3.31.35 \(\int \frac {e^{-x} (1+e^x x-x \log (x)+(4-16 x^2 \log (5)-8 x^4 \log ^2(5)) \log (e^{-4-4 x^2 \log (5)-x^4 \log ^2(5)} x^2)-x \log ^2(e^{-4-4 x^2 \log (5)-x^4 \log ^2(5)} x^2))}{x} \, dx\)

Optimal. Leaf size=32 \[ x+e^{-x} \left (\log (x)+\log ^2\left (e^{-\left (2+x^2 \log (5)\right )^2} x^2\right )\right ) \]

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Rubi [A]  time = 0.81, antiderivative size = 45, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 2, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6742, 2288} \begin {gather*} \frac {e^{-x} \left (x \log ^2\left (5^{-4 x^2} x^2 e^{x^4 \left (-\log ^2(5)\right )-4}\right )+x \log (x)\right )}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^x*x - x*Log[x] + (4 - 16*x^2*Log[5] - 8*x^4*Log[5]^2)*Log[E^(-4 - 4*x^2*Log[5] - x^4*Log[5]^2)*x^2]
- x*Log[E^(-4 - 4*x^2*Log[5] - x^4*Log[5]^2)*x^2]^2)/(E^x*x),x]

[Out]

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

Rule 2288

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {e^{-x} \left (1-x \log (x)+4 \log \left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )-16 x^2 \log (5) \log \left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )-8 x^4 \log ^2(5) \log \left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )-x \log ^2\left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )\right )}{x}\right ) \, dx\\ &=x+\int \frac {e^{-x} \left (1-x \log (x)+4 \log \left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )-16 x^2 \log (5) \log \left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )-8 x^4 \log ^2(5) \log \left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )-x \log ^2\left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )\right )}{x} \, dx\\ &=x+\frac {e^{-x} \left (x \log (x)+x \log ^2\left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 41, normalized size = 1.28 \begin {gather*} e^{-x} \left (e^x x+\log (x)+\log ^2\left (5^{-4 x^2} e^{-4-x^4 \log ^2(5)} x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^x*x - x*Log[x] + (4 - 16*x^2*Log[5] - 8*x^4*Log[5]^2)*Log[E^(-4 - 4*x^2*Log[5] - x^4*Log[5]^2
)*x^2] - x*Log[E^(-4 - 4*x^2*Log[5] - x^4*Log[5]^2)*x^2]^2)/(E^x*x),x]

[Out]

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

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fricas [B]  time = 1.02, size = 72, normalized size = 2.25 \begin {gather*} {\left (x^{8} \log \relax (5)^{4} + 8 \, x^{6} \log \relax (5)^{3} + 24 \, x^{4} \log \relax (5)^{2} + 32 \, x^{2} \log \relax (5) + x e^{x} - {\left (4 \, x^{4} \log \relax (5)^{2} + 16 \, x^{2} \log \relax (5) + 15\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} + 16\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x^2/exp(x^4*log(5)^2+4*x^2*log(5)+4))^2+(-8*x^4*log(5)^2-16*x^2*log(5)+4)*log(x^2/exp(x^4*lo
g(5)^2+4*x^2*log(5)+4))-x*log(x)+exp(x)*x+1)/exp(x)/x,x, algorithm="fricas")

[Out]

(x^8*log(5)^4 + 8*x^6*log(5)^3 + 24*x^4*log(5)^2 + 32*x^2*log(5) + x*e^x - (4*x^4*log(5)^2 + 16*x^2*log(5) + 1
5)*log(x) + 4*log(x)^2 + 16)*e^(-x)

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giac [B]  time = 0.44, size = 103, normalized size = 3.22 \begin {gather*} x^{8} e^{\left (-x\right )} \log \relax (5)^{4} + 8 \, x^{6} e^{\left (-x\right )} \log \relax (5)^{3} - 4 \, x^{4} e^{\left (-x\right )} \log \relax (5)^{2} \log \relax (x) + 24 \, x^{4} e^{\left (-x\right )} \log \relax (5)^{2} - 16 \, x^{2} e^{\left (-x\right )} \log \relax (5) \log \relax (x) + 32 \, x^{2} e^{\left (-x\right )} \log \relax (5) + 4 \, e^{\left (-x\right )} \log \relax (x)^{2} - 15 \, e^{\left (-x\right )} \log \relax (x) + x + 16 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x^2/exp(x^4*log(5)^2+4*x^2*log(5)+4))^2+(-8*x^4*log(5)^2-16*x^2*log(5)+4)*log(x^2/exp(x^4*lo
g(5)^2+4*x^2*log(5)+4))-x*log(x)+exp(x)*x+1)/exp(x)/x,x, algorithm="giac")

[Out]

x^8*e^(-x)*log(5)^4 + 8*x^6*e^(-x)*log(5)^3 - 4*x^4*e^(-x)*log(5)^2*log(x) + 24*x^4*e^(-x)*log(5)^2 - 16*x^2*e
^(-x)*log(5)*log(x) + 32*x^2*e^(-x)*log(5) + 4*e^(-x)*log(x)^2 - 15*e^(-x)*log(x) + x + 16*e^(-x)

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maple [C]  time = 0.55, size = 1827, normalized size = 57.09

method	result	size
risch	\({\mathrm e}^{-x} \ln \left (625^{x^{2}} {\mathrm e}^{x^{4} \ln \relax (5)^{2}+4}\right )^{2}-\left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}+i \pi \,\mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}+4 \ln \relax (x )\right ) {\mathrm e}^{-x} \ln \left (625^{x^{2}} {\mathrm e}^{x^{4} \ln \relax (5)^{2}+4}\right )+\frac {\left (4 \ln \relax (x )+16 \ln \relax (x )^{2}+4 \,{\mathrm e}^{x} x -8 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} \ln \relax (x )+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}-8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}-\pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}-\pi ^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{6}+4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}-6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}-\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{5}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{3}-\pi ^{2} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2} \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{5}-8 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \ln \relax (x )-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )+8 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}+8 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )^{2}-8 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right ) \mathrm {csgn}\left (i x^{2} 625^{-x^{2}} {\mathrm e}^{-x^{4} \ln \relax (5)^{2}-4}\right )+16 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \ln \relax (x )\right ) {\mathrm e}^{-x}}{4}\)	\(1827\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*ln(x^2/exp(x^4*ln(5)^2+4*x^2*ln(5)+4))^2+(-8*x^4*ln(5)^2-16*x^2*ln(5)+4)*ln(x^2/exp(x^4*ln(5)^2+4*x^2*
ln(5)+4))-x*ln(x)+exp(x)*x+1)/exp(x)/x,x,method=_RETURNVERBOSE)

[Out]

exp(-x)*ln(625^(x^2)*exp(x^4*ln(5)^2+4))^2-(-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*
csgn(I*x^2)^3-I*Pi*csgn(I*x^2)*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2
-4))+I*Pi*csgn(I*x^2)*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2+I*Pi*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4
))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2-I*Pi*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^3+4*ln(x))*e
xp(-x)*ln(625^(x^2)*exp(x^4*ln(5)^2+4))+1/4*(4*ln(x)+16*ln(x)^2-2*Pi^2*csgn(I*x)^2*csgn(I*x^2)^2*csgn(I/(625^(
x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))+2*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I/(
625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2+4*exp(x)*x-8*I*Pi*csgn(I*x^2)^3*
ln(x)-Pi^2*csgn(I*x^2)^6+4*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^3-8*I*ln(x
)*Pi*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^3-2*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I*x^2/(625^(x^2))*exp(-
x^4*ln(5)^2-4))^3-4*Pi^2*csgn(I*x)*csgn(I*x^2)^3*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2-4*Pi^2*csgn(I*x
^2)*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^4+2*Pi^2*csgn(I*x)^2*c
sgn(I*x^2)^2*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2+2*Pi^2*csgn(I*x^2)^2*csgn(I/(625^(x^2))*exp(-x^4*ln
(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^3+2*Pi^2*csgn(I*x^2)*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2
-4))^2*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^3+2*Pi^2*csgn(I*x^2)^3*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-
4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2-Pi^2*csgn(I*x^2)^2*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2
*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2-2*Pi^2*csgn(I*x^2)^4*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*cs
gn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))-Pi^2*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^6-Pi^2*csgn(I*x)^4*
csgn(I*x^2)^2+4*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3-6*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4+4*Pi^2*csgn(I*x)*csgn(I*x^2)^5
-4*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^
2-4))^2-Pi^2*csgn(I*x^2)^2*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^4+2*Pi^2*csgn(I*x^2)*csgn(I*x^2/(625^(x
^2))*exp(-x^4*ln(5)^2-4))^5+2*Pi^2*csgn(I*x^2)^4*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2-2*Pi^2*csgn(I*x
^2)^3*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^3-8*I*Pi*csgn(I*x)^2*csgn(I*x^2)*ln(x)+16*I*Pi*csgn(I*x)*csg
n(I*x^2)^2*ln(x)+4*Pi^2*csgn(I*x)*csgn(I*x^2)^3*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))
*exp(-x^4*ln(5)^2-4))+8*I*ln(x)*Pi*csgn(I*x^2)*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2+8*I*ln(x)*Pi*csgn
(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^2-8*I*ln(x)*Pi*csgn(I*x^2)*csg
n(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))-Pi^2*csgn(I/(625^(x^2))*exp(-
x^4*ln(5)^2-4))^2*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^4+2*Pi^2*csgn(I/(625^(x^2))*exp(-x^4*ln(5)^2-4))
*csgn(I*x^2/(625^(x^2))*exp(-x^4*ln(5)^2-4))^5)*exp(-x)

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maxima [B]  time = 0.84, size = 76, normalized size = 2.38 \begin {gather*} {\left (x^{8} \log \relax (5)^{4} + 8 \, x^{6} \log \relax (5)^{3} + 24 \, x^{4} \log \relax (5)^{2} + 32 \, x^{2} \log \relax (5) - 4 \, {\left (x^{4} \log \relax (5)^{2} + 4 \, x^{2} \log \relax (5) + 4\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} + 16\right )} e^{\left (-x\right )} + e^{\left (-x\right )} \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x^2/exp(x^4*log(5)^2+4*x^2*log(5)+4))^2+(-8*x^4*log(5)^2-16*x^2*log(5)+4)*log(x^2/exp(x^4*lo
g(5)^2+4*x^2*log(5)+4))-x*log(x)+exp(x)*x+1)/exp(x)/x,x, algorithm="maxima")

[Out]

(x^8*log(5)^4 + 8*x^6*log(5)^3 + 24*x^4*log(5)^2 + 32*x^2*log(5) - 4*(x^4*log(5)^2 + 4*x^2*log(5) + 4)*log(x)
+ 4*log(x)^2 + 16)*e^(-x) + e^(-x)*log(x) + x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{-x}\,\left (x\,{\ln \left (x^2\,{\mathrm {e}}^{-{\ln \relax (5)}^2\,x^4-4\,\ln \relax (5)\,x^2-4}\right )}^2+\left (8\,{\ln \relax (5)}^2\,x^4+16\,\ln \relax (5)\,x^2-4\right )\,\ln \left (x^2\,{\mathrm {e}}^{-{\ln \relax (5)}^2\,x^4-4\,\ln \relax (5)\,x^2-4}\right )-x\,{\mathrm {e}}^x+x\,\ln \relax (x)-1\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(x*log(x^2*exp(- x^4*log(5)^2 - 4*x^2*log(5) - 4))^2 + log(x^2*exp(- x^4*log(5)^2 - 4*x^2*log(5)
- 4))*(8*x^4*log(5)^2 + 16*x^2*log(5) - 4) - x*exp(x) + x*log(x) - 1))/x,x)

[Out]

int(-(exp(-x)*(x*log(x^2*exp(- x^4*log(5)^2 - 4*x^2*log(5) - 4))^2 + log(x^2*exp(- x^4*log(5)^2 - 4*x^2*log(5)
- 4))*(8*x^4*log(5)^2 + 16*x^2*log(5) - 4) - x*exp(x) + x*log(x) - 1))/x, x)

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sympy [B]  time = 0.86, size = 80, normalized size = 2.50 \begin {gather*} x + \left (x^{8} \log {\relax (5 )}^{4} + 8 x^{6} \log {\relax (5 )}^{3} - 4 x^{4} \log {\relax (5 )}^{2} \log {\relax (x )} + 24 x^{4} \log {\relax (5 )}^{2} - 16 x^{2} \log {\relax (5 )} \log {\relax (x )} + 32 x^{2} \log {\relax (5 )} + 4 \log {\relax (x )}^{2} - 15 \log {\relax (x )} + 16\right ) e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*ln(x**2/exp(x**4*ln(5)**2+4*x**2*ln(5)+4))**2+(-8*x**4*ln(5)**2-16*x**2*ln(5)+4)*ln(x**2/exp(x**
4*ln(5)**2+4*x**2*ln(5)+4))-x*ln(x)+exp(x)*x+1)/exp(x)/x,x)

[Out]

x + (x**8*log(5)**4 + 8*x**6*log(5)**3 - 4*x**4*log(5)**2*log(x) + 24*x**4*log(5)**2 - 16*x**2*log(5)*log(x) +
32*x**2*log(5) + 4*log(x)**2 - 15*log(x) + 16)*exp(-x)

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