3.72.74 \(\int \frac {-160+(-4000+e^{7-x} (-1600-800 x)+e^{14-2 x} (-160-160 x)) \log (x)+160 \log (x) \log (4 \log ^2(x))}{x^3 \log (x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {80 \left (\left (5+e^{7-x}\right )^2-\log \left (4 \log ^2(x)\right )\right )}{x^2} \]

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Rubi [C]  time = 0.97, antiderivative size = 71, normalized size of antiderivative = 2.73, number of steps used = 12, number of rules used = 7, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6742, 2197, 2309, 2178, 2366, 6482, 2522} \begin {gather*} 4000 \log (x) \text {Ei}(-2 \log (x))-160 (25 \log (x)+1) \text {Ei}(-2 \log (x))+160 \text {Ei}(-2 \log (x))+\frac {80 e^{14-2 x}}{x^2}+\frac {800 e^{7-x}}{x^2}+\frac {2000}{x^2}-\frac {80 \log \left (4 \log ^2(x)\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-160 + (-4000 + E^(7 - x)*(-1600 - 800*x) + E^(14 - 2*x)*(-160 - 160*x))*Log[x] + 160*Log[x]*Log[4*Log[x]
^2])/(x^3*Log[x]),x]

[Out]

2000/x^2 + (80*E^(14 - 2*x))/x^2 + (800*E^(7 - x))/x^2 + 160*ExpIntegralEi[-2*Log[x]] + 4000*ExpIntegralEi[-2*
Log[x]]*Log[x] - 160*ExpIntegralEi[-2*Log[x]]*(1 + 25*Log[x]) - (80*Log[4*Log[x]^2])/x^2

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 2197

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 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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2522

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

Rule 6482

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*ExpIntegralEi[a + b*x])/b, x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, 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 (-\frac {160 e^{14-2 x} (1+x)}{x^3}-\frac {800 e^{7-x} (2+x)}{x^3}+\frac {160 \left (-1-25 \log (x)+\log (x) \log \left (4 \log ^2(x)\right )\right )}{x^3 \log (x)}\right ) \, dx\\ &=-\left (160 \int \frac {e^{14-2 x} (1+x)}{x^3} \, dx\right )+160 \int \frac {-1-25 \log (x)+\log (x) \log \left (4 \log ^2(x)\right )}{x^3 \log (x)} \, dx-800 \int \frac {e^{7-x} (2+x)}{x^3} \, dx\\ &=\frac {80 e^{14-2 x}}{x^2}+\frac {800 e^{7-x}}{x^2}+160 \int \left (\frac {-1-25 \log (x)}{x^3 \log (x)}+\frac {\log \left (4 \log ^2(x)\right )}{x^3}\right ) \, dx\\ &=\frac {80 e^{14-2 x}}{x^2}+\frac {800 e^{7-x}}{x^2}+160 \int \frac {-1-25 \log (x)}{x^3 \log (x)} \, dx+160 \int \frac {\log \left (4 \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {80 e^{14-2 x}}{x^2}+\frac {800 e^{7-x}}{x^2}-160 \text {Ei}(-2 \log (x)) (1+25 \log (x))-\frac {80 \log \left (4 \log ^2(x)\right )}{x^2}+160 \int \frac {1}{x^3 \log (x)} \, dx+4000 \int \frac {\text {Ei}(-2 \log (x))}{x} \, dx\\ &=\frac {80 e^{14-2 x}}{x^2}+\frac {800 e^{7-x}}{x^2}-160 \text {Ei}(-2 \log (x)) (1+25 \log (x))-\frac {80 \log \left (4 \log ^2(x)\right )}{x^2}+160 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right )+4000 \operatorname {Subst}(\int \text {Ei}(-2 x) \, dx,x,\log (x))\\ &=\frac {2000}{x^2}+\frac {80 e^{14-2 x}}{x^2}+\frac {800 e^{7-x}}{x^2}+160 \text {Ei}(-2 \log (x))+4000 \text {Ei}(-2 \log (x)) \log (x)-160 \text {Ei}(-2 \log (x)) (1+25 \log (x))-\frac {80 \log \left (4 \log ^2(x)\right )}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 32, normalized size = 1.23 \begin {gather*} \frac {80 \left (e^{-2 x} \left (e^7+5 e^x\right )^2-\log \left (4 \log ^2(x)\right )\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-160 + (-4000 + E^(7 - x)*(-1600 - 800*x) + E^(14 - 2*x)*(-160 - 160*x))*Log[x] + 160*Log[x]*Log[4*
Log[x]^2])/(x^3*Log[x]),x]

[Out]

(80*((E^7 + 5*E^x)^2/E^(2*x) - Log[4*Log[x]^2]))/x^2

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fricas [A]  time = 1.10, size = 30, normalized size = 1.15 \begin {gather*} \frac {80 \, {\left (10 \, e^{\left (-x + 7\right )} + e^{\left (-2 \, x + 14\right )} - \log \left (4 \, \log \relax (x)^{2}\right ) + 25\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*log(x)*log(4*log(x)^2)+((-160*x-160)*exp(-x+7)^2+(-800*x-1600)*exp(-x+7)-4000)*log(x)-160)/x^3/
log(x),x, algorithm="fricas")

[Out]

80*(10*e^(-x + 7) + e^(-2*x + 14) - log(4*log(x)^2) + 25)/x^2

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giac [A]  time = 0.20, size = 30, normalized size = 1.15 \begin {gather*} \frac {80 \, {\left (10 \, e^{\left (-x + 7\right )} + e^{\left (-2 \, x + 14\right )} - \log \left (4 \, \log \relax (x)^{2}\right ) + 25\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*log(x)*log(4*log(x)^2)+((-160*x-160)*exp(-x+7)^2+(-800*x-1600)*exp(-x+7)-4000)*log(x)-160)/x^3/
log(x),x, algorithm="giac")

[Out]

80*(10*e^(-x + 7) + e^(-2*x + 14) - log(4*log(x)^2) + 25)/x^2

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maple [C]  time = 0.14, size = 91, normalized size = 3.50

method	result	size
risch	\(-\frac {160 \ln \left (\ln \relax (x )\right )}{x^{2}}-\frac {40 \left (-i \pi \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}-50-i \pi \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}-2 \,{\mathrm e}^{-2 x +14}+4 \ln \relax (2)-20 \,{\mathrm e}^{-x +7}\right )}{x^{2}}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((160*ln(x)*ln(4*ln(x)^2)+((-160*x-160)*exp(-x+7)^2+(-800*x-1600)*exp(-x+7)-4000)*ln(x)-160)/x^3/ln(x),x,me
thod=_RETURNVERBOSE)

[Out]

-160*ln(ln(x))/x^2-40*(-I*Pi*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+2*I*Pi*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-50-I*Pi*cs
gn(I*ln(x)^2)^3-2*exp(-2*x+14)+4*ln(2)-20*exp(-x+7))/x^2

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maxima [C]  time = 0.40, size = 50, normalized size = 1.92 \begin {gather*} 320 \, e^{14} \Gamma \left (-1, 2 \, x\right ) + 800 \, e^{7} \Gamma \left (-1, x\right ) + 640 \, e^{14} \Gamma \left (-2, 2 \, x\right ) + 1600 \, e^{7} \Gamma \left (-2, x\right ) - \frac {80 \, \log \left (4 \, \log \relax (x)^{2}\right )}{x^{2}} + \frac {2000}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*log(x)*log(4*log(x)^2)+((-160*x-160)*exp(-x+7)^2+(-800*x-1600)*exp(-x+7)-4000)*log(x)-160)/x^3/
log(x),x, algorithm="maxima")

[Out]

320*e^14*gamma(-1, 2*x) + 800*e^7*gamma(-1, x) + 640*e^14*gamma(-2, 2*x) + 1600*e^7*gamma(-2, x) - 80*log(4*lo
g(x)^2)/x^2 + 2000/x^2

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mupad [B]  time = 4.39, size = 40, normalized size = 1.54 \begin {gather*} \frac {800\,{\mathrm {e}}^{7-x}}{x^2}-\frac {80\,\ln \left (4\,{\ln \relax (x)}^2\right )}{x^2}+\frac {80\,{\mathrm {e}}^{14-2\,x}}{x^2}+\frac {2000}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(exp(14 - 2*x)*(160*x + 160) + exp(7 - x)*(800*x + 1600) + 4000) - 160*log(4*log(x)^2)*log(x) + 1
60)/(x^3*log(x)),x)

[Out]

(800*exp(7 - x))/x^2 - (80*log(4*log(x)^2))/x^2 + (80*exp(14 - 2*x))/x^2 + 2000/x^2

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sympy [A]  time = 0.55, size = 42, normalized size = 1.62 \begin {gather*} - \frac {80 \log {\left (4 \log {\relax (x )}^{2} \right )}}{x^{2}} + \frac {2000}{x^{2}} + \frac {800 x^{2} e^{7 - x} + 80 x^{2} e^{14 - 2 x}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*ln(x)*ln(4*ln(x)**2)+((-160*x-160)*exp(-x+7)**2+(-800*x-1600)*exp(-x+7)-4000)*ln(x)-160)/x**3/l
n(x),x)

[Out]

-80*log(4*log(x)**2)/x**2 + 2000/x**2 + (800*x**2*exp(7 - x) + 80*x**2*exp(14 - 2*x))/x**4

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