3.13.1 \(\int \frac {-2+e^{-50+2 e^x-2 x} (4 x-4 e^x x)}{x \log ^3(\frac {e^{4-e^{-50+2 e^x-2 x}}}{x})} \, dx\)

Optimal. Leaf size=29 \[ 1-\frac {1}{\log ^2\left (\frac {e^{4-e^{-50+2 e^x-2 x}}}{x}\right )} \]

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Rubi [A]  time = 0.18, antiderivative size = 27, normalized size of antiderivative = 0.93, number of steps used = 1, number of rules used = 1, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6686} \begin {gather*} -\frac {1}{\log ^2\left (\frac {e^{4-e^{-2 x+2 e^x-50}}}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + E^(-50 + 2*E^x - 2*x)*(4*x - 4*E^x*x))/(x*Log[E^(4 - E^(-50 + 2*E^x - 2*x))/x]^3),x]

[Out]

-Log[E^(4 - E^(-50 + 2*E^x - 2*x))/x]^(-2)

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {1}{\log ^2\left (\frac {e^{4-e^{-50+2 e^x-2 x}}}{x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 27, normalized size = 0.93 \begin {gather*} -\frac {1}{\log ^2\left (\frac {e^{4-e^{-50+2 e^x-2 x}}}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^(-50 + 2*E^x - 2*x)*(4*x - 4*E^x*x))/(x*Log[E^(4 - E^(-50 + 2*E^x - 2*x))/x]^3),x]

[Out]

-Log[E^(4 - E^(-50 + 2*E^x - 2*x))/x]^(-2)

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fricas [A]  time = 0.61, size = 24, normalized size = 0.83 \begin {gather*} -\frac {1}{\log \left (\frac {e^{\left (-e^{\left (-2 \, x + 2 \, e^{x} - 50\right )} + 4\right )}}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)*x+4*x)*exp(exp(x)-x-25)^2-2)/x/log(exp(-exp(exp(x)-x-25)^2+4)/x)^3,x, algorithm="fricas"
)

[Out]

-1/log(e^(-e^(-2*x + 2*e^x - 50) + 4)/x)^2

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giac [B]  time = 1.93, size = 5629, normalized size = 194.10 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)*x+4*x)*exp(exp(x)-x-25)^2-2)/x/log(exp(-exp(exp(x)-x-25)^2+4)/x)^3,x, algorithm="giac")

[Out]

-(8*x^3*e^(5*x + 200)*log(x)^5 - 24*x^3*e^(4*x + 200)*log(x)^5 + 24*x^3*e^(3*x + 200)*log(x)^5 - 8*x^3*e^(2*x
+ 200)*log(x)^5 - 160*x^3*e^(5*x + 200)*log(x)^4 + 480*x^3*e^(4*x + 200)*log(x)^4 + 16*x^3*e^(3*x + 2*e^x + 15
0)*log(x)^4 - 480*x^3*e^(3*x + 200)*log(x)^4 - 48*x^3*e^(2*x + 2*e^x + 150)*log(x)^4 + 160*x^3*e^(2*x + 200)*l
og(x)^4 + 48*x^3*e^(x + 2*e^x + 150)*log(x)^4 - 16*x^3*e^(2*e^x + 150)*log(x)^4 + 1280*x^3*e^(5*x + 200)*log(x
)^3 - 3840*x^3*e^(4*x + 200)*log(x)^3 - 256*x^3*e^(3*x + 2*e^x + 150)*log(x)^3 + 3840*x^3*e^(3*x + 200)*log(x)
^3 + 768*x^3*e^(2*x + 2*e^x + 150)*log(x)^3 - 1280*x^3*e^(2*x + 200)*log(x)^3 + 8*x^3*e^(x + 4*e^x + 100)*log(
x)^3 - 768*x^3*e^(x + 2*e^x + 150)*log(x)^3 + 24*x^3*e^(-x + 4*e^x + 100)*log(x)^3 - 8*x^3*e^(-2*x + 4*e^x + 1
00)*log(x)^3 - 24*x^3*e^(4*e^x + 100)*log(x)^3 + 256*x^3*e^(2*e^x + 150)*log(x)^3 - 12*x^2*e^(4*x + 200)*log(x
)^4 + 24*x^2*e^(3*x + 200)*log(x)^4 - 12*x^2*e^(2*x + 200)*log(x)^4 - 5120*x^3*e^(5*x + 200)*log(x)^2 + 15360*
x^3*e^(4*x + 200)*log(x)^2 + 1536*x^3*e^(3*x + 2*e^x + 150)*log(x)^2 - 15360*x^3*e^(3*x + 200)*log(x)^2 - 4608
*x^3*e^(2*x + 2*e^x + 150)*log(x)^2 + 5120*x^3*e^(2*x + 200)*log(x)^2 - 96*x^3*e^(x + 4*e^x + 100)*log(x)^2 +
4608*x^3*e^(x + 2*e^x + 150)*log(x)^2 - 288*x^3*e^(-x + 4*e^x + 100)*log(x)^2 + 96*x^3*e^(-2*x + 4*e^x + 100)*
log(x)^2 + 288*x^3*e^(4*e^x + 100)*log(x)^2 - 1536*x^3*e^(2*e^x + 150)*log(x)^2 + 192*x^2*e^(4*x + 200)*log(x)
^3 - 384*x^2*e^(3*x + 200)*log(x)^3 - 24*x^2*e^(2*x + 2*e^x + 150)*log(x)^3 + 192*x^2*e^(2*x + 200)*log(x)^3 +
48*x^2*e^(x + 2*e^x + 150)*log(x)^3 - 24*x^2*e^(2*e^x + 150)*log(x)^3 + 10240*x^3*e^(5*x + 200)*log(x) - 3072
0*x^3*e^(4*x + 200)*log(x) - 4096*x^3*e^(3*x + 2*e^x + 150)*log(x) + 30720*x^3*e^(3*x + 200)*log(x) + 12288*x^
3*e^(2*x + 2*e^x + 150)*log(x) - 10240*x^3*e^(2*x + 200)*log(x) + 384*x^3*e^(x + 4*e^x + 100)*log(x) - 12288*x
^3*e^(x + 2*e^x + 150)*log(x) + 1152*x^3*e^(-x + 4*e^x + 100)*log(x) - 384*x^3*e^(-2*x + 4*e^x + 100)*log(x) -
1152*x^3*e^(4*e^x + 100)*log(x) + 4096*x^3*e^(2*e^x + 150)*log(x) - 1152*x^2*e^(4*x + 200)*log(x)^2 + 2304*x^
2*e^(3*x + 200)*log(x)^2 + 288*x^2*e^(2*x + 2*e^x + 150)*log(x)^2 - 1152*x^2*e^(2*x + 200)*log(x)^2 - 576*x^2*
e^(x + 2*e^x + 150)*log(x)^2 + 24*x^2*e^(-x + 4*e^x + 100)*log(x)^2 - 12*x^2*e^(-2*x + 4*e^x + 100)*log(x)^2 -
12*x^2*e^(4*e^x + 100)*log(x)^2 + 288*x^2*e^(2*e^x + 150)*log(x)^2 + 6*x*e^(3*x + 200)*log(x)^3 - 6*x*e^(2*x
+ 200)*log(x)^3 - 8192*x^3*e^(5*x + 200) + 24576*x^3*e^(4*x + 200) + 4096*x^3*e^(3*x + 2*e^x + 150) - 24576*x^
3*e^(3*x + 200) - 12288*x^3*e^(2*x + 2*e^x + 150) + 8192*x^3*e^(2*x + 200) - 512*x^3*e^(x + 4*e^x + 100) + 122
88*x^3*e^(x + 2*e^x + 150) - 1536*x^3*e^(-x + 4*e^x + 100) + 512*x^3*e^(-2*x + 4*e^x + 100) + 1536*x^3*e^(4*e^
x + 100) - 4096*x^3*e^(2*e^x + 150) + 3072*x^2*e^(4*x + 200)*log(x) - 6144*x^2*e^(3*x + 200)*log(x) - 1152*x^2
*e^(2*x + 2*e^x + 150)*log(x) + 3072*x^2*e^(2*x + 200)*log(x) + 2304*x^2*e^(x + 2*e^x + 150)*log(x) - 192*x^2*
e^(-x + 4*e^x + 100)*log(x) + 96*x^2*e^(-2*x + 4*e^x + 100)*log(x) + 96*x^2*e^(4*e^x + 100)*log(x) - 1152*x^2*
e^(2*e^x + 150)*log(x) - 72*x*e^(3*x + 200)*log(x)^2 + 72*x*e^(2*x + 200)*log(x)^2 + 12*x*e^(x + 2*e^x + 150)*
log(x)^2 - 12*x*e^(2*e^x + 150)*log(x)^2 - 3072*x^2*e^(4*x + 200) + 6144*x^2*e^(3*x + 200) + 1536*x^2*e^(2*x +
2*e^x + 150) - 3072*x^2*e^(2*x + 200) - 3072*x^2*e^(x + 2*e^x + 150) + 384*x^2*e^(-x + 4*e^x + 100) - 192*x^2
*e^(-2*x + 4*e^x + 100) - 192*x^2*e^(4*e^x + 100) + 1536*x^2*e^(2*e^x + 150) + 288*x*e^(3*x + 200)*log(x) - 28
8*x*e^(2*x + 200)*log(x) - 96*x*e^(x + 2*e^x + 150)*log(x) + 6*x*e^(-x + 4*e^x + 100)*log(x) - 6*x*e^(-2*x + 4
*e^x + 100)*log(x) + 96*x*e^(2*e^x + 150)*log(x) - e^(2*x + 200)*log(x)^2 - 384*x*e^(3*x + 200) + 384*x*e^(2*x
+ 200) + 192*x*e^(x + 2*e^x + 150) - 24*x*e^(-x + 4*e^x + 100) + 24*x*e^(-2*x + 4*e^x + 100) - 192*x*e^(2*e^x
+ 150) + 8*e^(2*x + 200)*log(x) - 2*e^(2*e^x + 150)*log(x) - 16*e^(2*x + 200) - e^(-2*x + 4*e^x + 100) + 8*e^
(2*e^x + 150))/(8*x^3*e^(5*x + 200)*log(x)^7 - 24*x^3*e^(4*x + 200)*log(x)^7 + 24*x^3*e^(3*x + 200)*log(x)^7 -
8*x^3*e^(2*x + 200)*log(x)^7 - 224*x^3*e^(5*x + 200)*log(x)^6 + 672*x^3*e^(4*x + 200)*log(x)^6 + 32*x^3*e^(3*
x + 2*e^x + 150)*log(x)^6 - 672*x^3*e^(3*x + 200)*log(x)^6 - 96*x^3*e^(2*x + 2*e^x + 150)*log(x)^6 + 224*x^3*e
^(2*x + 200)*log(x)^6 + 96*x^3*e^(x + 2*e^x + 150)*log(x)^6 - 32*x^3*e^(2*e^x + 150)*log(x)^6 + 2688*x^3*e^(5*
x + 200)*log(x)^5 - 8064*x^3*e^(4*x + 200)*log(x)^5 - 768*x^3*e^(3*x + 2*e^x + 150)*log(x)^5 + 8064*x^3*e^(3*x
+ 200)*log(x)^5 + 2304*x^3*e^(2*x + 2*e^x + 150)*log(x)^5 - 2688*x^3*e^(2*x + 200)*log(x)^5 + 48*x^3*e^(x + 4
*e^x + 100)*log(x)^5 - 2304*x^3*e^(x + 2*e^x + 150)*log(x)^5 + 144*x^3*e^(-x + 4*e^x + 100)*log(x)^5 - 48*x^3*
e^(-2*x + 4*e^x + 100)*log(x)^5 - 144*x^3*e^(4*e^x + 100)*log(x)^5 + 768*x^3*e^(2*e^x + 150)*log(x)^5 - 12*x^2
*e^(4*x + 200)*log(x)^6 + 24*x^2*e^(3*x + 200)*log(x)^6 - 12*x^2*e^(2*x + 200)*log(x)^6 - 17920*x^3*e^(5*x + 2
00)*log(x)^4 + 53760*x^3*e^(4*x + 200)*log(x)^4 + 7680*x^3*e^(3*x + 2*e^x + 150)*log(x)^4 - 53760*x^3*e^(3*x +
200)*log(x)^4 - 23040*x^3*e^(2*x + 2*e^x + 150)*log(x)^4 + 17920*x^3*e^(2*x + 200)*log(x)^4 - 960*x^3*e^(x +
4*e^x + 100)*log(x)^4 + 23040*x^3*e^(x + 2*e^x + 150)*log(x)^4 + 32*x^3*e^(-x + 6*e^x + 50)*log(x)^4 - 2880*x^
3*e^(-x + 4*e^x + 100)*log(x)^4 - 96*x^3*e^(-2*x + 6*e^x + 50)*log(x)^4 + 960*x^3*e^(-2*x + 4*e^x + 100)*log(x
)^4 + 96*x^3*e^(-3*x + 6*e^x + 50)*log(x)^4 - 32*x^3*e^(-4*x + 6*e^x + 50)*log(x)^4 + 2880*x^3*e^(4*e^x + 100)
*log(x)^4 - 7680*x^3*e^(2*e^x + 150)*log(x)^4 + 288*x^2*e^(4*x + 200)*log(x)^5 - 576*x^2*e^(3*x + 200)*log(x)^
5 - 48*x^2*e^(2*x + 2*e^x + 150)*log(x)^5 + 288*x^2*e^(2*x + 200)*log(x)^5 + 96*x^2*e^(x + 2*e^x + 150)*log(x)
^5 - 48*x^2*e^(2*e^x + 150)*log(x)^5 + 71680*x^3*e^(5*x + 200)*log(x)^3 - 215040*x^3*e^(4*x + 200)*log(x)^3 -
40960*x^3*e^(3*x + 2*e^x + 150)*log(x)^3 + 215040*x^3*e^(3*x + 200)*log(x)^3 + 122880*x^3*e^(2*x + 2*e^x + 150
)*log(x)^3 - 71680*x^3*e^(2*x + 200)*log(x)^3 + 7680*x^3*e^(x + 4*e^x + 100)*log(x)^3 - 122880*x^3*e^(x + 2*e^
x + 150)*log(x)^3 - 512*x^3*e^(-x + 6*e^x + 50)*log(x)^3 + 23040*x^3*e^(-x + 4*e^x + 100)*log(x)^3 + 1536*x^3*
e^(-2*x + 6*e^x + 50)*log(x)^3 - 7680*x^3*e^(-2*x + 4*e^x + 100)*log(x)^3 + 8*x^3*e^(-3*x + 8*e^x)*log(x)^3 -
1536*x^3*e^(-3*x + 6*e^x + 50)*log(x)^3 - 24*x^3*e^(-4*x + 8*e^x)*log(x)^3 + 512*x^3*e^(-4*x + 6*e^x + 50)*log
(x)^3 + 24*x^3*e^(-5*x + 8*e^x)*log(x)^3 - 8*x^3*e^(-6*x + 8*e^x)*log(x)^3 - 23040*x^3*e^(4*e^x + 100)*log(x)^
3 + 40960*x^3*e^(2*e^x + 150)*log(x)^3 - 2880*x^2*e^(4*x + 200)*log(x)^4 + 5760*x^2*e^(3*x + 200)*log(x)^4 + 9
60*x^2*e^(2*x + 2*e^x + 150)*log(x)^4 - 2880*x^2*e^(2*x + 200)*log(x)^4 - 1920*x^2*e^(x + 2*e^x + 150)*log(x)^
4 + 144*x^2*e^(-x + 4*e^x + 100)*log(x)^4 - 72*x^2*e^(-2*x + 4*e^x + 100)*log(x)^4 - 72*x^2*e^(4*e^x + 100)*lo
g(x)^4 + 960*x^2*e^(2*e^x + 150)*log(x)^4 + 6*x*e^(3*x + 200)*log(x)^5 - 6*x*e^(2*x + 200)*log(x)^5 - 172032*x
^3*e^(5*x + 200)*log(x)^2 + 516096*x^3*e^(4*x + 200)*log(x)^2 + 122880*x^3*e^(3*x + 2*e^x + 150)*log(x)^2 - 51
6096*x^3*e^(3*x + 200)*log(x)^2 - 368640*x^3*e^(2*x + 2*e^x + 150)*log(x)^2 + 172032*x^3*e^(2*x + 200)*log(x)^
2 - 30720*x^3*e^(x + 4*e^x + 100)*log(x)^2 + 368640*x^3*e^(x + 2*e^x + 150)*log(x)^2 + 3072*x^3*e^(-x + 6*e^x
+ 50)*log(x)^2 - 92160*x^3*e^(-x + 4*e^x + 100)*log(x)^2 - 9216*x^3*e^(-2*x + 6*e^x + 50)*log(x)^2 + 30720*x^3
*e^(-2*x + 4*e^x + 100)*log(x)^2 - 96*x^3*e^(-3*x + 8*e^x)*log(x)^2 + 9216*x^3*e^(-3*x + 6*e^x + 50)*log(x)^2
+ 288*x^3*e^(-4*x + 8*e^x)*log(x)^2 - 3072*x^3*e^(-4*x + 6*e^x + 50)*log(x)^2 - 288*x^3*e^(-5*x + 8*e^x)*log(x
)^2 + 96*x^3*e^(-6*x + 8*e^x)*log(x)^2 + 92160*x^3*e^(4*e^x + 100)*log(x)^2 - 122880*x^3*e^(2*e^x + 150)*log(x
)^2 + 15360*x^2*e^(4*x + 200)*log(x)^3 - 30720*x^2*e^(3*x + 200)*log(x)^3 - 7680*x^2*e^(2*x + 2*e^x + 150)*log
(x)^3 + 15360*x^2*e^(2*x + 200)*log(x)^3 + 15360*x^2*e^(x + 2*e^x + 150)*log(x)^3 - 2304*x^2*e^(-x + 4*e^x + 1
00)*log(x)^3 - 48*x^2*e^(-2*x + 6*e^x + 50)*log(x)^3 + 1152*x^2*e^(-2*x + 4*e^x + 100)*log(x)^3 + 96*x^2*e^(-3
*x + 6*e^x + 50)*log(x)^3 - 48*x^2*e^(-4*x + 6*e^x + 50)*log(x)^3 + 1152*x^2*e^(4*e^x + 100)*log(x)^3 - 7680*x
^2*e^(2*e^x + 150)*log(x)^3 - 120*x*e^(3*x + 200)*log(x)^4 + 120*x*e^(2*x + 200)*log(x)^4 + 24*x*e^(x + 2*e^x
+ 150)*log(x)^4 - 24*x*e^(2*e^x + 150)*log(x)^4 + 229376*x^3*e^(5*x + 200)*log(x) - 688128*x^3*e^(4*x + 200)*l
og(x) - 196608*x^3*e^(3*x + 2*e^x + 150)*log(x) + 688128*x^3*e^(3*x + 200)*log(x) + 589824*x^3*e^(2*x + 2*e^x
+ 150)*log(x) - 229376*x^3*e^(2*x + 200)*log(x) + 61440*x^3*e^(x + 4*e^x + 100)*log(x) - 589824*x^3*e^(x + 2*e
^x + 150)*log(x) - 8192*x^3*e^(-x + 6*e^x + 50)*log(x) + 184320*x^3*e^(-x + 4*e^x + 100)*log(x) + 24576*x^3*e^
(-2*x + 6*e^x + 50)*log(x) - 61440*x^3*e^(-2*x + 4*e^x + 100)*log(x) + 384*x^3*e^(-3*x + 8*e^x)*log(x) - 24576
*x^3*e^(-3*x + 6*e^x + 50)*log(x) - 1152*x^3*e^(-4*x + 8*e^x)*log(x) + 8192*x^3*e^(-4*x + 6*e^x + 50)*log(x) +
1152*x^3*e^(-5*x + 8*e^x)*log(x) - 384*x^3*e^(-6*x + 8*e^x)*log(x) - 184320*x^3*e^(4*e^x + 100)*log(x) + 1966
08*x^3*e^(2*e^x + 150)*log(x) - 46080*x^2*e^(4*x + 200)*log(x)^2 + 92160*x^2*e^(3*x + 200)*log(x)^2 + 30720*x^
2*e^(2*x + 2*e^x + 150)*log(x)^2 - 46080*x^2*e^(2*x + 200)*log(x)^2 - 61440*x^2*e^(x + 2*e^x + 150)*log(x)^2 +
13824*x^2*e^(-x + 4*e^x + 100)*log(x)^2 + 576*x^2*e^(-2*x + 6*e^x + 50)*log(x)^2 - 6912*x^2*e^(-2*x + 4*e^x +
100)*log(x)^2 - 1152*x^2*e^(-3*x + 6*e^x + 50)*log(x)^2 - 12*x^2*e^(-4*x + 8*e^x)*log(x)^2 + 576*x^2*e^(-4*x
+ 6*e^x + 50)*log(x)^2 + 24*x^2*e^(-5*x + 8*e^x)*log(x)^2 - 12*x^2*e^(-6*x + 8*e^x)*log(x)^2 - 6912*x^2*e^(4*e
^x + 100)*log(x)^2 + 30720*x^2*e^(2*e^x + 150)*log(x)^2 + 960*x*e^(3*x + 200)*log(x)^3 - 960*x*e^(2*x + 200)*l
og(x)^3 - 384*x*e^(x + 2*e^x + 150)*log(x)^3 + 36*x*e^(-x + 4*e^x + 100)*log(x)^3 - 36*x*e^(-2*x + 4*e^x + 100
)*log(x)^3 + 384*x*e^(2*e^x + 150)*log(x)^3 - e^(2*x + 200)*log(x)^4 - 131072*x^3*e^(5*x + 200) + 393216*x^3*e
^(4*x + 200) + 131072*x^3*e^(3*x + 2*e^x + 150) - 393216*x^3*e^(3*x + 200) - 393216*x^3*e^(2*x + 2*e^x + 150)
+ 131072*x^3*e^(2*x + 200) - 49152*x^3*e^(x + 4*e^x + 100) + 393216*x^3*e^(x + 2*e^x + 150) + 8192*x^3*e^(-x +
6*e^x + 50) - 147456*x^3*e^(-x + 4*e^x + 100) - 24576*x^3*e^(-2*x + 6*e^x + 50) + 49152*x^3*e^(-2*x + 4*e^x +
100) - 512*x^3*e^(-3*x + 8*e^x) + 24576*x^3*e^(-3*x + 6*e^x + 50) + 1536*x^3*e^(-4*x + 8*e^x) - 8192*x^3*e^(-
4*x + 6*e^x + 50) - 1536*x^3*e^(-5*x + 8*e^x) + 512*x^3*e^(-6*x + 8*e^x) + 147456*x^3*e^(4*e^x + 100) - 131072
*x^3*e^(2*e^x + 150) + 73728*x^2*e^(4*x + 200)*log(x) - 147456*x^2*e^(3*x + 200)*log(x) - 61440*x^2*e^(2*x + 2
*e^x + 150)*log(x) + 73728*x^2*e^(2*x + 200)*log(x) + 122880*x^2*e^(x + 2*e^x + 150)*log(x) - 36864*x^2*e^(-x
+ 4*e^x + 100)*log(x) - 2304*x^2*e^(-2*x + 6*e^x + 50)*log(x) + 18432*x^2*e^(-2*x + 4*e^x + 100)*log(x) + 4608
*x^2*e^(-3*x + 6*e^x + 50)*log(x) + 96*x^2*e^(-4*x + 8*e^x)*log(x) - 2304*x^2*e^(-4*x + 6*e^x + 50)*log(x) - 1
92*x^2*e^(-5*x + 8*e^x)*log(x) + 96*x^2*e^(-6*x + 8*e^x)*log(x) + 18432*x^2*e^(4*e^x + 100)*log(x) - 61440*x^2
*e^(2*e^x + 150)*log(x) - 3840*x*e^(3*x + 200)*log(x)^2 + 3840*x*e^(2*x + 200)*log(x)^2 + 2304*x*e^(x + 2*e^x
+ 150)*log(x)^2 - 432*x*e^(-x + 4*e^x + 100)*log(x)^2 + 432*x*e^(-2*x + 4*e^x + 100)*log(x)^2 + 24*x*e^(-3*x +
6*e^x + 50)*log(x)^2 - 24*x*e^(-4*x + 6*e^x + 50)*log(x)^2 - 2304*x*e^(2*e^x + 150)*log(x)^2 + 16*e^(2*x + 20
0)*log(x)^3 - 4*e^(2*e^x + 150)*log(x)^3 - 49152*x^2*e^(4*x + 200) + 98304*x^2*e^(3*x + 200) + 49152*x^2*e^(2*
x + 2*e^x + 150) - 49152*x^2*e^(2*x + 200) - 98304*x^2*e^(x + 2*e^x + 150) + 36864*x^2*e^(-x + 4*e^x + 100) +
3072*x^2*e^(-2*x + 6*e^x + 50) - 18432*x^2*e^(-2*x + 4*e^x + 100) - 6144*x^2*e^(-3*x + 6*e^x + 50) - 192*x^2*e
^(-4*x + 8*e^x) + 3072*x^2*e^(-4*x + 6*e^x + 50) + 384*x^2*e^(-5*x + 8*e^x) - 192*x^2*e^(-6*x + 8*e^x) - 18432
*x^2*e^(4*e^x + 100) + 49152*x^2*e^(2*e^x + 150) + 7680*x*e^(3*x + 200)*log(x) - 7680*x*e^(2*x + 200)*log(x) -
6144*x*e^(x + 2*e^x + 150)*log(x) + 1728*x*e^(-x + 4*e^x + 100)*log(x) - 1728*x*e^(-2*x + 4*e^x + 100)*log(x)
- 192*x*e^(-3*x + 6*e^x + 50)*log(x) + 192*x*e^(-4*x + 6*e^x + 50)*log(x) + 6*x*e^(-5*x + 8*e^x)*log(x) - 6*x
*e^(-6*x + 8*e^x)*log(x) + 6144*x*e^(2*e^x + 150)*log(x) - 96*e^(2*x + 200)*log(x)^2 - 6*e^(-2*x + 4*e^x + 100
)*log(x)^2 + 48*e^(2*e^x + 150)*log(x)^2 - 6144*x*e^(3*x + 200) + 6144*x*e^(2*x + 200) + 6144*x*e^(x + 2*e^x +
150) - 2304*x*e^(-x + 4*e^x + 100) + 2304*x*e^(-2*x + 4*e^x + 100) + 384*x*e^(-3*x + 6*e^x + 50) - 384*x*e^(-
4*x + 6*e^x + 50) - 24*x*e^(-5*x + 8*e^x) + 24*x*e^(-6*x + 8*e^x) - 6144*x*e^(2*e^x + 150) + 256*e^(2*x + 200)
*log(x) + 48*e^(-2*x + 4*e^x + 100)*log(x) - 4*e^(-4*x + 6*e^x + 50)*log(x) - 192*e^(2*e^x + 150)*log(x) - 256
*e^(2*x + 200) - 96*e^(-2*x + 4*e^x + 100) + 16*e^(-4*x + 6*e^x + 50) - e^(-6*x + 8*e^x) + 256*e^(2*e^x + 150)
)

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maple [C]  time = 0.18, size = 186, normalized size = 6.41

method	result	size
risch	\(\frac {4}{\left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}}{x}\right )^{3}-2 i \ln \relax (x )+2 i \ln \left ({\mathrm e}^{-{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x -50}+4}\right )\right )^{2}}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(x)*x+4*x)*exp(exp(x)-x-25)^2-2)/x/ln(exp(-exp(exp(x)-x-25)^2+4)/x)^3,x,method=_RETURNVERBOSE)

[Out]

4/(Pi*csgn(I/x)*csgn(I*exp(-exp(2*exp(x)-2*x-50)+4))*csgn(I/x*exp(-exp(2*exp(x)-2*x-50)+4))-Pi*csgn(I/x)*csgn(
I/x*exp(-exp(2*exp(x)-2*x-50)+4))^2-Pi*csgn(I*exp(-exp(2*exp(x)-2*x-50)+4))*csgn(I/x*exp(-exp(2*exp(x)-2*x-50)
+4))^2+Pi*csgn(I/x*exp(-exp(2*exp(x)-2*x-50)+4))^3-2*I*ln(x)+2*I*ln(exp(-exp(2*exp(x)-2*x-50)+4)))^2

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maxima [B]  time = 0.81, size = 60, normalized size = 2.07 \begin {gather*} -\frac {e^{\left (4 \, x + 100\right )}}{{\left (e^{100} \log \relax (x)^{2} - 8 \, e^{100} \log \relax (x) + 16 \, e^{100}\right )} e^{\left (4 \, x\right )} + 2 \, {\left (e^{50} \log \relax (x) - 4 \, e^{50}\right )} e^{\left (2 \, x + 2 \, e^{x}\right )} + e^{\left (4 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)*x+4*x)*exp(exp(x)-x-25)^2-2)/x/log(exp(-exp(exp(x)-x-25)^2+4)/x)^3,x, algorithm="maxima"
)

[Out]

-e^(4*x + 100)/((e^100*log(x)^2 - 8*e^100*log(x) + 16*e^100)*e^(4*x) + 2*(e^50*log(x) - 4*e^50)*e^(2*x + 2*e^x
) + e^(4*e^x))

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mupad [B]  time = 0.95, size = 25, normalized size = 0.86 \begin {gather*} -\frac {1}{{\ln \left (\frac {{\mathrm {e}}^{-{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-50}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^4}{x}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(x) - 2*x - 50)*(4*x - 4*x*exp(x)) - 2)/(x*log(exp(4 - exp(2*exp(x) - 2*x - 50))/x)^3),x)

[Out]

-1/log((exp(-exp(-2*x)*exp(-50)*exp(2*exp(x)))*exp(4))/x)^2

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sympy [A]  time = 0.62, size = 22, normalized size = 0.76 \begin {gather*} - \frac {1}{\log {\left (\frac {e^{4 - e^{- 2 x + 2 e^{x} - 50}}}{x} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)*x+4*x)*exp(exp(x)-x-25)**2-2)/x/ln(exp(-exp(exp(x)-x-25)**2+4)/x)**3,x)

[Out]

-1/log(exp(4 - exp(-2*x + 2*exp(x) - 50))/x)**2

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