*log(log(2)) + x*log(2) + I*
pi*x) + 4*x*log(2)/(x*log(log(2)) + x*log(2) + I*pi*x) + 2*I*pi*x*log(2)*log(log(2))/(x*log(log(2)) + x*log(2)
+ I*pi*x) + I*pi*x*log(log(2))**2/(x*log(log(2)) + x*log(2) + I*pi*x) + I*pi*x*log(2)**2/(x*log(log(2)) + x*l
og(2) + I*pi*x) + 4*I*pi*x/(x*log(log(2)) + x*log(2) + I*pi*x) - 4/(x*log(log(2)) + x*log(2) + I*pi*x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (27) = 54\).
time = 1.22, size = 443, normalized size = 14.77 \begin {gather*} \frac {1}{2} \, e^{x} \log \left (\pi ^{2} x^{2} \log \left (2\right )^{4} + x^{2} \log \left (2\right )^{6} + 4 \, \pi ^{2} x^{2} \log \left (2\right )^{3} \log \left (\log \left (2\right )\right ) + 6 \, x^{2} \log \left (2\right )^{5} \log \left (\log \left (2\right )\right ) + 6 \, \pi ^{2} x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right )^{2} + 15 \, x^{2} \log \left (2\right )^{4} \log \left (\log \left (2\right )\right )^{2} + 4 \, \pi ^{2} x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 20 \, x^{2} \log \left (2\right )^{3} \log \left (\log \left (2\right )\right )^{3} + \pi ^{2} x^{2} \log \left (\log \left (2\right )\right )^{4} + 15 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right )^{4} + 6 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{5} + x^{2} \log \left (\log \left (2\right )\right )^{6} + 8 \, \pi ^{2} x^{2} \log \left (2\right )^{2} + 8 \, x^{2} \log \left (2\right )^{4} + 16 \, \pi ^{2} x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 32 \, x^{2} \log \left (2\right )^{3} \log \left (\log \left (2\right )\right ) + 8 \, \pi ^{2} x^{2} \log \left (\log \left (2\right )\right )^{2} + 48 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right )^{2} + 32 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 8 \, x^{2} \log \left (\log \left (2\right )\right )^{4} + 2 \, x^{2} \log \left (2\right )^{3} + 6 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right ) + 6 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 2 \, x^{2} \log \left (\log \left (2\right )\right )^{3} + 16 \, \pi ^{2} x^{2} + 16 \, x^{2} \log \left (2\right )^{2} - 8 \, x \log \left (2\right )^{3} + 32 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right ) - 24 \, x \log \left (2\right )^{2} \log \left (\log \left (2\right )\right ) + 16 \, x^{2} \log \left (\log \left (2\right )\right )^{2} - 24 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} - 8 \, x \log \left (\log \left (2\right )\right )^{3} + 8 \, x^{2} \log \left (2\right ) + 8 \, x^{2} \log \left (\log \left (2\right )\right ) + x^{2} - 32 \, x \log \left (2\right ) - 32 \, x \log \left (\log \left (2\right )\right ) - 8 \, x + 16\right ) - \frac {1}{2} \, e^{x} \log \left (\pi ^{2} x^{2} + x^{2} \log \left (2\right )^{2} + 2 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right ) + x^{2} \log \left (\log \left (2\right )\right )^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^x*log(pi^2*x^2*log(2)^4 + x^2*log(2)^6 + 4*pi^2*x^2*log(2)^3*log(log(2)) + 6*x^2*log(2)^5*log(log(2)) +
6*pi^2*x^2*log(2)^2*log(log(2))^2 + 15*x^2*log(2)^4*log(log(2))^2 + 4*pi^2*x^2*log(2)*log(log(2))^3 + 20*x^2*l
og(2)^3*log(log(2))^3 + pi^2*x^2*log(log(2))^4 + 15*x^2*log(2)^2*log(log(2))^4 + 6*x^2*log(2)*log(log(2))^5 +
x^2*log(log(2))^6 + 8*pi^2*x^2*log(2)^2 + 8*x^2*log(2)^4 + 16*pi^2*x^2*log(2)*log(log(2)) + 32*x^2*log(2)^3*lo
g(log(2)) + 8*pi^2*x^2*log(log(2))^2 + 48*x^2*log(2)^2*log(log(2))^2 + 32*x^2*log(2)*log(log(2))^3 + 8*x^2*log
(log(2))^4 + 2*x^2*log(2)^3 + 6*x^2*log(2)^2*log(log(2)) + 6*x^2*log(2)*log(log(2))^2 + 2*x^2*log(log(2))^3 +
16*pi^2*x^2 + 16*x^2*log(2)^2 - 8*x*log(2)^3 + 32*x^2*log(2)*log(log(2)) - 24*x*log(2)^2*log(log(2)) + 16*x^2*
log(log(2))^2 - 24*x*log(2)*log(log(2))^2 - 8*x*log(log(2))^3 + 8*x^2*log(2) + 8*x^2*log(log(2)) + x^2 - 32*x*
log(2) - 32*x*log(log(2)) - 8*x + 16) - 1/2*e^x*log(pi^2*x^2 + x^2*log(2)^2 + 2*x^2*log(2)*log(log(2)) + x^2*l
og(log(2))^2)

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Mupad [B]
time = 6.51, size = 38, normalized size = 1.27 \begin {gather*} {\mathrm {e}}^x\,\ln \left (\frac {x+4\,x\,\ln \left (-\ln \left (4\right )\right )+x\,\ln \left (-\ln \left (4\right )\right )\,{\ln \left (\ln \left (4\right )\right )}^2-4}{x\,\ln \left (-\ln \left (4\right )\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*log((x + 4*x*log(-log(4)) + x*log(-log(4))*log(log(4))^2 - 4)/(x*log(-log(4))))

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