3.47.14 \(\int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log (x^2)+((9+9 x-9 x^2) \log (3)+(3+3 x-3 x^2) \log (3) \log (x^2)) \log (\frac {9+3 \log (x^2)}{x})}{3 e^x+e^x \log (x^2)} \, dx\)

Optimal. Leaf size=26 \[ 3 e^{-x} \left (x+x^2\right ) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \]

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Rubi [A]  time = 1.08, antiderivative size = 47, normalized size of antiderivative = 1.81, number of steps used = 18, number of rules used = 6, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6741, 6742, 2194, 2176, 2196, 2555} \begin {gather*} 3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right )+3 e^{-x} x \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-3 - 3*x)*Log[3] + (-3 - 3*x)*Log[3]*Log[x^2] + ((9 + 9*x - 9*x^2)*Log[3] + (3 + 3*x - 3*x^2)*Log[3]*Log
[x^2])*Log[(9 + 3*Log[x^2])/x])/(3*E^x + E^x*Log[x^2]),x]

[Out]

(3*x*Log[3]*Log[(3*(3 + Log[x^2]))/x])/E^x + (3*x^2*Log[3]*Log[(3*(3 + Log[x^2]))/x])/E^x

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 2555

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify
[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]

Rule 6741

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

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 \frac {e^{-x} \left ((-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )\right )}{3+\log \left (x^2\right )} \, dx\\ &=\int \left (-\frac {3 e^{-x} (1+x) \log (3) \left (1+\log \left (x^2\right )\right )}{3+\log \left (x^2\right )}-3 e^{-x} \left (-1-x+x^2\right ) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )\right ) \, dx\\ &=-\left ((3 \log (3)) \int \frac {e^{-x} (1+x) \left (1+\log \left (x^2\right )\right )}{3+\log \left (x^2\right )} \, dx\right )-(3 \log (3)) \int e^{-x} \left (-1-x+x^2\right ) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \, dx\\ &=3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+(3 \log (3)) \int \frac {e^{-x} (1+x) \left (1+\log \left (x^2\right )\right )}{3+\log \left (x^2\right )} \, dx-(3 \log (3)) \int \left (e^{-x}+e^{-x} x-\frac {2 e^{-x} (1+x)}{3+\log \left (x^2\right )}\right ) \, dx\\ &=3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )-(3 \log (3)) \int e^{-x} \, dx-(3 \log (3)) \int e^{-x} x \, dx+(3 \log (3)) \int \left (e^{-x}+e^{-x} x-\frac {2 e^{-x} (1+x)}{3+\log \left (x^2\right )}\right ) \, dx+(6 \log (3)) \int \frac {e^{-x} (1+x)}{3+\log \left (x^2\right )} \, dx\\ &=3 e^{-x} \log (3)+3 e^{-x} x \log (3)+3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+(3 \log (3)) \int e^{-x} x \, dx-(6 \log (3)) \int \frac {e^{-x} (1+x)}{3+\log \left (x^2\right )} \, dx+(6 \log (3)) \int \left (\frac {e^{-x}}{3+\log \left (x^2\right )}+\frac {e^{-x} x}{3+\log \left (x^2\right )}\right ) \, dx\\ &=3 e^{-x} \log (3)+3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+(3 \log (3)) \int e^{-x} \, dx+(6 \log (3)) \int \frac {e^{-x}}{3+\log \left (x^2\right )} \, dx+(6 \log (3)) \int \frac {e^{-x} x}{3+\log \left (x^2\right )} \, dx-(6 \log (3)) \int \left (\frac {e^{-x}}{3+\log \left (x^2\right )}+\frac {e^{-x} x}{3+\log \left (x^2\right )}\right ) \, dx\\ &=3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 25, normalized size = 0.96 \begin {gather*} 3 e^{-x} x (1+x) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3 - 3*x)*Log[3] + (-3 - 3*x)*Log[3]*Log[x^2] + ((9 + 9*x - 9*x^2)*Log[3] + (3 + 3*x - 3*x^2)*Log[
3]*Log[x^2])*Log[(9 + 3*Log[x^2])/x])/(3*E^x + E^x*Log[x^2]),x]

[Out]

(3*x*(1 + x)*Log[3]*Log[(3*(3 + Log[x^2]))/x])/E^x

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fricas [A]  time = 0.49, size = 25, normalized size = 0.96 \begin {gather*} 3 \, {\left (x^{2} + x\right )} e^{\left (-x\right )} \log \relax (3) \log \left (\frac {3 \, {\left (\log \left (x^{2}\right ) + 3\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*log(x^2)+9)/x)+(-3*x-3)*log(3)*log(x^
2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3*exp(x)),x, algorithm="fricas")

[Out]

3*(x^2 + x)*e^(-x)*log(3)*log(3*(log(x^2) + 3)/x)

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giac [B]  time = 0.31, size = 83, normalized size = 3.19 \begin {gather*} 3 \, x^{2} e^{\left (-x\right )} \log \relax (3)^{2} - 3 \, x^{2} e^{\left (-x\right )} \log \relax (3) \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \relax (3) \log \left (\log \left (x^{2}\right ) + 3\right ) + 3 \, x e^{\left (-x\right )} \log \relax (3)^{2} - 3 \, x e^{\left (-x\right )} \log \relax (3) \log \relax (x) + 3 \, x e^{\left (-x\right )} \log \relax (3) \log \left (\log \left (x^{2}\right ) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*log(x^2)+9)/x)+(-3*x-3)*log(3)*log(x^
2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3*exp(x)),x, algorithm="giac")

[Out]

3*x^2*e^(-x)*log(3)^2 - 3*x^2*e^(-x)*log(3)*log(x) + 3*x^2*e^(-x)*log(3)*log(log(x^2) + 3) + 3*x*e^(-x)*log(3)
^2 - 3*x*e^(-x)*log(3)*log(x) + 3*x*e^(-x)*log(3)*log(log(x^2) + 3)

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maple [C]  time = 0.31, size = 1616, normalized size = 62.15

method	result	size
risch	\(3 \ln \relax (3) x \left (x +1\right ) {\mathrm e}^{-x} \ln \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )+\frac {3 \ln \relax (3) x \left (-i \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )+i x \pi \,\mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}+i x \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}-i x \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )-i x \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{3}-i x \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )-i x \pi +i x \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{3}-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )+i \pi \,\mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{3}+i x \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}-i x \pi \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{3}-i \pi +2 x \ln \left (\frac {3}{2}\right )-2 x \ln \relax (x )+2 \ln \left (\frac {3}{2}\right )-2 \ln \relax (x )\right ) {\mathrm e}^{-x}}{2}\)	\(1616\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x^2+3*x+3)*ln(3)*ln(x^2)+(-9*x^2+9*x+9)*ln(3))*ln((3*ln(x^2)+9)/x)+(-3*x-3)*ln(3)*ln(x^2)+(-3*x-3)*l
n(3))/(exp(x)*ln(x^2)+3*exp(x)),x,method=_RETURNVERBOSE)

[Out]

3*ln(3)*x*(x+1)*exp(-x)*ln(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2
)^2)+3/2*ln(3)*x*(-I*Pi*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csg
n(I*x^2)^2))*csgn(1/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)
)+I*x*Pi*csgn(I*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn
(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2+I*x*Pi*csgn(I
/x)*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2+I*Pi*
csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(1/x*(6
*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2-I*x*Pi*csgn(I/x*(6*I
+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(1/x*(6*I+Pi*csgn(I*
x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))-I*x*Pi*csgn(1/x*(6*I+Pi*csgn(I*x^2)
^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^3-I*x*Pi*csgn(I/x)*csgn(I*(6*I+Pi*csgn(
I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*
I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))-I*x*Pi+I*x*Pi*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3
+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(1/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+
Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2-I*Pi*csgn(1/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*c
sgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^3-I*Pi*csgn(I/x)*csgn(I*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+
Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x
)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))+I*Pi*csgn(I*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn
(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*P
i*csgn(I*x)*csgn(I*x^2)^2))^2-I*Pi*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*cs
gn(I*x)*csgn(I*x^2)^2))^3+I*x*Pi*csgn(1/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn
(I*x)*csgn(I*x^2)^2))^2+I*Pi*csgn(I/x)*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*P
i*csgn(I*x)*csgn(I*x^2)^2))^2+I*Pi*csgn(1/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*cs
gn(I*x)*csgn(I*x^2)^2))^2-I*x*Pi*csgn(I/x*(6*I+Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn
(I*x)*csgn(I*x^2)^2))^3-I*Pi+2*x*ln(3/2)-2*x*ln(x)+2*ln(3/2)-2*ln(x))*exp(-x)

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maxima [B]  time = 0.49, size = 61, normalized size = 2.35 \begin {gather*} 3 \, {\left (x^{2} \log \relax (3) + x \log \relax (3)\right )} e^{\left (-x\right )} \log \left (2 \, \log \relax (x) + 3\right ) + 3 \, {\left (x^{2} \log \relax (3)^{2} + x \log \relax (3)^{2} - {\left (x^{2} \log \relax (3) + x \log \relax (3)\right )} \log \relax (x)\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*log(x^2)+9)/x)+(-3*x-3)*log(3)*log(x^
2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3*exp(x)),x, algorithm="maxima")

[Out]

3*(x^2*log(3) + x*log(3))*e^(-x)*log(2*log(x) + 3) + 3*(x^2*log(3)^2 + x*log(3)^2 - (x^2*log(3) + x*log(3))*lo
g(x))*e^(-x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \relax (3)\,\left (3\,x+3\right )-\ln \left (\frac {3\,\ln \left (x^2\right )+9}{x}\right )\,\left (\ln \relax (3)\,\left (-9\,x^2+9\,x+9\right )+\ln \left (x^2\right )\,\ln \relax (3)\,\left (-3\,x^2+3\,x+3\right )\right )+\ln \left (x^2\right )\,\ln \relax (3)\,\left (3\,x+3\right )}{3\,{\mathrm {e}}^x+\ln \left (x^2\right )\,{\mathrm {e}}^x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(3)*(3*x + 3) - log((3*log(x^2) + 9)/x)*(log(3)*(9*x - 9*x^2 + 9) + log(x^2)*log(3)*(3*x - 3*x^2 + 3)
) + log(x^2)*log(3)*(3*x + 3))/(3*exp(x) + log(x^2)*exp(x)),x)

[Out]

int(-(log(3)*(3*x + 3) - log((3*log(x^2) + 9)/x)*(log(3)*(9*x - 9*x^2 + 9) + log(x^2)*log(3)*(3*x - 3*x^2 + 3)
) + log(x^2)*log(3)*(3*x + 3))/(3*exp(x) + log(x^2)*exp(x)), x)

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sympy [A]  time = 3.97, size = 41, normalized size = 1.58 \begin {gather*} \left (3 x^{2} \log {\relax (3 )} \log {\left (\frac {3 \log {\left (x^{2} \right )} + 9}{x} \right )} + 3 x \log {\relax (3 )} \log {\left (\frac {3 \log {\left (x^{2} \right )} + 9}{x} \right )}\right ) e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**2+3*x+3)*ln(3)*ln(x**2)+(-9*x**2+9*x+9)*ln(3))*ln((3*ln(x**2)+9)/x)+(-3*x-3)*ln(3)*ln(x**2)
+(-3*x-3)*ln(3))/(exp(x)*ln(x**2)+3*exp(x)),x)

[Out]

(3*x**2*log(3)*log((3*log(x**2) + 9)/x) + 3*x*log(3)*log((3*log(x**2) + 9)/x))*exp(-x)

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