3.24.7 \(\int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+(-4 \log (-x)+4 e^x x \log (-x) \log (x)) \log (-e^x+\log (\log (x)))+(2 e^x \log (x)-2 \log (x) \log (\log (x))) \log ^2(-e^x+\log (\log (x)))+(4-4 e^x x \log (x)) \log ^3(-e^x+\log (\log (x)))}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx\)
Optimal. Leaf size=21 \[ \left (-\log (-x)+\log ^2\left (-e^x+\log (\log (x))\right )\right )^2 \]
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Rubi [A] time = 0.63, antiderivative size = 21, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 123, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used
= {6688, 12, 6686} \begin {gather*} \left (\log (-x)-\log ^2\left (\log (\log (x))-e^x\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-2*E^x*Log[-x]*Log[x] + 2*Log[-x]*Log[x]*Log[Log[x]] + (-4*Log[-x] + 4*E^x*x*Log[-x]*Log[x])*Log[-E^x + L
og[Log[x]]] + (2*E^x*Log[x] - 2*Log[x]*Log[Log[x]])*Log[-E^x + Log[Log[x]]]^2 + (4 - 4*E^x*x*Log[x])*Log[-E^x
+ Log[Log[x]]]^3)/(-(E^x*x*Log[x]) + x*Log[x]*Log[Log[x]]),x]
[Out]
(Log[-x] - Log[-E^x + Log[Log[x]]]^2)^2
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 6686
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /; !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Rule 6688
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 {2 \left (\log (-x)-\log ^2\left (-e^x+\log (\log (x))\right )\right ) \left (2 \log \left (-e^x+\log (\log (x))\right )-\log (x) \left (\log (\log (x))+e^x \left (-1+2 x \log \left (-e^x+\log (\log (x))\right )\right )\right )\right )}{x \log (x) \left (e^x-\log (\log (x))\right )} \, dx\\ &=2 \int \frac {\left (\log (-x)-\log ^2\left (-e^x+\log (\log (x))\right )\right ) \left (2 \log \left (-e^x+\log (\log (x))\right )-\log (x) \left (\log (\log (x))+e^x \left (-1+2 x \log \left (-e^x+\log (\log (x))\right )\right )\right )\right )}{x \log (x) \left (e^x-\log (\log (x))\right )} \, dx\\ &=\left (\log (-x)-\log ^2\left (-e^x+\log (\log (x))\right )\right )^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 21, normalized size = 1.00 \begin {gather*} \left (-\log (-x)+\log ^2\left (-e^x+\log (\log (x))\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-2*E^x*Log[-x]*Log[x] + 2*Log[-x]*Log[x]*Log[Log[x]] + (-4*Log[-x] + 4*E^x*x*Log[-x]*Log[x])*Log[-E
^x + Log[Log[x]]] + (2*E^x*Log[x] - 2*Log[x]*Log[Log[x]])*Log[-E^x + Log[Log[x]]]^2 + (4 - 4*E^x*x*Log[x])*Log
[-E^x + Log[Log[x]]]^3)/(-(E^x*x*Log[x]) + x*Log[x]*Log[Log[x]]),x]
[Out]
(-Log[-x] + Log[-E^x + Log[Log[x]]]^2)^2
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fricas [B] time = 0.75, size = 454, normalized size = 21.62 \begin {gather*} 16 \, \arctan \left (\frac {\sqrt {16 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )^{2} - 4 \, e^{x} \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right ) + \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )^{2} + 4 \, e^{\left (2 \, x\right )}} + 2 \, e^{x} - \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )}{4 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )}\right )^{4} + \frac {1}{16} \, \log \left (4 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )^{2} - e^{x} \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right ) + \frac {1}{4} \, \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )^{2} + e^{\left (2 \, x\right )}\right )^{4} - 2 \, {\left (3 \, \log \left (4 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )^{2} - e^{x} \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right ) + \frac {1}{4} \, \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )^{2} + e^{\left (2 \, x\right )}\right )^{2} - 4 \, \log \left (-x\right )\right )} \arctan \left (\frac {\sqrt {16 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )^{2} - 4 \, e^{x} \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right ) + \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )^{2} + 4 \, e^{\left (2 \, x\right )}} + 2 \, e^{x} - \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )}{4 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )}\right )^{2} - \frac {1}{2} \, \log \left (4 \, \arctan \left (\frac {\sqrt {\pi ^{2} + \log \left (-x\right )^{2}} - \log \left (-x\right )}{\pi }\right )^{2} - e^{x} \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right ) + \frac {1}{4} \, \log \left (\pi ^{2} + \log \left (-x\right )^{2}\right )^{2} + e^{\left (2 \, x\right )}\right )^{2} \log \left (-x\right ) + \log \left (-x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x*exp(x)*log(x)+4)*log(log(log(x))-exp(x))^3+(-2*log(x)*log(log(x))+2*exp(x)*log(x))*log(log(lo
g(x))-exp(x))^2+(4*x*exp(x)*log(-x)*log(x)-4*log(-x))*log(log(log(x))-exp(x))+2*log(-x)*log(x)*log(log(x))-2*e
xp(x)*log(-x)*log(x))/(x*log(x)*log(log(x))-x*exp(x)*log(x)),x, algorithm="fricas")
[Out]
16*arctan(1/4*(sqrt(16*arctan((sqrt(pi^2 + log(-x)^2) - log(-x))/pi)^2 - 4*e^x*log(pi^2 + log(-x)^2) + log(pi^
2 + log(-x)^2)^2 + 4*e^(2*x)) + 2*e^x - log(pi^2 + log(-x)^2))/arctan((sqrt(pi^2 + log(-x)^2) - log(-x))/pi))^
4 + 1/16*log(4*arctan((sqrt(pi^2 + log(-x)^2) - log(-x))/pi)^2 - e^x*log(pi^2 + log(-x)^2) + 1/4*log(pi^2 + lo
g(-x)^2)^2 + e^(2*x))^4 - 2*(3*log(4*arctan((sqrt(pi^2 + log(-x)^2) - log(-x))/pi)^2 - e^x*log(pi^2 + log(-x)^
2) + 1/4*log(pi^2 + log(-x)^2)^2 + e^(2*x))^2 - 4*log(-x))*arctan(1/4*(sqrt(16*arctan((sqrt(pi^2 + log(-x)^2)
- log(-x))/pi)^2 - 4*e^x*log(pi^2 + log(-x)^2) + log(pi^2 + log(-x)^2)^2 + 4*e^(2*x)) + 2*e^x - log(pi^2 + log
(-x)^2))/arctan((sqrt(pi^2 + log(-x)^2) - log(-x))/pi))^2 - 1/2*log(4*arctan((sqrt(pi^2 + log(-x)^2) - log(-x)
)/pi)^2 - e^x*log(pi^2 + log(-x)^2) + 1/4*log(pi^2 + log(-x)^2)^2 + e^(2*x))^2*log(-x) + log(-x)^2
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giac [B] time = 2.00, size = 3110, normalized size = 148.10 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x*exp(x)*log(x)+4)*log(log(log(x))-exp(x))^3+(-2*log(x)*log(log(x))+2*exp(x)*log(x))*log(log(lo
g(x))-exp(x))^2+(4*x*exp(x)*log(-x)*log(x)-4*log(-x))*log(log(log(x))-exp(x))+2*log(-x)*log(x)*log(log(x))-2*e
xp(x)*log(-x)*log(x))/(x*log(x)*log(log(x))-x*exp(x)*log(x)),x, algorithm="giac")
[Out]
-2*pi^3*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - p
i*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi +
pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs(log(x)))) - 2*pi*arctan(1/2*(pi*sg
n(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(
e^x - log(abs(log(x)))))^3*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-
1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs(log(x)))) + 3/2*pi*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sg
n(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x))
)))*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(
pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - p
i*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))^2*sgn(pi*sgn(-pi + pi*sgn(x)
)*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs(
log(x)))) - 1/2*pi^2*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) +
pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + ar
ctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))*sgn(pi*sgn(
-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn
(x)*sgn(e^x - log(abs(log(x)))) - 2*pi^3*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + p
i*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))*sgn(pi*sgn(-pi + pi*sgn(x))
*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))) - 2*pi*arctan(1/2*(
pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)
)))/(e^x - log(abs(log(x)))))^3*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arc
tan(-1/2*(pi - pi*sgn(x))/log(abs(x)))) + 3/2*pi*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn
(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))*log(-pi*arctan(-1/2
*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs
(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^
2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))^2*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi
*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))) - 1/2*pi^2*log(-pi*arctan(-1/2*(pi - pi*s
gn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-
pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*l
og(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi
*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(x) + 1/2*pi^4*sgn(e^x - log(abs(log(x)))) + 3*pi^2
*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x
))/log(abs(x))))/(e^x - log(abs(log(x)))))^2*sgn(e^x - log(abs(log(x)))) - 3/4*pi^2*log(-pi*arctan(-1/2*(pi -
pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*s
gn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e
^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))^2*sgn(e^x - log(abs(log(x)))) + 3/2*pi^2*log(-pi*arctan(-1
/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(a
bs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))
)^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi
*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs(log(x)))) - 2*pi*arctan
(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(
abs(x))))/(e^x - log(abs(log(x)))))*log(abs(x))*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi
*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs(log(x)))) + 1/2*pi^4 + 3*pi^2*arctan
(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(
abs(x))))/(e^x - log(abs(log(x)))))^2 + arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi
*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))^4 - 3/4*pi^2*log(-pi*arctan(
-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log
(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x
)))^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))^2 - 3/2*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn
(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))
))^2*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*
(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi -
pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))^2 + 1/16*log(-pi*arctan(-1/
2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(ab
s(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))
^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x))^4 + 3/2*pi^2*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/
log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi
*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(
log(x))) + log(abs(log(x)))^2 + e^(2*x))*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)
) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))) - 2*pi*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) -
pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))*log(abs(x))*
sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(ab
s(x)))) + pi*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(p
i - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-
pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^
2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(x))) + log(abs
(log(x)))^2 + e^(2*x))*sgn(x) + pi^2*log(abs(x))*sgn(e^x - log(abs(log(x)))) - 3*pi*arctan(1/2*(pi*sgn(-pi + p
i*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log
(abs(log(x)))))*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*a
rctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(
-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(x))) + log(abs(log(x)))^2 + e^(2*x)) + pi^2*log(abs(x
)) + 2*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi
*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))^2*log(abs(x)) - 1/2*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log(
abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn
(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(
x))) + log(abs(log(x)))^2 + e^(2*x))^2*log(abs(x)) + 3/2*pi^2*sgn(x) - 3/2*pi^2 + log(abs(x))^2
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maple [C] time = 0.17, size = 92, normalized size = 4.38
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result |
size |
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\(\ln \left (\ln \left (\ln \relax (x )\right )-{\mathrm e}^{x}\right )^{4}+\left (-2 i \pi \mathrm {csgn}\left (i x \right )^{3}+2 i \pi \mathrm {csgn}\left (i x \right )^{2}-2 i \pi -2 \ln \relax (x )\right ) \ln \left (\ln \left (\ln \relax (x )\right )-{\mathrm e}^{x}\right )^{2}+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{3}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \ln \relax (x )+\ln \relax (x )^{2}\) |
\(92\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-4*x*exp(x)*ln(x)+4)*ln(ln(ln(x))-exp(x))^3+(-2*ln(x)*ln(ln(x))+2*exp(x)*ln(x))*ln(ln(ln(x))-exp(x))^2+(
4*x*exp(x)*ln(-x)*ln(x)-4*ln(-x))*ln(ln(ln(x))-exp(x))+2*ln(-x)*ln(x)*ln(ln(x))-2*exp(x)*ln(-x)*ln(x))/(x*ln(x
)*ln(ln(x))-x*exp(x)*ln(x)),x,method=_RETURNVERBOSE)
[Out]
ln(ln(ln(x))-exp(x))^4+(-2*I*Pi*csgn(I*x)^3+2*I*Pi*csgn(I*x)^2-2*I*Pi-2*ln(x))*ln(ln(ln(x))-exp(x))^2+2*I*Pi*l
n(x)*csgn(I*x)^3-2*I*Pi*ln(x)*csgn(I*x)^2+2*I*Pi*ln(x)+ln(x)^2
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maxima [C] time = 0.64, size = 40, normalized size = 1.90 \begin {gather*} \log \left (-e^{x} + \log \left (\log \relax (x)\right )\right )^{4} - 2 \, {\left (i \, \pi + \log \relax (x)\right )} \log \left (-e^{x} + \log \left (\log \relax (x)\right )\right )^{2} + 2 i \, \pi \log \relax (x) + \log \relax (x)^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x*exp(x)*log(x)+4)*log(log(log(x))-exp(x))^3+(-2*log(x)*log(log(x))+2*exp(x)*log(x))*log(log(lo
g(x))-exp(x))^2+(4*x*exp(x)*log(-x)*log(x)-4*log(-x))*log(log(log(x))-exp(x))+2*log(-x)*log(x)*log(log(x))-2*e
xp(x)*log(-x)*log(x))/(x*log(x)*log(log(x))-x*exp(x)*log(x)),x, algorithm="maxima")
[Out]
log(-e^x + log(log(x)))^4 - 2*(I*pi + log(x))*log(-e^x + log(log(x)))^2 + 2*I*pi*log(x) + log(x)^2
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mupad [B] time = 1.73, size = 43, normalized size = 2.05 \begin {gather*} {\ln \left (\ln \left (\ln \relax (x)\right )-{\mathrm {e}}^x\right )}^4-2\,\ln \left (-x\right )\,{\ln \left (\ln \left (\ln \relax (x)\right )-{\mathrm {e}}^x\right )}^2-{\ln \relax (x)}^2+2\,\ln \left (-x\right )\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(log(log(log(x)) - exp(x))^3*(4*x*exp(x)*log(x) - 4) - log(log(log(x)) - exp(x))^2*(2*exp(x)*log(x) - 2*l
og(log(x))*log(x)) + log(log(log(x)) - exp(x))*(4*log(-x) - 4*x*log(-x)*exp(x)*log(x)) + 2*log(-x)*exp(x)*log(
x) - 2*log(-x)*log(log(x))*log(x))/(x*log(log(x))*log(x) - x*exp(x)*log(x)),x)
[Out]
2*log(-x)*log(x) - 2*log(-x)*log(log(log(x)) - exp(x))^2 - log(x)^2 + log(log(log(x)) - exp(x))^4
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sympy [C] time = 3.58, size = 48, normalized size = 2.29 \begin {gather*} \left (- 2 \log {\relax (x )} - 2 i \pi \right ) \log {\left (- e^{x} + \log {\left (\log {\relax (x )} \right )} \right )}^{2} + \log {\relax (x )}^{2} + 2 i \pi \log {\relax (x )} + \log {\left (- e^{x} + \log {\left (\log {\relax (x )} \right )} \right )}^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x*exp(x)*ln(x)+4)*ln(ln(ln(x))-exp(x))**3+(-2*ln(x)*ln(ln(x))+2*exp(x)*ln(x))*ln(ln(ln(x))-exp(
x))**2+(4*x*exp(x)*ln(-x)*ln(x)-4*ln(-x))*ln(ln(ln(x))-exp(x))+2*ln(-x)*ln(x)*ln(ln(x))-2*exp(x)*ln(-x)*ln(x))
/(x*ln(x)*ln(ln(x))-x*exp(x)*ln(x)),x)
[Out]
(-2*log(x) - 2*I*pi)*log(-exp(x) + log(log(x)))**2 + log(x)**2 + 2*I*pi*log(x) + log(-exp(x) + log(log(x)))**4
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