3.54.22 \(\int \frac {e^{-3+x} x (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3)))}{x^3+25 x \log ^4(3)+2 x^2 (i \pi +\log (3))+x (i \pi +\log (3))^2+\log ^2(3) (-10 x^2-10 x (i \pi +\log (3)))} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^{-3+x} x}{i \pi +x+\log (3)-5 \log ^2(3)} \]

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Rubi [A]  time = 0.26, antiderivative size = 45, normalized size of antiderivative = 1.88, number of steps used = 9, number of rules used = 7, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6, 1586, 27, 2199, 2194, 2177, 2178} \begin {gather*} e^{x-3}-\frac {e^{x-3} (\log (3) (1-\log (243))+i \pi )}{x+i \pi -5 \log ^2(3)+\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-3 + x)*x*(x^2 + (-5 - 5*x)*Log[3]^2 + (1 + x)*(I*Pi + Log[3])))/(x^3 + 25*x*Log[3]^4 + 2*x^2*(I*Pi +
Log[3]) + x*(I*Pi + Log[3])^2 + Log[3]^2*(-10*x^2 - 10*x*(I*Pi + Log[3]))),x]

[Out]

E^(-3 + x) - (E^(-3 + x)*(I*Pi + Log[3]*(1 - Log[243])))/(I*Pi + x + Log[3] - 5*Log[3]^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2177

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

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$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 2199

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-3+x} x \left (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3))\right )}{x^3+2 x^2 (i \pi +\log (3))+\log ^2(3) \left (-10 x^2-10 x (i \pi +\log (3))\right )+x \left (25 \log ^4(3)+(i \pi +\log (3))^2\right )} \, dx\\ &=\int \frac {e^{-3+x} \left (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3))\right )}{-\pi ^2+x^2+2 i \pi \log (3)+\log ^2(3)-10 i \pi \log ^2(3)-10 \log ^3(3)+25 \log ^4(3)+x \left (2 i \pi +2 \log (3)-10 \log ^2(3)\right )} \, dx\\ &=\int \frac {e^{-3+x} \left (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3))\right )}{\left (x+i \left (\pi -i \log (3)+5 i \log ^2(3)\right )\right )^2} \, dx\\ &=\int \left (e^{-3+x}+\frac {i e^{-3+x} (\pi -i \log (3) (1-\log (243)))}{\left (i \pi +x+\log (3)-5 \log ^2(3)\right )^2}+\frac {e^{-3+x} (-i \pi -\log (3)+\log (3) \log (243))}{i \pi +x+\log (3)-5 \log ^2(3)}\right ) \, dx\\ &=(-i \pi -\log (3) (1-\log (243))) \int \frac {e^{-3+x}}{i \pi +x+\log (3)-5 \log ^2(3)} \, dx+(i \pi +\log (3)-\log (3) \log (243)) \int \frac {e^{-3+x}}{\left (i \pi +x+\log (3)-5 \log ^2(3)\right )^2} \, dx+\int e^{-3+x} \, dx\\ &=e^{-3+x}+\frac {1}{3} e^{-3+5 \log ^2(3)} \text {Ei}\left (i \pi +x+\log (3)-5 \log ^2(3)\right ) (i \pi +\log (3) (1-\log (243)))-\frac {e^{-3+x} (i \pi +\log (3) (1-\log (243)))}{i \pi +x+\log (3)-5 \log ^2(3)}+(i \pi +\log (3)-\log (3) \log (243)) \int \frac {e^{-3+x}}{i \pi +x+\log (3)-5 \log ^2(3)} \, dx\\ &=e^{-3+x}+\frac {1}{3} e^{-3+5 \log ^2(3)} \text {Ei}\left (i \pi +x+\log (3)-5 \log ^2(3)\right ) (i \pi +\log (3) (1-\log (243)))-\frac {e^{-3+x} (i \pi +\log (3) (1-\log (243)))}{i \pi +x+\log (3)-5 \log ^2(3)}-\frac {1}{3} e^{-3+5 \log ^2(3)} \text {Ei}\left (i \pi +x+\log (3)-5 \log ^2(3)\right ) (i \pi +\log (3)-\log (3) \log (243))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 39, normalized size = 1.62 \begin {gather*} \frac {e^{-3+x} x (i \pi +x+\log (3)-\log (3) \log (243))}{\left (i \pi +x+\log (3)-5 \log ^2(3)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-3 + x)*x*(x^2 + (-5 - 5*x)*Log[3]^2 + (1 + x)*(I*Pi + Log[3])))/(x^3 + 25*x*Log[3]^4 + 2*x^2*(I
*Pi + Log[3]) + x*(I*Pi + Log[3])^2 + Log[3]^2*(-10*x^2 - 10*x*(I*Pi + Log[3]))),x]

[Out]

(E^(-3 + x)*x*(I*Pi + x + Log[3] - Log[3]*Log[243]))/(I*Pi + x + Log[3] - 5*Log[3]^2)^2

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fricas [A]  time = 0.60, size = 27, normalized size = 1.12 \begin {gather*} -\frac {e^{\left (x + \log \relax (x) - 3\right )}}{-i \, \pi + 5 \, \log \relax (3)^{2} - x - \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-5)*log(3)^2+(x+1)*(log(3)+I*pi)+x^2)*exp(log(x)-3+x)/(25*x*log(3)^4+(-10*x*(log(3)+I*pi)-10*x
^2)*log(3)^2+x*(log(3)+I*pi)^2+2*x^2*(log(3)+I*pi)+x^3),x, algorithm="fricas")

[Out]

-e^(x + log(x) - 3)/(-I*pi + 5*log(3)^2 - x - log(3))

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giac [A]  time = 0.22, size = 31, normalized size = 1.29 \begin {gather*} -\frac {i \, x e^{x}}{5 i \, e^{3} \log \relax (3)^{2} + \pi e^{3} - i \, x e^{3} - i \, e^{3} \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-5)*log(3)^2+(x+1)*(log(3)+I*pi)+x^2)*exp(log(x)-3+x)/(25*x*log(3)^4+(-10*x*(log(3)+I*pi)-10*x
^2)*log(3)^2+x*(log(3)+I*pi)^2+2*x^2*(log(3)+I*pi)+x^3),x, algorithm="giac")

[Out]

-I*x*e^x/(5*I*e^3*log(3)^2 + pi*e^3 - I*x*e^3 - I*e^3*log(3))

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maple [A]  time = 0.61, size = 29, normalized size = 1.21

method	result	size
risch	\(-\frac {i x \,{\mathrm e}^{x -3}}{\pi +5 i \ln \relax (3)^{2}-i \ln \relax (3)-i x}\)	\(29\)
norman	\(\frac {x \,{\mathrm e}^{\ln \relax (x )-3+x}+\left (-5 \ln \relax (3)^{2}+\ln \relax (3)-i \pi \right ) {\mathrm e}^{\ln \relax (x )-3+x}}{25 \ln \relax (3)^{4}-10 \ln \relax (3)^{3}-10 x \ln \relax (3)^{2}+\pi ^{2}+\ln \relax (3)^{2}+2 x \ln \relax (3)+x^{2}}\)	\(68\)
gosper	\(-\frac {\left (\pi +5 i \ln \relax (3)^{2}-i \ln \relax (3)-i x \right ) \left (i x \pi -5 x \ln \relax (3)^{2}+i \pi -5 \ln \relax (3)^{2}+x \ln \relax (3)+x^{2}+\ln \relax (3)\right ) {\mathrm e}^{\ln \relax (x )-3+x}}{\left (\pi x +5 i \ln \relax (3)^{2} x +\pi +5 i \ln \relax (3)^{2}-i x \ln \relax (3)-i x^{2}-i \ln \relax (3)\right ) \left (10 i \pi \ln \relax (3)^{2}-25 \ln \relax (3)^{4}-2 i \pi \ln \relax (3)-2 i x \pi +10 \ln \relax (3)^{3}+10 x \ln \relax (3)^{2}+\pi ^{2}-\ln \relax (3)^{2}-2 x \ln \relax (3)-x^{2}\right )}\)	\(158\)
default	\(-\frac {30 i {\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi \ln \relax (3)^{3}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}+\frac {75 i {\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi \ln \relax (3)^{4}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}+\frac {10 i {\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi \ln \relax (3)^{2}}{-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi }+\frac {3 i {\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi \ln \relax (3)^{2}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}-\frac {2 i {\mathrm e}^{x -3} \pi \ln \relax (3)}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right ) \left (\ln \relax (3)+i \pi -5 \ln \relax (3)^{2}+x \right )}-\frac {2 i {\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi \ln \relax (3)}{-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi }+\frac {10 i {\mathrm e}^{x -3} \pi \ln \relax (3)^{2}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right ) \left (\ln \relax (3)+i \pi -5 \ln \relax (3)^{2}+x \right )}-\frac {125 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{6}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}+\frac {75 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{5}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}-\frac {15 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{4}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}+\frac {{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{3}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}+\frac {{\mathrm e}^{x -3} \pi ^{2}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right ) \left (\ln \relax (3)+i \pi -5 \ln \relax (3)^{2}+x \right )}-\frac {{\mathrm e}^{x -3} \ln \relax (3)^{2}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right ) \left (\ln \relax (3)+i \pi -5 \ln \relax (3)^{2}+x \right )}-\frac {25 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{4}}{-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi }+\frac {10 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{3}}{-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi }+\frac {{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi ^{2}}{-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi }-\frac {{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{2}}{-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi }-\frac {25 \,{\mathrm e}^{x -3} \ln \relax (3)^{4}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right ) \left (\ln \relax (3)+i \pi -5 \ln \relax (3)^{2}+x \right )}+\frac {10 \,{\mathrm e}^{x -3} \ln \relax (3)^{3}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right ) \left (\ln \relax (3)+i \pi -5 \ln \relax (3)^{2}+x \right )}+{\mathrm e}^{x -3}-\frac {i {\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \pi ^{3}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}+\frac {15 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3)^{2} \pi ^{2}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}-\frac {3 \,{\mathrm e}^{i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )} \expIntegralEi \left (1, -x +3+i \left (-5 i \ln \relax (3)^{2}+i \ln \relax (3)-\pi +3 i\right )\right ) \ln \relax (3) \pi ^{2}}{\left (-5 \ln \relax (3)^{2}+\ln \relax (3)+i \pi \right )^{2}}\)	\(1397\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x-5)*ln(3)^2+(x+1)*(ln(3)+I*Pi)+x^2)*exp(ln(x)-3+x)/(25*x*ln(3)^4+(-10*x*(ln(3)+I*Pi)-10*x^2)*ln(3)^2
+x*(ln(3)+I*Pi)^2+2*x^2*(ln(3)+I*Pi)+x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.63, size = 26, normalized size = 1.08 \begin {gather*} \frac {x e^{x}}{{\left (i \, \pi - 5 \, \log \relax (3)^{2} + \log \relax (3)\right )} e^{3} + x e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-5)*log(3)^2+(x+1)*(log(3)+I*pi)+x^2)*exp(log(x)-3+x)/(25*x*log(3)^4+(-10*x*(log(3)+I*pi)-10*x
^2)*log(3)^2+x*(log(3)+I*pi)^2+2*x^2*(log(3)+I*pi)+x^3),x, algorithm="maxima")

[Out]

x*e^x/((I*pi - 5*log(3)^2 + log(3))*e^3 + x*e^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{x+\ln \relax (x)-3}\,\left (x^2+\left (\ln \relax (3)+\Pi \,1{}\mathrm {i}\right )\,\left (x+1\right )-{\ln \relax (3)}^2\,\left (5\,x+5\right )\right )}{25\,x\,{\ln \relax (3)}^4-{\ln \relax (3)}^2\,\left (10\,x\,\left (\ln \relax (3)+\Pi \,1{}\mathrm {i}\right )+10\,x^2\right )+x\,{\left (\ln \relax (3)+\Pi \,1{}\mathrm {i}\right )}^2+2\,x^2\,\left (\ln \relax (3)+\Pi \,1{}\mathrm {i}\right )+x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + log(x) - 3)*((Pi*1i + log(3))*(x + 1) - log(3)^2*(5*x + 5) + x^2))/(25*x*log(3)^4 - log(3)^2*(10*
x*(Pi*1i + log(3)) + 10*x^2) + x*(Pi*1i + log(3))^2 + 2*x^2*(Pi*1i + log(3)) + x^3),x)

[Out]

int((exp(x + log(x) - 3)*((Pi*1i + log(3))*(x + 1) - log(3)^2*(5*x + 5) + x^2))/(25*x*log(3)^4 - log(3)^2*(10*
x*(Pi*1i + log(3)) + 10*x^2) + x*(Pi*1i + log(3))^2 + 2*x^2*(Pi*1i + log(3)) + x^3), x)

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sympy [A]  time = 0.62, size = 34, normalized size = 1.42 \begin {gather*} - \frac {x e^{x}}{- x e^{3} - e^{3} \log {\relax (3 )} + 5 e^{3} \log {\relax (3 )}^{2} - i \pi e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-5)*ln(3)**2+(x+1)*(ln(3)+I*pi)+x**2)*exp(ln(x)-3+x)/(25*x*ln(3)**4+(-10*x*(ln(3)+I*pi)-10*x**
2)*ln(3)**2+x*(ln(3)+I*pi)**2+2*x**2*(ln(3)+I*pi)+x**3),x)

[Out]

-x*exp(x)/(-x*exp(3) - exp(3)*log(3) + 5*exp(3)*log(3)**2 - I*pi*exp(3))

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