3.20.41 \(\int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log (x^2)+(e^{2 x} x^2+4 x^7) \log ^2(x^2)}{4 x^6 \log ^2(x^2)}} (2 \log ^2(x)+((-1+2 e^x x) \log (x)+3 \log ^2(x)) \log (x^2)+(-e^x x+e^x (5 x-x^2) \log (x)) \log ^2(x^2)+(2 x^6-2 x^7+e^{2 x} (2 x^2-x^3)) \log ^3(x^2))}{2 x^6 \log ^3(x^2)} \, dx\)

Optimal. Leaf size=33 \[ e^{-x-\frac {\left (e^x+\frac {\log (x)}{x \log \left (x^2\right )}\right )^2}{4 x^4}} x \]

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Rubi [F]  time = 27.95, 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 {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*Log[x]^2 + ((-1 + 2*E^x*x)*Log[x] + 3*Log[x]^2)*Log[x^2] + (-(E^x*x) + E^x*(5*x - x^2)*Log[x])*Log[x^2]
^2 + (2*x^6 - 2*x^7 + E^(2*x)*(2*x^2 - x^3))*Log[x^2]^3)/(2*E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*
x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3),x]

[Out]

Defer[Int][E^(-1/4*(Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(x^6*Log[x^2]^2)),
x] + Defer[Int][E^(2*x - (Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^
2]^2))/x^4, x] - Defer[Int][E^(2*x - (Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(
4*x^6*Log[x^2]^2))/x^3, x]/2 - Defer[Int][x/E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log
[x^2]^2)/(4*x^6*Log[x^2]^2)), x] + Defer[Int][Log[x]^2/(E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2
+ 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3), x] - Defer[Int][Log[x]/(E^((Log[x]^2 + 2*E^x*x*Log[x
]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^2), x]/2 + Defer[Int][Log[x]/(
E^((Log[x] + E^x*x*Log[x^2])^2/(4*x^6*Log[x^2]^2))*x^5*Log[x^2]^2), x] + (3*Defer[Int][Log[x]^2/(E^((Log[x]^2
+ 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^2), x])/2 - Def
er[Int][1/(E^((Log[x] + E^x*x*Log[x^2])^2/(4*x^6*Log[x^2]^2))*x^5*Log[x^2]), x]/2 + (5*Defer[Int][Log[x]/(E^((
Log[x] + E^x*x*Log[x^2])^2/(4*x^6*Log[x^2]^2))*x^5*Log[x^2]), x])/2 - Defer[Int][Log[x]/(E^((Log[x] + E^x*x*Lo
g[x^2])^2/(4*x^6*Log[x^2]^2))*x^4*Log[x^2]), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-2+x)}{x^4}-\frac {\exp \left (x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )}+\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)-\log (x) \log \left (x^2\right )+3 \log ^2(x) \log \left (x^2\right )+2 x^6 \log ^3\left (x^2\right )-2 x^7 \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-2+x)}{x^4} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx+\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)-\log (x) \log \left (x^2\right )+3 \log ^2(x) \log \left (x^2\right )+2 x^6 \log ^3\left (x^2\right )-2 x^7 \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {2 \exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4}+\frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3}\right ) \, dx\right )+\frac {1}{2} \int \left (-2 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-1+x)+\frac {2 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )}+\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x) (-1+3 \log (x))}{x^6 \log ^2\left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x) (-1+3 \log (x))}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (\log \left (x^2\right )+\log (x) \left (-2+(-5+x) \log \left (x^2\right )\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-1+x) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )+\frac {1}{2} \int \left (-\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )}+\frac {3 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \left (-\frac {2 \exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )}+\frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) (1-5 \log (x)+x \log (x))}{x^5 \log \left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \left (-\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )+\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x\right ) \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) (1-5 \log (x)+x \log (x))}{x^5 \log \left (x^2\right )} \, dx+\frac {3}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )} \, dx+\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx+\int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.47, size = 56, normalized size = 1.70 \begin {gather*} e^{-\frac {e^{2 x}}{4 x^4}-x-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}} x^{1-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*Log[x]^2 + ((-1 + 2*E^x*x)*Log[x] + 3*Log[x]^2)*Log[x^2] + (-(E^x*x) + E^x*(5*x - x^2)*Log[x])*Lo
g[x^2]^2 + (2*x^6 - 2*x^7 + E^(2*x)*(2*x^2 - x^3))*Log[x^2]^3)/(2*E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^
(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3),x]

[Out]

E^(-1/4*E^(2*x)/x^4 - x - Log[x]^2/(4*x^6*Log[x^2]^2))*x^(1 - E^x/(2*x^5*Log[x^2]))

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fricas [A]  time = 0.63, size = 29, normalized size = 0.88 \begin {gather*} x e^{\left (-\frac {16 \, x^{7} + 4 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{x} + 1}{16 \, x^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(
3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2
+2*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="fricas")

[Out]

x*e^(-1/16*(16*x^7 + 4*x^2*e^(2*x) + 4*x*e^x + 1)/x^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (2 \, x^{7} - 2 \, x^{6} + {\left (x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{3} + {\left ({\left (x^{2} - 5 \, x\right )} e^{x} \log \relax (x) + x e^{x}\right )} \log \left (x^{2}\right )^{2} - {\left ({\left (2 \, x e^{x} - 1\right )} \log \relax (x) + 3 \, \log \relax (x)^{2}\right )} \log \left (x^{2}\right ) - 2 \, \log \relax (x)^{2}\right )} e^{\left (-\frac {2 \, x e^{x} \log \left (x^{2}\right ) \log \relax (x) + {\left (4 \, x^{7} + x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{2} + \log \relax (x)^{2}}{4 \, x^{6} \log \left (x^{2}\right )^{2}}\right )}}{2 \, x^{6} \log \left (x^{2}\right )^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(
3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2
+2*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="giac")

[Out]

integrate(-1/2*((2*x^7 - 2*x^6 + (x^3 - 2*x^2)*e^(2*x))*log(x^2)^3 + ((x^2 - 5*x)*e^x*log(x) + x*e^x)*log(x^2)
^2 - ((2*x*e^x - 1)*log(x) + 3*log(x)^2)*log(x^2) - 2*log(x)^2)*e^(-1/4*(2*x*e^x*log(x^2)*log(x) + (4*x^7 + x^
2*e^(2*x))*log(x^2)^2 + log(x)^2)/(x^6*log(x^2)^2))/(x^6*log(x^2)^3), x)

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maple [C]  time = 1.39, size = 549, normalized size = 16.64

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {-4 \mathrm {csgn}\left (i x^{2}\right )^{6} \pi ^{2} x^{7}+16 \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right ) \pi ^{2} x^{7}-24 \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2} \pi ^{2} x^{7}+16 \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3} \pi ^{2} x^{7}-4 \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4} \pi ^{2} x^{7}-\mathrm {csgn}\left (i x^{2}\right )^{6} \pi ^{2} {\mathrm e}^{2 x} x^{2}+4 \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right ) \pi ^{2} {\mathrm e}^{2 x} x^{2}-6 \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2} \pi ^{2} {\mathrm e}^{2 x} x^{2}+4 \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3} \pi ^{2} {\mathrm e}^{2 x} x^{2}-\mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4} \pi ^{2} {\mathrm e}^{2 x} x^{2}-32 i \mathrm {csgn}\left (i x^{2}\right )^{3} \pi \ln \relax (x ) x^{7}+16 i \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi \ln \relax (x ) {\mathrm e}^{2 x} x^{2}-8 i \mathrm {csgn}\left (i x^{2}\right )^{3} \pi \ln \relax (x ) {\mathrm e}^{2 x} x^{2}+64 i \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi \ln \relax (x ) x^{7}+8 i \ln \relax (x ) {\mathrm e}^{x} x \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi -4 i \ln \relax (x ) {\mathrm e}^{x} x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi +64 x^{7} \ln \relax (x )^{2}-4 i \ln \relax (x ) {\mathrm e}^{x} x \mathrm {csgn}\left (i x^{2}\right )^{3} \pi -8 i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi \ln \relax (x ) {\mathrm e}^{2 x} x^{2}-32 i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi \ln \relax (x ) x^{7}+16 \ln \relax (x )^{2} {\mathrm e}^{2 x} x^{2}+16 x \,{\mathrm e}^{x} \ln \relax (x )^{2}+4 \ln \relax (x )^{2}}{4 x^{6} \left (4 \ln \relax (x )-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )^{2}}}\)	\(549\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*ln(x^2)^3+((-x^2+5*x)*exp(x)*ln(x)-exp(x)*x)*ln(x^2)^2+(3*ln(x)^2
+(2*exp(x)*x-1)*ln(x))*ln(x^2)+2*ln(x)^2)/x^6/ln(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*ln(x^2)^2+2*x*exp(x)*ln(
x)*ln(x^2)+ln(x)^2)/x^6/ln(x^2)^2),x,method=_RETURNVERBOSE)

[Out]

x*exp(-1/4*(-4*csgn(I*x^2)^6*Pi^2*x^7+16*csgn(I*x^2)^5*csgn(I*x)*Pi^2*x^7-24*csgn(I*x^2)^4*csgn(I*x)^2*Pi^2*x^
7+16*csgn(I*x^2)^3*csgn(I*x)^3*Pi^2*x^7-4*csgn(I*x^2)^2*csgn(I*x)^4*Pi^2*x^7-csgn(I*x^2)^6*Pi^2*exp(2*x)*x^2+4
*csgn(I*x^2)^5*csgn(I*x)*Pi^2*exp(2*x)*x^2-6*csgn(I*x^2)^4*csgn(I*x)^2*Pi^2*exp(2*x)*x^2+4*csgn(I*x^2)^3*csgn(
I*x)^3*Pi^2*exp(2*x)*x^2-csgn(I*x^2)^2*csgn(I*x)^4*Pi^2*exp(2*x)*x^2-32*I*csgn(I*x^2)^3*Pi*ln(x)*x^7+16*I*csgn
(I*x^2)^2*csgn(I*x)*Pi*ln(x)*exp(2*x)*x^2-8*I*csgn(I*x^2)^3*Pi*ln(x)*exp(2*x)*x^2+64*I*csgn(I*x^2)^2*csgn(I*x)
*Pi*ln(x)*x^7+8*I*ln(x)*exp(x)*x*csgn(I*x^2)^2*csgn(I*x)*Pi-4*I*ln(x)*exp(x)*x*csgn(I*x^2)*csgn(I*x)^2*Pi+64*x
^7*ln(x)^2-4*I*ln(x)*exp(x)*x*csgn(I*x^2)^3*Pi-8*I*csgn(I*x^2)*csgn(I*x)^2*Pi*ln(x)*exp(2*x)*x^2-32*I*csgn(I*x
^2)*csgn(I*x)^2*Pi*ln(x)*x^7+16*ln(x)^2*exp(2*x)*x^2+16*x*exp(x)*ln(x)^2+4*ln(x)^2)/x^6/(4*ln(x)-I*Pi*csgn(I*x
^2)*csgn(I*x)^2+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*x^2)^3)^2)

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maxima [A]  time = 1.18, size = 28, normalized size = 0.85 \begin {gather*} x e^{\left (-x - \frac {e^{\left (2 \, x\right )}}{4 \, x^{4}} - \frac {e^{x}}{4 \, x^{5}} - \frac {1}{16 \, x^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(
3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2
+2*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="maxima")

[Out]

x*e^(-x - 1/4*e^(2*x)/x^4 - 1/4*e^x/x^5 - 1/16/x^6)

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mupad [B]  time = 2.13, size = 48, normalized size = 1.45 \begin {gather*} x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x\,\ln \relax (x)}{2\,x^5\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-\frac {{\ln \relax (x)}^2}{4\,x^6\,{\ln \left (x^2\right )}^2}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{2\,x}}{4\,x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-((log(x^2)^2*(x^2*exp(2*x) + 4*x^7))/4 + log(x)^2/4 + (x*log(x^2)*exp(x)*log(x))/2)/(x^6*log(x^2)^2)
)*((log(x^2)^3*(exp(2*x)*(2*x^2 - x^3) + 2*x^6 - 2*x^7))/2 + log(x)^2 - (log(x^2)^2*(x*exp(x) - exp(x)*log(x)*
(5*x - x^2)))/2 + (log(x^2)*(3*log(x)^2 + log(x)*(2*x*exp(x) - 1)))/2))/(x^6*log(x^2)^3),x)

[Out]

x*exp(-(exp(x)*log(x))/(2*x^5*log(x^2)))*exp(-x)*exp(-log(x)^2/(4*x^6*log(x^2)^2))*exp(-exp(2*x)/(4*x^4))

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sympy [A]  time = 3.70, size = 48, normalized size = 1.45 \begin {gather*} x e^{- \frac {x e^{x} \log {\relax (x )}^{2} + \left (4 x^{7} + x^{2} e^{2 x}\right ) \log {\relax (x )}^{2} + \frac {\log {\relax (x )}^{2}}{4}}{4 x^{6} \log {\relax (x )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x**3+2*x**2)*exp(x)**2-2*x**7+2*x**6)*ln(x**2)**3+((-x**2+5*x)*exp(x)*ln(x)-exp(x)*x)*ln(x**
2)**2+(3*ln(x)**2+(2*exp(x)*x-1)*ln(x))*ln(x**2)+2*ln(x)**2)/x**6/ln(x**2)**3/exp(1/4*((exp(x)**2*x**2+4*x**7)
*ln(x**2)**2+2*x*exp(x)*ln(x)*ln(x**2)+ln(x)**2)/x**6/ln(x**2)**2),x)

[Out]

x*exp(-(x*exp(x)*log(x)**2 + (4*x**7 + x**2*exp(2*x))*log(x)**2 + log(x)**2/4)/(4*x**6*log(x)**2))

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