∫ e2x(−2+8x)+4 log(5)+ex(−4+8x+(2−4x) log(5))____________________________________________________ (e2xx−2x log(5)+ex(2x−x log(5))) log(e4x+e3x(4−2 log(5))+4 log2(5)+e2x(4−8 log(5)+log2(5))+ex(−8 log(5)+4 log2(5)) 2x ) log(log2(e4x+e3x(4−2 log(5))+4 log2(5)+e2x(4−8 log(5)+log2(5))+ex(−8 log(5)+4 log2(5)) 2x )) dx

3.12.60 \(\int \frac {e^{2 x} (-2+8 x)+4 \log (5)+e^x (-4+8 x+(2-4 x) \log (5))}{(e^{2 x} x-2 x \log (5)+e^x (2 x-x \log (5))) \log (\frac {e^{4 x}+e^{3 x} (4-2 \log (5))+4 \log ^2(5)+e^{2 x} (4-8 \log (5)+\log ^2(5))+e^x (-8 \log (5)+4 \log ^2(5))}{2 x}) \log (\log ^2(\frac {e^{4 x}+e^{3 x} (4-2 \log (5))+4 \log ^2(5)+e^{2 x} (4-8 \log (5)+\log ^2(5))+e^x (-8 \log (5)+4 \log ^2(5))}{2 x}))} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\log \left (\log ^2\left (\frac {\left (2+e^x\right )^2 \left (e^x-\log (5)\right )^2}{2 x}\right )\right )\right ) \]

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Rubi [B] time = 0.49, antiderivative size = 66, normalized size of antiderivative = 2.28, number of steps used = 1, number of rules used = 1, integrand size = 195, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {6684} \begin {gather*} \log \left (\log \left (\log ^2\left (\frac {e^{4 x}+e^{2 x} \left (4+\log ^2(5)-8 \log (5)\right )+2 e^{3 x} (2-\log (5))-4 e^x (2-\log (5)) \log (5)+4 \log ^2(5)}{2 x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(2*x)*(-2 + 8*x) + 4*Log[5] + E^x*(-4 + 8*x + (2 - 4*x)*Log[5]))/((E^(2*x)*x - 2*x*Log[5] + E^x*(2*x -

x*Log[5]))*Log[(E^(4*x) + E^(3*x)*(4 - 2*Log[5]) + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5] + Log[5]^2) + E^x*(-8*Lo

g[5] + 4*Log[5]^2))/(2*x)]*Log[Log[(E^(4*x) + E^(3*x)*(4 - 2*Log[5]) + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5] + Lo

g[5]^2) + E^x*(-8*Log[5] + 4*Log[5]^2))/(2*x)]^2]),x]

[Out]

Log[Log[Log[(E^(4*x) + 2*E^(3*x)*(2 - Log[5]) - 4*E^x*(2 - Log[5])*Log[5] + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5]

+ Log[5]^2))/(2*x)]^2]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !Fa

lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (\log ^2\left (\frac {e^{4 x}+2 e^{3 x} (2-\log (5))-4 e^x (2-\log (5)) \log (5)+4 \log ^2(5)+e^{2 x} \left (4-8 \log (5)+\log ^2(5)\right )}{2 x}\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [F] time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{2 x} (-2+8 x)+4 \log (5)+e^x (-4+8 x+(2-4 x) \log (5))}{\left (e^{2 x} x-2 x \log (5)+e^x (2 x-x \log (5))\right ) \log \left (\frac {e^{4 x}+e^{3 x} (4-2 \log (5))+4 \log ^2(5)+e^{2 x} \left (4-8 \log (5)+\log ^2(5)\right )+e^x \left (-8 \log (5)+4 \log ^2(5)\right )}{2 x}\right ) \log \left (\log ^2\left (\frac {e^{4 x}+e^{3 x} (4-2 \log (5))+4 \log ^2(5)+e^{2 x} \left (4-8 \log (5)+\log ^2(5)\right )+e^x \left (-8 \log (5)+4 \log ^2(5)\right )}{2 x}\right )\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Integrate[(E^(2*x)*(-2 + 8*x) + 4*Log[5] + E^x*(-4 + 8*x + (2 - 4*x)*Log[5]))/((E^(2*x)*x - 2*x*Log[5] + E^x*(

2*x - x*Log[5]))*Log[(E^(4*x) + E^(3*x)*(4 - 2*Log[5]) + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5] + Log[5]^2) + E^x*

(-8*Log[5] + 4*Log[5]^2))/(2*x)]*Log[Log[(E^(4*x) + E^(3*x)*(4 - 2*Log[5]) + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5

] + Log[5]^2) + E^x*(-8*Log[5] + 4*Log[5]^2))/(2*x)]^2]),x]

[Out]

Integrate[(E^(2*x)*(-2 + 8*x) + 4*Log[5] + E^x*(-4 + 8*x + (2 - 4*x)*Log[5]))/((E^(2*x)*x - 2*x*Log[5] + E^x*(

2*x - x*Log[5]))*Log[(E^(4*x) + E^(3*x)*(4 - 2*Log[5]) + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5] + Log[5]^2) + E^x*

(-8*Log[5] + 4*Log[5]^2))/(2*x)]*Log[Log[(E^(4*x) + E^(3*x)*(4 - 2*Log[5]) + 4*Log[5]^2 + E^(2*x)*(4 - 8*Log[5

] + Log[5]^2) + E^x*(-8*Log[5] + 4*Log[5]^2))/(2*x)]^2]), x]

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fricas [B] time = 0.98, size = 62, normalized size = 2.14 \begin {gather*} \log \left (\log \left (\log \left (-\frac {2 \, {\left (\log \relax (5) - 2\right )} e^{\left (3 \, x\right )} - {\left (\log \relax (5)^{2} - 8 \, \log \relax (5) + 4\right )} e^{\left (2 \, x\right )} - 4 \, {\left (\log \relax (5)^{2} - 2 \, \log \relax (5)\right )} e^{x} - 4 \, \log \relax (5)^{2} - e^{\left (4 \, x\right )}}{2 \, x}\right )^{2}\right )\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-2)*exp(x)^2+((-4*x+2)*log(5)+8*x-4)*exp(x)+4*log(5))/(x*exp(x)^2+(-x*log(5)+2*x)*exp(x)-2*x*lo

g(5))/log(1/2*(exp(x)^4+(-2*log(5)+4)*exp(x)^3+(log(5)^2-8*log(5)+4)*exp(x)^2+(4*log(5)^2-8*log(5))*exp(x)+4*l

og(5)^2)/x)/log(log(1/2*(exp(x)^4+(-2*log(5)+4)*exp(x)^3+(log(5)^2-8*log(5)+4)*exp(x)^2+(4*log(5)^2-8*log(5))*

exp(x)+4*log(5)^2)/x)^2),x, algorithm="fricas")

[Out]

log(log(log(-1/2*(2*(log(5) - 2)*e^(3*x) - (log(5)^2 - 8*log(5) + 4)*e^(2*x) - 4*(log(5)^2 - 2*log(5))*e^x - 4

*log(5)^2 - e^(4*x))/x)^2))

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giac [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(((8*x-2)*exp(x)^2+((-4*x+2)*log(5)+8*x-4)*exp(x)+4*log(5))/(x*exp(x)^2+(-x*log(5)+2*x)*exp(x)-2*x*lo

g(5))/log(1/2*(exp(x)^4+(-2*log(5)+4)*exp(x)^3+(log(5)^2-8*log(5)+4)*exp(x)^2+(4*log(5)^2-8*log(5))*exp(x)+4*l

og(5)^2)/x)/log(log(1/2*(exp(x)^4+(-2*log(5)+4)*exp(x)^3+(log(5)^2-8*log(5)+4)*exp(x)^2+(4*log(5)^2-8*log(5))*

exp(x)+4*log(5)^2)/x)^2),x, algorithm="giac")

[Out]

Timed out

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maple [C] time = 2.28, size = 3290, normalized size = 113.45


method	result	size

risch	\(\text {Expression too large to display}\)	\(3290\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x-2)*exp(x)^2+((-4*x+2)*ln(5)+8*x-4)*exp(x)+4*ln(5))/(x*exp(x)^2+(-x*ln(5)+2*x)*exp(x)-2*x*ln(5))/ln(1

/2*(exp(x)^4+(-2*ln(5)+4)*exp(x)^3+(ln(5)^2-8*ln(5)+4)*exp(x)^2+(4*ln(5)^2-8*ln(5))*exp(x)+4*ln(5)^2)/x)/ln(ln

(1/2*(exp(x)^4+(-2*ln(5)+4)*exp(x)^3+(ln(5)^2-8*ln(5)+4)*exp(x)^2+(4*ln(5)^2-8*ln(5))*exp(x)+4*ln(5)^2)/x)^2),

x,method=_RETURNVERBOSE)

[Out]

ln(ln(4*I*ln(exp(x)+2)+Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)+

Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)-2*I*ln(x)+4*I*ln(-e

xp(x)+ln(5))+Pi*csgn(I*(exp(x)-ln(5))^2)^3+Pi*csgn(I*(exp(x)+2)^2)^3+Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^

3+Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^3-2*I*ln(2)+Pi*csgn(I*(exp(x)-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)+

2*Pi*csgn(I*(exp(x)-ln(5)))*csgn(I*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2))^2-2*Pi*csgn(

I*(exp(x)+2)^2)^2*csgn(I*(exp(x)+2))-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2-Pi*csgn(I

*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2-Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x

)^2-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2)-1/4*I*(2*Pi*csgn(I*(-4

*I*ln(exp(x)+2)-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)-Pi*csgn

(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)+2*I*ln(x)-4*I*ln(-exp(x)+l

n(5))-Pi*csgn(I*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^3-Pi*cs

gn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^3+2*I*ln(2)-Pi*csgn(I*(exp(x)-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)-2*Pi*cs

gn(I*(exp(x)-ln(5)))*csgn(I*(exp(x)-ln(5))^2)^2-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2))^2+2*Pi*csgn(I*(exp(

x)+2)^2)^2*csgn(I*(exp(x)+2))+Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x

)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2+Pi*

csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2)^2)^2+Pi*csgn(I*(-4*I*ln(exp(x

)+2)-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)-Pi*csgn(I/x)*csgn(

I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)+2*I*ln(x)-4*I*ln(-exp(x)+ln(5))-Pi*cs

gn(I*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x

)+2)^2*(exp(x)-ln(5))^2/x)^3+2*I*ln(2)-Pi*csgn(I*(exp(x)-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)-2*Pi*csgn(I*(exp(x

)-ln(5)))*csgn(I*(exp(x)-ln(5))^2)^2-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2))^2+2*Pi*csgn(I*(exp(x)+2)^2)^2*

csgn(I*(exp(x)+2))+Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x)-ln(5))^2)

*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2+Pi*csgn(I*(exp

(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2))^2*csgn(I*(-4*I*ln(exp(x)+2)-Pi*csgn(I*(

exp(x)+2)^2)*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)-Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(

exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)+2*I*ln(x)-4*I*ln(-exp(x)+ln(5))-Pi*csgn(I*(exp(x)-ln(

5))^2)^3-Pi*csgn(I*(exp(x)+2)^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-l

n(5))^2/x)^3+2*I*ln(2)-Pi*csgn(I*(exp(x)-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)-2*Pi*csgn(I*(exp(x)-ln(5)))*csgn(I

*(exp(x)-ln(5))^2)^2-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2))^2+2*Pi*csgn(I*(exp(x)+2)^2)^2*csgn(I*(exp(x)+2

))+Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+

2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2+Pi*csgn(I*(exp(x)+2)^2*(exp(x)

-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2)^2)+2*Pi*csgn(I*(-4*I*ln(exp(x)+2)-Pi*csgn(I*(exp(x)+2)^2

)*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)-Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5

))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)+2*I*ln(x)-4*I*ln(-exp(x)+ln(5))-Pi*csgn(I*(exp(x)-ln(5))^2)^3-Pi

*csgn(I*(exp(x)+2)^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^

3+2*I*ln(2)-Pi*csgn(I*(exp(x)-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)-2*Pi*csgn(I*(exp(x)-ln(5)))*csgn(I*(exp(x)-ln

(5))^2)^2-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)+2))^2+2*Pi*csgn(I*(exp(x)+2)^2)^2*csgn(I*(exp(x)+2))+Pi*csgn(

I*(exp(x)+2)^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x

)-ln(5))^2)^2+Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2+Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*

csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2))*csgn(I*(-4*I*ln(exp(x)+2)-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)-l

n(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)-Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(

x)+2)^2*(exp(x)-ln(5))^2/x)+2*I*ln(x)-4*I*ln(-exp(x)+ln(5))-Pi*csgn(I*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)

^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^3+2*I*ln(2)-Pi*csg

n(I*(exp(x)-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)-2*Pi*csgn(I*(exp(x)-ln(5)))*csgn(I*(exp(x)-ln(5))^2)^2-Pi*csgn(

I*(exp(x)+2)^2)*csgn(I*(exp(x)+2))^2+2*Pi*csgn(I*(exp(x)+2)^2)^2*csgn(I*(exp(x)+2))+Pi*csgn(I*(exp(x)+2)^2)*cs

gn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*c

sgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2+Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^

2*(exp(x)-ln(5))^2/x)^2)^2)^2-Pi*csgn(I*(-4*I*ln(exp(x)+2)-Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x)-ln(5))^2)*cs

gn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)-Pi*csgn(I/x)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(ex

p(x)-ln(5))^2/x)+2*I*ln(x)-4*I*ln(-exp(x)+ln(5))-Pi*csgn(I*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2)^3-Pi*cs

gn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^3-Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^3+2*I*ln(2)-Pi*csgn(I*(exp(x)

-ln(5)))^2*csgn(I*(exp(x)-ln(5))^2)-2*Pi*csgn(I*(exp(x)-ln(5)))*csgn(I*(exp(x)-ln(5))^2)^2-Pi*csgn(I*(exp(x)+2

)^2)*csgn(I*(exp(x)+2))^2+2*Pi*csgn(I*(exp(x)+2)^2)^2*csgn(I*(exp(x)+2))+Pi*csgn(I*(exp(x)+2)^2)*csgn(I*(exp(x

)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)^2+Pi*csgn(I/x)*cs

gn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2/x)^2+Pi*csgn(I*(exp(x)+2)^2*(exp(x)-ln(5))^2)*csgn(I*(exp(x)+2)^2*(exp(x)-l

n(5))^2/x)^2)^2)^3-4*I*ln(2)-2*Pi))

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maxima [A] time = 6.08, size = 28, normalized size = 0.97 \begin {gather*} \log \left (\log \left (-\log \relax (2) - \log \relax (x) + 2 \, \log \left (e^{x} - \log \relax (5)\right ) + 2 \, \log \left (e^{x} + 2\right )\right )\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-2)*exp(x)^2+((-4*x+2)*log(5)+8*x-4)*exp(x)+4*log(5))/(x*exp(x)^2+(-x*log(5)+2*x)*exp(x)-2*x*lo

g(5))/log(1/2*(exp(x)^4+(-2*log(5)+4)*exp(x)^3+(log(5)^2-8*log(5)+4)*exp(x)^2+(4*log(5)^2-8*log(5))*exp(x)+4*l

og(5)^2)/x)/log(log(1/2*(exp(x)^4+(-2*log(5)+4)*exp(x)^3+(log(5)^2-8*log(5)+4)*exp(x)^2+(4*log(5)^2-8*log(5))*

exp(x)+4*log(5)^2)/x)^2),x, algorithm="maxima")

[Out]

log(log(-log(2) - log(x) + 2*log(e^x - log(5)) + 2*log(e^x + 2)))

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mupad [B] time = 3.63, size = 65, normalized size = 2.24 \begin {gather*} \ln \left (\ln \left ({\ln \left (\frac {\frac {{\mathrm {e}}^{4\,x}}{2}-\frac {{\mathrm {e}}^x\,\left (8\,\ln \relax (5)-4\,{\ln \relax (5)}^2\right )}{2}+\frac {{\mathrm {e}}^{2\,x}\,\left ({\ln \relax (5)}^2-8\,\ln \relax (5)+4\right )}{2}-\frac {{\mathrm {e}}^{3\,x}\,\left (2\,\ln \relax (5)-4\right )}{2}+2\,{\ln \relax (5)}^2}{x}\right )}^2\right )\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(5) - exp(x)*(log(5)*(4*x - 2) - 8*x + 4) + exp(2*x)*(8*x - 2))/(log(log((exp(4*x)/2 - (exp(x)*(8*lo

g(5) - 4*log(5)^2))/2 + (exp(2*x)*(log(5)^2 - 8*log(5) + 4))/2 - (exp(3*x)*(2*log(5) - 4))/2 + 2*log(5)^2)/x)^

2)*log((exp(4*x)/2 - (exp(x)*(8*log(5) - 4*log(5)^2))/2 + (exp(2*x)*(log(5)^2 - 8*log(5) + 4))/2 - (exp(3*x)*(

2*log(5) - 4))/2 + 2*log(5)^2)/x)*(x*exp(2*x) - 2*x*log(5) + exp(x)*(2*x - x*log(5)))),x)

[Out]

log(log(log((exp(4*x)/2 - (exp(x)*(8*log(5) - 4*log(5)^2))/2 + (exp(2*x)*(log(5)^2 - 8*log(5) + 4))/2 - (exp(3

*x)*(2*log(5) - 4))/2 + 2*log(5)^2)/x)^2))

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sympy [B] time = 32.96, size = 70, normalized size = 2.41 \begin {gather*} \log {\left (\log {\left (\log {\left (\frac {\frac {e^{4 x}}{2} + \frac {\left (4 - 2 \log {\relax (5 )}\right ) e^{3 x}}{2} + \frac {\left (- 8 \log {\relax (5 )} + \log {\relax (5 )}^{2} + 4\right ) e^{2 x}}{2} + \frac {\left (- 8 \log {\relax (5 )} + 4 \log {\relax (5 )}^{2}\right ) e^{x}}{2} + 2 \log {\relax (5 )}^{2}}{x} \right )}^{2} \right )} \right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-2)*exp(x)**2+((-4*x+2)*ln(5)+8*x-4)*exp(x)+4*ln(5))/(x*exp(x)**2+(-x*ln(5)+2*x)*exp(x)-2*x*ln(

5))/ln(1/2*(exp(x)**4+(-2*ln(5)+4)*exp(x)**3+(ln(5)**2-8*ln(5)+4)*exp(x)**2+(4*ln(5)**2-8*ln(5))*exp(x)+4*ln(5

)**2)/x)/ln(ln(1/2*(exp(x)**4+(-2*ln(5)+4)*exp(x)**3+(ln(5)**2-8*ln(5)+4)*exp(x)**2+(4*ln(5)**2-8*ln(5))*exp(x

)+4*ln(5)**2)/x)**2),x)

[Out]

log(log(log((exp(4*x)/2 + (4 - 2*log(5))*exp(3*x)/2 + (-8*log(5) + log(5)**2 + 4)*exp(2*x)/2 + (-8*log(5) + 4*

log(5)**2)*exp(x)/2 + 2*log(5)**2)/x)**2))

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