3.9.10 \(\int \frac {e^2 (16 x^6+4 x^7)+e^2 (28 x^6+6 x^7) \log (x)+e^2 (-32 x^6-12 x^7) \log ^2(x)+e^2 (-104 x^6-32 x^7) \log ^3(x)+e^2 (-80 x^6-24 x^7) \log ^4(x)+(-4+e^2 (-20 x^6-6 x^7)) \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {(4+x) \left (\frac {1}{e^2}-x^2 \left (x+\frac {x}{\log (x)}\right )^4\right )}{x} \]

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Rubi [B]  time = 1.22, antiderivative size = 91, normalized size of antiderivative = 3.50, number of steps used = 55, number of rules used = 7, integrand size = 123, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 6742, 14, 2353, 2306, 2309, 2178} \begin {gather*} -x^6-\frac {x^6}{\log ^4(x)}-\frac {4 x^6}{\log ^3(x)}-\frac {6 x^6}{\log ^2(x)}-\frac {4 x^6}{\log (x)}-4 x^5-\frac {4 x^5}{\log ^4(x)}-\frac {16 x^5}{\log ^3(x)}-\frac {24 x^5}{\log ^2(x)}-\frac {16 x^5}{\log (x)}+\frac {4}{e^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^2*(16*x^6 + 4*x^7) + E^2*(28*x^6 + 6*x^7)*Log[x] + E^2*(-32*x^6 - 12*x^7)*Log[x]^2 + E^2*(-104*x^6 - 32
*x^7)*Log[x]^3 + E^2*(-80*x^6 - 24*x^7)*Log[x]^4 + (-4 + E^2*(-20*x^6 - 6*x^7))*Log[x]^5)/(E^2*x^2*Log[x]^5),x
]

[Out]

4/(E^2*x) - 4*x^5 - x^6 - (4*x^5)/Log[x]^4 - x^6/Log[x]^4 - (16*x^5)/Log[x]^3 - (4*x^6)/Log[x]^3 - (24*x^5)/Lo
g[x]^2 - (6*x^6)/Log[x]^2 - (16*x^5)/Log[x] - (4*x^6)/Log[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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^2 \left (16 x^6+4 x^7\right )+e^2 \left (28 x^6+6 x^7\right ) \log (x)+e^2 \left (-32 x^6-12 x^7\right ) \log ^2(x)+e^2 \left (-104 x^6-32 x^7\right ) \log ^3(x)+e^2 \left (-80 x^6-24 x^7\right ) \log ^4(x)+\left (-4+e^2 \left (-20 x^6-6 x^7\right )\right ) \log ^5(x)}{x^2 \log ^5(x)} \, dx}{e^2}\\ &=\frac {\int \left (-\frac {2 \left (2+10 e^2 x^6+3 e^2 x^7\right )}{x^2}+\frac {4 e^2 x^4 (4+x)}{\log ^5(x)}+\frac {2 e^2 x^4 (14+3 x)}{\log ^4(x)}-\frac {4 e^2 x^4 (8+3 x)}{\log ^3(x)}-\frac {8 e^2 x^4 (13+4 x)}{\log ^2(x)}-\frac {8 e^2 x^4 (10+3 x)}{\log (x)}\right ) \, dx}{e^2}\\ &=2 \int \frac {x^4 (14+3 x)}{\log ^4(x)} \, dx+4 \int \frac {x^4 (4+x)}{\log ^5(x)} \, dx-4 \int \frac {x^4 (8+3 x)}{\log ^3(x)} \, dx-8 \int \frac {x^4 (13+4 x)}{\log ^2(x)} \, dx-8 \int \frac {x^4 (10+3 x)}{\log (x)} \, dx-\frac {2 \int \frac {2+10 e^2 x^6+3 e^2 x^7}{x^2} \, dx}{e^2}\\ &=2 \int \left (\frac {14 x^4}{\log ^4(x)}+\frac {3 x^5}{\log ^4(x)}\right ) \, dx+4 \int \left (\frac {4 x^4}{\log ^5(x)}+\frac {x^5}{\log ^5(x)}\right ) \, dx-4 \int \left (\frac {8 x^4}{\log ^3(x)}+\frac {3 x^5}{\log ^3(x)}\right ) \, dx-8 \int \left (\frac {13 x^4}{\log ^2(x)}+\frac {4 x^5}{\log ^2(x)}\right ) \, dx-8 \int \left (\frac {10 x^4}{\log (x)}+\frac {3 x^5}{\log (x)}\right ) \, dx-\frac {2 \int \left (\frac {2}{x^2}+10 e^2 x^4+3 e^2 x^5\right ) \, dx}{e^2}\\ &=\frac {4}{e^2 x}-4 x^5-x^6+4 \int \frac {x^5}{\log ^5(x)} \, dx+6 \int \frac {x^5}{\log ^4(x)} \, dx-12 \int \frac {x^5}{\log ^3(x)} \, dx+16 \int \frac {x^4}{\log ^5(x)} \, dx-24 \int \frac {x^5}{\log (x)} \, dx+28 \int \frac {x^4}{\log ^4(x)} \, dx-32 \int \frac {x^4}{\log ^3(x)} \, dx-32 \int \frac {x^5}{\log ^2(x)} \, dx-80 \int \frac {x^4}{\log (x)} \, dx-104 \int \frac {x^4}{\log ^2(x)} \, dx\\ &=\frac {4}{e^2 x}-4 x^5-x^6-\frac {4 x^5}{\log ^4(x)}-\frac {x^6}{\log ^4(x)}-\frac {28 x^5}{3 \log ^3(x)}-\frac {2 x^6}{\log ^3(x)}+\frac {16 x^5}{\log ^2(x)}+\frac {6 x^6}{\log ^2(x)}+\frac {104 x^5}{\log (x)}+\frac {32 x^6}{\log (x)}+6 \int \frac {x^5}{\log ^4(x)} \, dx+12 \int \frac {x^5}{\log ^3(x)} \, dx+20 \int \frac {x^4}{\log ^4(x)} \, dx-24 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-36 \int \frac {x^5}{\log ^2(x)} \, dx+\frac {140}{3} \int \frac {x^4}{\log ^3(x)} \, dx-80 \int \frac {x^4}{\log ^2(x)} \, dx-80 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-192 \int \frac {x^5}{\log (x)} \, dx-520 \int \frac {x^4}{\log (x)} \, dx\\ &=\frac {4}{e^2 x}-4 x^5-x^6-80 \text {Ei}(5 \log (x))-24 \text {Ei}(6 \log (x))-\frac {4 x^5}{\log ^4(x)}-\frac {x^6}{\log ^4(x)}-\frac {16 x^5}{\log ^3(x)}-\frac {4 x^6}{\log ^3(x)}-\frac {22 x^5}{3 \log ^2(x)}+\frac {184 x^5}{\log (x)}+\frac {68 x^6}{\log (x)}+12 \int \frac {x^5}{\log ^3(x)} \, dx+\frac {100}{3} \int \frac {x^4}{\log ^3(x)} \, dx+36 \int \frac {x^5}{\log ^2(x)} \, dx+\frac {350}{3} \int \frac {x^4}{\log ^2(x)} \, dx-192 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-216 \int \frac {x^5}{\log (x)} \, dx-400 \int \frac {x^4}{\log (x)} \, dx-520 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4}{e^2 x}-4 x^5-x^6-600 \text {Ei}(5 \log (x))-216 \text {Ei}(6 \log (x))-\frac {4 x^5}{\log ^4(x)}-\frac {x^6}{\log ^4(x)}-\frac {16 x^5}{\log ^3(x)}-\frac {4 x^6}{\log ^3(x)}-\frac {24 x^5}{\log ^2(x)}-\frac {6 x^6}{\log ^2(x)}+\frac {202 x^5}{3 \log (x)}+\frac {32 x^6}{\log (x)}+36 \int \frac {x^5}{\log ^2(x)} \, dx+\frac {250}{3} \int \frac {x^4}{\log ^2(x)} \, dx+216 \int \frac {x^5}{\log (x)} \, dx-216 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-400 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+\frac {1750}{3} \int \frac {x^4}{\log (x)} \, dx\\ &=\frac {4}{e^2 x}-4 x^5-x^6-1000 \text {Ei}(5 \log (x))-432 \text {Ei}(6 \log (x))-\frac {4 x^5}{\log ^4(x)}-\frac {x^6}{\log ^4(x)}-\frac {16 x^5}{\log ^3(x)}-\frac {4 x^6}{\log ^3(x)}-\frac {24 x^5}{\log ^2(x)}-\frac {6 x^6}{\log ^2(x)}-\frac {16 x^5}{\log (x)}-\frac {4 x^6}{\log (x)}+216 \int \frac {x^5}{\log (x)} \, dx+216 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )+\frac {1250}{3} \int \frac {x^4}{\log (x)} \, dx+\frac {1750}{3} \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4}{e^2 x}-4 x^5-x^6-\frac {1250}{3} \text {Ei}(5 \log (x))-216 \text {Ei}(6 \log (x))-\frac {4 x^5}{\log ^4(x)}-\frac {x^6}{\log ^4(x)}-\frac {16 x^5}{\log ^3(x)}-\frac {4 x^6}{\log ^3(x)}-\frac {24 x^5}{\log ^2(x)}-\frac {6 x^6}{\log ^2(x)}-\frac {16 x^5}{\log (x)}-\frac {4 x^6}{\log (x)}+216 \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )+\frac {1250}{3} \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4}{e^2 x}-4 x^5-x^6-\frac {4 x^5}{\log ^4(x)}-\frac {x^6}{\log ^4(x)}-\frac {16 x^5}{\log ^3(x)}-\frac {4 x^6}{\log ^3(x)}-\frac {24 x^5}{\log ^2(x)}-\frac {6 x^6}{\log ^2(x)}-\frac {16 x^5}{\log (x)}-\frac {4 x^6}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.28, size = 65, normalized size = 2.50 \begin {gather*} \frac {4}{e^2 x}-x^5 (4+x)-\frac {x^5 (4+x)}{\log ^4(x)}-\frac {4 x^5 (4+x)}{\log ^3(x)}-\frac {6 x^5 (4+x)}{\log ^2(x)}-\frac {4 x^5 (4+x)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(16*x^6 + 4*x^7) + E^2*(28*x^6 + 6*x^7)*Log[x] + E^2*(-32*x^6 - 12*x^7)*Log[x]^2 + E^2*(-104*x^
6 - 32*x^7)*Log[x]^3 + E^2*(-80*x^6 - 24*x^7)*Log[x]^4 + (-4 + E^2*(-20*x^6 - 6*x^7))*Log[x]^5)/(E^2*x^2*Log[x
]^5),x]

[Out]

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

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fricas [B]  time = 0.58, size = 92, normalized size = 3.54 \begin {gather*} -\frac {{\left (4 \, {\left (x^{7} + 4 \, x^{6}\right )} e^{2} \log \relax (x)^{3} + {\left ({\left (x^{7} + 4 \, x^{6}\right )} e^{2} - 4\right )} \log \relax (x)^{4} + 6 \, {\left (x^{7} + 4 \, x^{6}\right )} e^{2} \log \relax (x)^{2} + 4 \, {\left (x^{7} + 4 \, x^{6}\right )} e^{2} \log \relax (x) + {\left (x^{7} + 4 \, x^{6}\right )} e^{2}\right )} e^{\left (-2\right )}}{x \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^7-20*x^6)*exp(2)-4)*log(x)^5+(-24*x^7-80*x^6)*exp(2)*log(x)^4+(-32*x^7-104*x^6)*exp(2)*log(x
)^3+(-12*x^7-32*x^6)*exp(2)*log(x)^2+(6*x^7+28*x^6)*exp(2)*log(x)+(4*x^7+16*x^6)*exp(2))/x^2/exp(2)/log(x)^5,x
, algorithm="fricas")

[Out]

-(4*(x^7 + 4*x^6)*e^2*log(x)^3 + ((x^7 + 4*x^6)*e^2 - 4)*log(x)^4 + 6*(x^7 + 4*x^6)*e^2*log(x)^2 + 4*(x^7 + 4*
x^6)*e^2*log(x) + (x^7 + 4*x^6)*e^2)*e^(-2)/(x*log(x)^4)

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giac [B]  time = 0.40, size = 114, normalized size = 4.38 \begin {gather*} -\frac {{\left (x^{7} e^{2} \log \relax (x)^{4} + 4 \, x^{7} e^{2} \log \relax (x)^{3} + 4 \, x^{6} e^{2} \log \relax (x)^{4} + 6 \, x^{7} e^{2} \log \relax (x)^{2} + 16 \, x^{6} e^{2} \log \relax (x)^{3} + 4 \, x^{7} e^{2} \log \relax (x) + 24 \, x^{6} e^{2} \log \relax (x)^{2} + x^{7} e^{2} + 16 \, x^{6} e^{2} \log \relax (x) + 4 \, x^{6} e^{2} - 4 \, \log \relax (x)^{4}\right )} e^{\left (-2\right )}}{x \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^7-20*x^6)*exp(2)-4)*log(x)^5+(-24*x^7-80*x^6)*exp(2)*log(x)^4+(-32*x^7-104*x^6)*exp(2)*log(x
)^3+(-12*x^7-32*x^6)*exp(2)*log(x)^2+(6*x^7+28*x^6)*exp(2)*log(x)+(4*x^7+16*x^6)*exp(2))/x^2/exp(2)/log(x)^5,x
, algorithm="giac")

[Out]

-(x^7*e^2*log(x)^4 + 4*x^7*e^2*log(x)^3 + 4*x^6*e^2*log(x)^4 + 6*x^7*e^2*log(x)^2 + 16*x^6*e^2*log(x)^3 + 4*x^
7*e^2*log(x) + 24*x^6*e^2*log(x)^2 + x^7*e^2 + 16*x^6*e^2*log(x) + 4*x^6*e^2 - 4*log(x)^4)*e^(-2)/(x*log(x)^4)

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maple [B]  time = 0.04, size = 71, normalized size = 2.73

method	result	size
risch	\(-\frac {{\mathrm e}^{-2} \left (x^{7} {\mathrm e}^{2}+4 x^{6} {\mathrm e}^{2}-4\right )}{x}-\frac {x^{5} \left (4 x \ln \relax (x )^{3}+6 x \ln \relax (x )^{2}+16 \ln \relax (x )^{3}+4 x \ln \relax (x )+24 \ln \relax (x )^{2}+x +16 \ln \relax (x )+4\right )}{\ln \relax (x )^{4}}\)	\(71\)
default	\({\mathrm e}^{-2} \left (-x^{6} {\mathrm e}^{2}-4 \,{\mathrm e}^{2} x^{5}+24 \,{\mathrm e}^{2} \expIntegralEi \left (1, -6 \ln \relax (x )\right )+80 \,{\mathrm e}^{2} \expIntegralEi \left (1, -5 \ln \relax (x )\right )-32 \,{\mathrm e}^{2} \left (-\frac {x^{6}}{\ln \relax (x )}-6 \expIntegralEi \left (1, -6 \ln \relax (x )\right )\right )-104 \,{\mathrm e}^{2} \left (-\frac {x^{5}}{\ln \relax (x )}-5 \expIntegralEi \left (1, -5 \ln \relax (x )\right )\right )-12 \,{\mathrm e}^{2} \left (-\frac {x^{6}}{2 \ln \relax (x )^{2}}-\frac {3 x^{6}}{\ln \relax (x )}-18 \expIntegralEi \left (1, -6 \ln \relax (x )\right )\right )-32 \,{\mathrm e}^{2} \left (-\frac {x^{5}}{2 \ln \relax (x )^{2}}-\frac {5 x^{5}}{2 \ln \relax (x )}-\frac {25 \expIntegralEi \left (1, -5 \ln \relax (x )\right )}{2}\right )+6 \,{\mathrm e}^{2} \left (-\frac {x^{6}}{3 \ln \relax (x )^{3}}-\frac {x^{6}}{\ln \relax (x )^{2}}-\frac {6 x^{6}}{\ln \relax (x )}-36 \expIntegralEi \left (1, -6 \ln \relax (x )\right )\right )+28 \,{\mathrm e}^{2} \left (-\frac {x^{5}}{3 \ln \relax (x )^{3}}-\frac {5 x^{5}}{6 \ln \relax (x )^{2}}-\frac {25 x^{5}}{6 \ln \relax (x )}-\frac {125 \expIntegralEi \left (1, -5 \ln \relax (x )\right )}{6}\right )+4 \,{\mathrm e}^{2} \left (-\frac {x^{6}}{4 \ln \relax (x )^{4}}-\frac {x^{6}}{2 \ln \relax (x )^{3}}-\frac {3 x^{6}}{2 \ln \relax (x )^{2}}-\frac {9 x^{6}}{\ln \relax (x )}-54 \expIntegralEi \left (1, -6 \ln \relax (x )\right )\right )+16 \,{\mathrm e}^{2} \left (-\frac {x^{5}}{4 \ln \relax (x )^{4}}-\frac {5 x^{5}}{12 \ln \relax (x )^{3}}-\frac {25 x^{5}}{24 \ln \relax (x )^{2}}-\frac {125 x^{5}}{24 \ln \relax (x )}-\frac {625 \expIntegralEi \left (1, -5 \ln \relax (x )\right )}{24}\right )+\frac {4}{x}\right )\)	\(330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x^7-20*x^6)*exp(2)-4)*ln(x)^5+(-24*x^7-80*x^6)*exp(2)*ln(x)^4+(-32*x^7-104*x^6)*exp(2)*ln(x)^3+(-12*
x^7-32*x^6)*exp(2)*ln(x)^2+(6*x^7+28*x^6)*exp(2)*ln(x)+(4*x^7+16*x^6)*exp(2))/x^2/exp(2)/ln(x)^5,x,method=_RET
URNVERBOSE)

[Out]

-exp(-2)*(x^7*exp(2)+4*x^6*exp(2)-4)/x-x^5*(4*x*ln(x)^3+6*x*ln(x)^2+16*ln(x)^3+4*x*ln(x)+24*ln(x)^2+x+16*ln(x)
+4)/ln(x)^4

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maxima [C]  time = 0.77, size = 121, normalized size = 4.65 \begin {gather*} -{\left (x^{6} e^{2} + 4 \, x^{5} e^{2} + 24 \, {\rm Ei}\left (6 \, \log \relax (x)\right ) e^{2} + 80 \, {\rm Ei}\left (5 \, \log \relax (x)\right ) e^{2} + 520 \, e^{2} \Gamma \left (-1, -5 \, \log \relax (x)\right ) + 192 \, e^{2} \Gamma \left (-1, -6 \, \log \relax (x)\right ) - 800 \, e^{2} \Gamma \left (-2, -5 \, \log \relax (x)\right ) - 432 \, e^{2} \Gamma \left (-2, -6 \, \log \relax (x)\right ) - 3500 \, e^{2} \Gamma \left (-3, -5 \, \log \relax (x)\right ) - 1296 \, e^{2} \Gamma \left (-3, -6 \, \log \relax (x)\right ) + 10000 \, e^{2} \Gamma \left (-4, -5 \, \log \relax (x)\right ) + 5184 \, e^{2} \Gamma \left (-4, -6 \, \log \relax (x)\right ) - \frac {4}{x}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^7-20*x^6)*exp(2)-4)*log(x)^5+(-24*x^7-80*x^6)*exp(2)*log(x)^4+(-32*x^7-104*x^6)*exp(2)*log(x
)^3+(-12*x^7-32*x^6)*exp(2)*log(x)^2+(6*x^7+28*x^6)*exp(2)*log(x)+(4*x^7+16*x^6)*exp(2))/x^2/exp(2)/log(x)^5,x
, algorithm="maxima")

[Out]

-(x^6*e^2 + 4*x^5*e^2 + 24*Ei(6*log(x))*e^2 + 80*Ei(5*log(x))*e^2 + 520*e^2*gamma(-1, -5*log(x)) + 192*e^2*gam
ma(-1, -6*log(x)) - 800*e^2*gamma(-2, -5*log(x)) - 432*e^2*gamma(-2, -6*log(x)) - 3500*e^2*gamma(-3, -5*log(x)
) - 1296*e^2*gamma(-3, -6*log(x)) + 10000*e^2*gamma(-4, -5*log(x)) + 5184*e^2*gamma(-4, -6*log(x)) - 4/x)*e^(-
2)

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mupad [B]  time = 0.75, size = 84, normalized size = 3.23 \begin {gather*} -\frac {x^8+4\,x^7-4\,{\mathrm {e}}^{-2}\,x}{x^2}-\frac {\ln \relax (x)\,\left (4\,x^8+16\,x^7\right )+{\ln \relax (x)}^3\,\left (4\,x^8+16\,x^7\right )+{\ln \relax (x)}^2\,\left (6\,x^8+24\,x^7\right )+4\,x^7+x^8}{x^2\,{\ln \relax (x)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2)*(log(x)^5*(exp(2)*(20*x^6 + 6*x^7) + 4) - exp(2)*(16*x^6 + 4*x^7) - exp(2)*log(x)*(28*x^6 + 6*x^
7) + exp(2)*log(x)^2*(32*x^6 + 12*x^7) + exp(2)*log(x)^4*(80*x^6 + 24*x^7) + exp(2)*log(x)^3*(104*x^6 + 32*x^7
)))/(x^2*log(x)^5),x)

[Out]

- (4*x^7 - 4*x*exp(-2) + x^8)/x^2 - (log(x)*(16*x^7 + 4*x^8) + log(x)^3*(16*x^7 + 4*x^8) + log(x)^2*(24*x^7 +
6*x^8) + 4*x^7 + x^8)/(x^2*log(x)^4)

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sympy [B]  time = 0.18, size = 83, normalized size = 3.19 \begin {gather*} \frac {- x^{6} e^{2} - 4 x^{5} e^{2} + \frac {4}{x}}{e^{2}} + \frac {- x^{6} - 4 x^{5} + \left (- 6 x^{6} - 24 x^{5}\right ) \log {\relax (x )}^{2} + \left (- 4 x^{6} - 16 x^{5}\right ) \log {\relax (x )}^{3} + \left (- 4 x^{6} - 16 x^{5}\right ) \log {\relax (x )}}{\log {\relax (x )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x**7-20*x**6)*exp(2)-4)*ln(x)**5+(-24*x**7-80*x**6)*exp(2)*ln(x)**4+(-32*x**7-104*x**6)*exp(2)
*ln(x)**3+(-12*x**7-32*x**6)*exp(2)*ln(x)**2+(6*x**7+28*x**6)*exp(2)*ln(x)+(4*x**7+16*x**6)*exp(2))/x**2/exp(2
)/ln(x)**5,x)

[Out]

(-x**6*exp(2) - 4*x**5*exp(2) + 4/x)*exp(-2) + (-x**6 - 4*x**5 + (-6*x**6 - 24*x**5)*log(x)**2 + (-4*x**6 - 16
*x**5)*log(x)**3 + (-4*x**6 - 16*x**5)*log(x))/log(x)**4

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