3.101.91 \(\int \frac {e^{2 x} (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} (-2 x^2+6 x^3)+e^{2 x} (6 x-8 x^2+2 x^3-8 x^4)) (3-\log ^2(3))^2}{x^5} \, dx\) [10091]

Optimal. Leaf size=37 \[ \frac {e^{2 x} \left (e^{2 x}-\frac {1+x^2}{x}\right )^2 \left (3-\log ^2(3)\right )^2}{x^2} \]

[Out]

(exp(x)^2-(x^2+1)/x)^2/x^2*exp(ln(-ln(3)^2+3)+x)^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(37)=74\).
time = 1.49, antiderivative size = 115, normalized size of antiderivative = 3.11, number of steps used = 33, number of rules used = 7, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 6874, 2228, 2230, 2208, 2209, 2225} \begin {gather*} \frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+e^{2 x} \left (3-\log ^2(3)\right )^2-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(-4 + 2*x - 4*x^2 + 4*x^3 + 2*x^5 + E^(4*x)*(-2*x^2 + 6*x^3) + E^(2*x)*(6*x - 8*x^2 + 2*x^3 - 8*x
^4))*(3 - Log[3]^2)^2)/x^5,x]

[Out]

E^(2*x)*(3 - Log[3]^2)^2 + (E^(2*x)*(3 - Log[3]^2)^2)/x^4 - (2*E^(4*x)*(3 - Log[3]^2)^2)/x^3 + (2*E^(2*x)*(3 -
Log[3]^2)^2)/x^2 + (E^(6*x)*(3 - Log[3]^2)^2)/x^2 - (2*E^(4*x)*(3 - Log[3]^2)^2)/x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2208

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] &&  !TrueQ[$UseGamm
a]

Rule 2209

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

Rule 2225

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 2228

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[g*u^(m + 1)*(F^(c*v)/(b*c*
e*Log[F])), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 2230

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] &&  !TrueQ[$UseGamma]

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (3-\log ^2(3)\right )^2 \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right )}{x^5} \, dx\\ &=\left (3-\log ^2(3)\right )^2 \int \left (\frac {2 e^{6 x} (-1+3 x)}{x^3}-\frac {2 e^{4 x} \left (-3+4 x-x^2+4 x^3\right )}{x^4}+\frac {2 e^{2 x} \left (-2+x-2 x^2+2 x^3+x^5\right )}{x^5}\right ) \, dx\\ &=\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{6 x} (-1+3 x)}{x^3} \, dx-\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x} \left (-3+4 x-x^2+4 x^3\right )}{x^4} \, dx+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x} \left (-2+x-2 x^2+2 x^3+x^5\right )}{x^5} \, dx\\ &=\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \left (e^{2 x}-\frac {2 e^{2 x}}{x^5}+\frac {e^{2 x}}{x^4}-\frac {2 e^{2 x}}{x^3}+\frac {2 e^{2 x}}{x^2}\right ) \, dx-\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \left (-\frac {3 e^{4 x}}{x^4}+\frac {4 e^{4 x}}{x^3}-\frac {e^{4 x}}{x^2}+\frac {4 e^{4 x}}{x}\right ) \, dx\\ &=\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int e^{2 x} \, dx+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^4} \, dx+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^2} \, dx-\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^5} \, dx-\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^3} \, dx+\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx+\left (6 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^4} \, dx-\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^3} \, dx-\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{3 x^3}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {4 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {4 e^{2 x} \left (3-\log ^2(3)\right )^2}{x}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}-8 \text {Ei}(4 x) \left (3-\log ^2(3)\right )^2+\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^3} \, dx-\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^4} \, dx-\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx+\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^3} \, dx+\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx+\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx-\left (16 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^2} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {4 e^{2 x} \left (3-\log ^2(3)\right )^2}{3 x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {14 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}+8 \text {Ei}(2 x) \left (3-\log ^2(3)\right )^2-\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^3} \, dx+\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx-\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx+\left (16 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^2} \, dx-\left (64 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {4 e^{2 x} \left (3-\log ^2(3)\right )^2}{3 x}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}-64 \text {Ei}(4 x) \left (3-\log ^2(3)\right )^2-\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx+\frac {1}{3} \left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx+\left (64 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}+\frac {8}{3} \text {Ei}(2 x) \left (3-\log ^2(3)\right )^2-\frac {1}{3} \left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 2.86, size = 32, normalized size = 0.86 \begin {gather*} \frac {e^{2 x} \left (1-e^{2 x} x+x^2\right )^2 \left (-3+\log ^2(3)\right )^2}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-4 + 2*x - 4*x^2 + 4*x^3 + 2*x^5 + E^(4*x)*(-2*x^2 + 6*x^3) + E^(2*x)*(6*x - 8*x^2 + 2*x^3
- 8*x^4))*(3 - Log[3]^2)^2)/x^5,x]

[Out]

(E^(2*x)*(1 - E^(2*x)*x + x^2)^2*(-3 + Log[3]^2)^2)/x^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs. \(2(35)=70\).
time = 0.07, size = 200, normalized size = 5.41

method	result	size
risch	\(\frac {\left (-\ln \left (3\right )^{2}+3\right )^{2} {\mathrm e}^{6 x}}{x^{2}}-\frac {2 \left (-\ln \left (3\right )^{2}+3\right )^{2} \left (x^{2}+1\right ) {\mathrm e}^{4 x}}{x^{3}}+\frac {\left (-\ln \left (3\right )^{2}+3\right )^{2} \left (x^{4}+2 x^{2}+1\right ) {\mathrm e}^{2 x}}{x^{4}}\)	\(72\)
default	\(-6 \ln \left (3\right )^{2} {\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{2 x}}{x^{4}}+\frac {18 \,{\mathrm e}^{2 x}}{x^{2}}-\frac {6 \ln \left (3\right )^{2} {\mathrm e}^{2 x}}{x^{4}}+\frac {{\mathrm e}^{6 x} \ln \left (3\right )^{4}}{x^{2}}-\frac {6 \,{\mathrm e}^{6 x} \ln \left (3\right )^{2}}{x^{2}}-\frac {2 \,{\mathrm e}^{4 x} \ln \left (3\right )^{4}}{x}+\frac {12 \,{\mathrm e}^{4 x} \ln \left (3\right )^{2}}{x}-\frac {2 \,{\mathrm e}^{4 x} \ln \left (3\right )^{4}}{x^{3}}+\frac {12 \,{\mathrm e}^{4 x} \ln \left (3\right )^{2}}{x^{3}}+\frac {2 \ln \left (3\right )^{4} {\mathrm e}^{2 x}}{x^{2}}-\frac {12 \ln \left (3\right )^{2} {\mathrm e}^{2 x}}{x^{2}}+\frac {\ln \left (3\right )^{4} {\mathrm e}^{2 x}}{x^{4}}+\ln \left (3\right )^{4} {\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{6 x}}{x^{2}}-\frac {18 \,{\mathrm e}^{4 x}}{x}-\frac {18 \,{\mathrm e}^{4 x}}{x^{3}}\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+4*x^3-4*x^2+2*x-4)*exp(ln(-ln(3)^2+3)+x)^2
/x^5,x,method=_RETURNVERBOSE)

[Out]

-6*ln(3)^2*exp(x)^2+9*exp(x)^2+9*exp(x)^2/x^4+18*exp(x)^2/x^2-6/x^4*ln(3)^2*exp(x)^2+1/x^2*exp(x)^6*ln(3)^4-6/
x^2*exp(x)^6*ln(3)^2-2/x*exp(x)^4*ln(3)^4+12/x*exp(x)^4*ln(3)^2-2/x^3*exp(x)^4*ln(3)^4+12/x^3*exp(x)^4*ln(3)^2
+2/x^2*ln(3)^4*exp(x)^2-12/x^2*ln(3)^2*exp(x)^2+1/x^4*ln(3)^4*exp(x)^2+ln(3)^4*exp(x)^2+9/x^2*exp(x)^6-18/x*ex
p(x)^4-18/x^3*exp(x)^4

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.30, size = 86, normalized size = 2.32 \begin {gather*} -{\left (\log \left (3\right )^{2} - 3\right )}^{2} {\left (8 \, {\rm Ei}\left (4 \, x\right ) - e^{\left (2 \, x\right )} - 8 \, \Gamma \left (-1, -2 \, x\right ) - 8 \, \Gamma \left (-1, -4 \, x\right ) - 36 \, \Gamma \left (-1, -6 \, x\right ) - 16 \, \Gamma \left (-2, -2 \, x\right ) - 128 \, \Gamma \left (-2, -4 \, x\right ) - 72 \, \Gamma \left (-2, -6 \, x\right ) - 16 \, \Gamma \left (-3, -2 \, x\right ) - 384 \, \Gamma \left (-3, -4 \, x\right ) - 64 \, \Gamma \left (-4, -2 \, x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+4*x^3-4*x^2+2*x-4)*exp(log(-log(3)^2
+3)+x)^2/x^5,x, algorithm="maxima")

[Out]

-(log(3)^2 - 3)^2*(8*Ei(4*x) - e^(2*x) - 8*gamma(-1, -2*x) - 8*gamma(-1, -4*x) - 36*gamma(-1, -6*x) - 16*gamma
(-2, -2*x) - 128*gamma(-2, -4*x) - 72*gamma(-2, -6*x) - 16*gamma(-3, -2*x) - 384*gamma(-3, -4*x) - 64*gamma(-4
, -2*x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (38) = 76\).
time = 0.44, size = 205, normalized size = 5.54 \begin {gather*} \frac {x^{2} e^{\left (6 \, x + 6 \, \log \left (-\log \left (3\right )^{2} + 3\right )\right )} - 2 \, {\left ({\left (x^{3} + x\right )} \log \left (3\right )^{4} + 9 \, x^{3} - 6 \, {\left (x^{3} + x\right )} \log \left (3\right )^{2} + 9 \, x\right )} e^{\left (4 \, x + 4 \, \log \left (-\log \left (3\right )^{2} + 3\right )\right )} + {\left ({\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{8} - 12 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{6} + 54 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{4} + 81 \, x^{4} - 108 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (3\right )^{2} + 162 \, x^{2} + 81\right )} e^{\left (2 \, x + 2 \, \log \left (-\log \left (3\right )^{2} + 3\right )\right )}}{x^{4} \log \left (3\right )^{8} - 12 \, x^{4} \log \left (3\right )^{6} + 54 \, x^{4} \log \left (3\right )^{4} - 108 \, x^{4} \log \left (3\right )^{2} + 81 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+4*x^3-4*x^2+2*x-4)*exp(log(-log(3)^2
+3)+x)^2/x^5,x, algorithm="fricas")

[Out]

(x^2*e^(6*x + 6*log(-log(3)^2 + 3)) - 2*((x^3 + x)*log(3)^4 + 9*x^3 - 6*(x^3 + x)*log(3)^2 + 9*x)*e^(4*x + 4*l
og(-log(3)^2 + 3)) + ((x^4 + 2*x^2 + 1)*log(3)^8 - 12*(x^4 + 2*x^2 + 1)*log(3)^6 + 54*(x^4 + 2*x^2 + 1)*log(3)
^4 + 81*x^4 - 108*(x^4 + 2*x^2 + 1)*log(3)^2 + 162*x^2 + 81)*e^(2*x + 2*log(-log(3)^2 + 3)))/(x^4*log(3)^8 - 1
2*x^4*log(3)^6 + 54*x^4*log(3)^4 - 108*x^4*log(3)^2 + 81*x^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (29) = 58\).
time = 0.17, size = 165, normalized size = 4.46 \begin {gather*} \frac {\left (- 6 x^{7} \log {\left (3 \right )}^{2} + x^{7} \log {\left (3 \right )}^{4} + 9 x^{7}\right ) e^{6 x} + \left (- 18 x^{8} - 2 x^{8} \log {\left (3 \right )}^{4} + 12 x^{8} \log {\left (3 \right )}^{2} - 18 x^{6} - 2 x^{6} \log {\left (3 \right )}^{4} + 12 x^{6} \log {\left (3 \right )}^{2}\right ) e^{4 x} + \left (- 6 x^{9} \log {\left (3 \right )}^{2} + x^{9} \log {\left (3 \right )}^{4} + 9 x^{9} - 12 x^{7} \log {\left (3 \right )}^{2} + 2 x^{7} \log {\left (3 \right )}^{4} + 18 x^{7} - 6 x^{5} \log {\left (3 \right )}^{2} + x^{5} \log {\left (3 \right )}^{4} + 9 x^{5}\right ) e^{2 x}}{x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**3-2*x**2)*exp(x)**4+(-8*x**4+2*x**3-8*x**2+6*x)*exp(x)**2+2*x**5+4*x**3-4*x**2+2*x-4)*exp(ln(
-ln(3)**2+3)+x)**2/x**5,x)

[Out]

((-6*x**7*log(3)**2 + x**7*log(3)**4 + 9*x**7)*exp(6*x) + (-18*x**8 - 2*x**8*log(3)**4 + 12*x**8*log(3)**2 - 1
8*x**6 - 2*x**6*log(3)**4 + 12*x**6*log(3)**2)*exp(4*x) + (-6*x**9*log(3)**2 + x**9*log(3)**4 + 9*x**9 - 12*x*
*7*log(3)**2 + 2*x**7*log(3)**4 + 18*x**7 - 6*x**5*log(3)**2 + x**5*log(3)**4 + 9*x**5)*exp(2*x))/x**9

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (38) = 76\).
time = 0.41, size = 197, normalized size = 5.32 \begin {gather*} \frac {x^{4} e^{\left (2 \, x\right )} \log \left (3\right )^{4} - 2 \, x^{3} e^{\left (4 \, x\right )} \log \left (3\right )^{4} - 6 \, x^{4} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + x^{2} e^{\left (6 \, x\right )} \log \left (3\right )^{4} + 2 \, x^{2} e^{\left (2 \, x\right )} \log \left (3\right )^{4} + 12 \, x^{3} e^{\left (4 \, x\right )} \log \left (3\right )^{2} - 2 \, x e^{\left (4 \, x\right )} \log \left (3\right )^{4} + 9 \, x^{4} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{\left (6 \, x\right )} \log \left (3\right )^{2} - 12 \, x^{2} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + e^{\left (2 \, x\right )} \log \left (3\right )^{4} - 18 \, x^{3} e^{\left (4 \, x\right )} + 12 \, x e^{\left (4 \, x\right )} \log \left (3\right )^{2} + 9 \, x^{2} e^{\left (6 \, x\right )} + 18 \, x^{2} e^{\left (2 \, x\right )} - 6 \, e^{\left (2 \, x\right )} \log \left (3\right )^{2} - 18 \, x e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^3-2*x^2)*exp(x)^4+(-8*x^4+2*x^3-8*x^2+6*x)*exp(x)^2+2*x^5+4*x^3-4*x^2+2*x-4)*exp(log(-log(3)^2
+3)+x)^2/x^5,x, algorithm="giac")

[Out]

(x^4*e^(2*x)*log(3)^4 - 2*x^3*e^(4*x)*log(3)^4 - 6*x^4*e^(2*x)*log(3)^2 + x^2*e^(6*x)*log(3)^4 + 2*x^2*e^(2*x)
*log(3)^4 + 12*x^3*e^(4*x)*log(3)^2 - 2*x*e^(4*x)*log(3)^4 + 9*x^4*e^(2*x) - 6*x^2*e^(6*x)*log(3)^2 - 12*x^2*e
^(2*x)*log(3)^2 + e^(2*x)*log(3)^4 - 18*x^3*e^(4*x) + 12*x*e^(4*x)*log(3)^2 + 9*x^2*e^(6*x) + 18*x^2*e^(2*x) -
6*e^(2*x)*log(3)^2 - 18*x*e^(4*x) + 9*e^(2*x))/x^4

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Mupad [B]
time = 7.53, size = 30, normalized size = 0.81 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,{\left ({\ln \left (3\right )}^2-3\right )}^2\,{\left (x^2-x\,{\mathrm {e}}^{2\,x}+1\right )}^2}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x + 2*log(3 - log(3)^2))*(2*x - exp(4*x)*(2*x^2 - 6*x^3) + exp(2*x)*(6*x - 8*x^2 + 2*x^3 - 8*x^4) -
4*x^2 + 4*x^3 + 2*x^5 - 4))/x^5,x)

[Out]

(exp(2*x)*(log(3)^2 - 3)^2*(x^2 - x*exp(2*x) + 1)^2)/x^4

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