3.30.94 \(\int \frac {-380+76 e^{e^5}-3 x^2+(380-76 e^{e^5}+9 x^2) \log (2 x)}{76 \log ^2(2 x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x \left (5-e^{e^5}+\frac {3 x^2}{76}\right )}{\log (2 x)} \]

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Rubi [A]  time = 0.35, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {12, 6741, 6742, 2330, 2297, 2298, 2306, 2309, 2178} \begin {gather*} \frac {3 x^3}{76 \log (2 x)}+\frac {\left (5-e^{e^5}\right ) x}{\log (2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-380 + 76*E^E^5 - 3*x^2 + (380 - 76*E^E^5 + 9*x^2)*Log[2*x])/(76*Log[2*x]^2),x]

[Out]

((5 - E^E^5)*x)/Log[2*x] + (3*x^3)/(76*Log[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 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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

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 2330

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

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} &=\frac {1}{76} \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{\log ^2(2 x)} \, dx\\ &=\frac {1}{76} \int \frac {-380 \left (1-\frac {e^{e^5}}{5}\right )-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{\log ^2(2 x)} \, dx\\ &=\frac {1}{76} \int \left (\frac {-380+76 e^{e^5}-3 x^2}{\log ^2(2 x)}-\frac {-380+76 e^{e^5}-9 x^2}{\log (2 x)}\right ) \, dx\\ &=\frac {1}{76} \int \frac {-380+76 e^{e^5}-3 x^2}{\log ^2(2 x)} \, dx-\frac {1}{76} \int \frac {-380+76 e^{e^5}-9 x^2}{\log (2 x)} \, dx\\ &=\frac {1}{76} \int \left (-\frac {380 \left (1-\frac {e^{e^5}}{5}\right )}{\log ^2(2 x)}-\frac {3 x^2}{\log ^2(2 x)}\right ) \, dx-\frac {1}{76} \int \left (-\frac {380 \left (1-\frac {e^{e^5}}{5}\right )}{\log (2 x)}-\frac {9 x^2}{\log (2 x)}\right ) \, dx\\ &=-\left (\frac {3}{76} \int \frac {x^2}{\log ^2(2 x)} \, dx\right )+\frac {9}{76} \int \frac {x^2}{\log (2 x)} \, dx+\left (-5+e^{e^5}\right ) \int \frac {1}{\log ^2(2 x)} \, dx-\left (-5+e^{e^5}\right ) \int \frac {1}{\log (2 x)} \, dx\\ &=\frac {\left (5-e^{e^5}\right ) x}{\log (2 x)}+\frac {3 x^3}{76 \log (2 x)}+\frac {1}{2} \left (5-e^{e^5}\right ) \text {li}(2 x)+\frac {9}{608} \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (2 x)\right )-\frac {9}{76} \int \frac {x^2}{\log (2 x)} \, dx+\left (-5+e^{e^5}\right ) \int \frac {1}{\log (2 x)} \, dx\\ &=\frac {9}{608} \text {Ei}(3 \log (2 x))+\frac {\left (5-e^{e^5}\right ) x}{\log (2 x)}+\frac {3 x^3}{76 \log (2 x)}-\frac {9}{608} \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (2 x)\right )\\ &=\frac {\left (5-e^{e^5}\right ) x}{\log (2 x)}+\frac {3 x^3}{76 \log (2 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 25, normalized size = 1.04 \begin {gather*} \frac {x \left (380-76 e^{e^5}+3 x^2\right )}{76 \log (2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-380 + 76*E^E^5 - 3*x^2 + (380 - 76*E^E^5 + 9*x^2)*Log[2*x])/(76*Log[2*x]^2),x]

[Out]

(x*(380 - 76*E^E^5 + 3*x^2))/(76*Log[2*x])

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fricas [A]  time = 0.54, size = 23, normalized size = 0.96 \begin {gather*} \frac {3 \, x^{3} - 76 \, x e^{\left (e^{5}\right )} + 380 \, x}{76 \, \log \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/76*((-76*exp(exp(5))+9*x^2+380)*log(2*x)+76*exp(exp(5))-3*x^2-380)/log(2*x)^2,x, algorithm="fricas
")

[Out]

1/76*(3*x^3 - 76*x*e^(e^5) + 380*x)/log(2*x)

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giac [A]  time = 0.33, size = 33, normalized size = 1.38 \begin {gather*} \frac {3 \, x^{3}}{76 \, \log \left (2 \, x\right )} - \frac {x e^{\left (e^{5}\right )}}{\log \left (2 \, x\right )} + \frac {5 \, x}{\log \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/76*((-76*exp(exp(5))+9*x^2+380)*log(2*x)+76*exp(exp(5))-3*x^2-380)/log(2*x)^2,x, algorithm="giac")

[Out]

3/76*x^3/log(2*x) - x*e^(e^5)/log(2*x) + 5*x/log(2*x)

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maple [A]  time = 0.07, size = 22, normalized size = 0.92

method	result	size
risch	\(-\frac {x \left (-3 x^{2}+76 \,{\mathrm e}^{{\mathrm e}^{5}}-380\right )}{76 \ln \left (2 x \right )}\)	\(22\)
norman	\(\frac {\left (-{\mathrm e}^{{\mathrm e}^{5}}+5\right ) x +\frac {3 x^{3}}{76}}{\ln \left (2 x \right )}\)	\(23\)
derivativedivides	\(\frac {{\mathrm e}^{{\mathrm e}^{5}} \expIntegralEi \left (1, -\ln \left (2 x \right )\right )}{2}+\frac {3 x^{3}}{76 \ln \left (2 x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{5}} \left (-\frac {2 x}{\ln \left (2 x \right )}-\expIntegralEi \left (1, -\ln \left (2 x \right )\right )\right )}{2}+\frac {5 x}{\ln \left (2 x \right )}\)	\(60\)
default	\(\frac {{\mathrm e}^{{\mathrm e}^{5}} \expIntegralEi \left (1, -\ln \left (2 x \right )\right )}{2}+\frac {3 x^{3}}{76 \ln \left (2 x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{5}} \left (-\frac {2 x}{\ln \left (2 x \right )}-\expIntegralEi \left (1, -\ln \left (2 x \right )\right )\right )}{2}+\frac {5 x}{\ln \left (2 x \right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/76*((-76*exp(exp(5))+9*x^2+380)*ln(2*x)+76*exp(exp(5))-3*x^2-380)/ln(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/76*x*(-3*x^2+76*exp(exp(5))-380)/ln(2*x)

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maxima [C]  time = 0.80, size = 60, normalized size = 2.50 \begin {gather*} -\frac {1}{2} \, {\rm Ei}\left (\log \left (2 \, x\right )\right ) e^{\left (e^{5}\right )} + \frac {1}{2} \, e^{\left (e^{5}\right )} \Gamma \left (-1, -\log \left (2 \, x\right )\right ) + \frac {9}{608} \, {\rm Ei}\left (3 \, \log \left (2 \, x\right )\right ) + \frac {5}{2} \, {\rm Ei}\left (\log \left (2 \, x\right )\right ) - \frac {5}{2} \, \Gamma \left (-1, -\log \left (2 \, x\right )\right ) - \frac {9}{608} \, \Gamma \left (-1, -3 \, \log \left (2 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/76*((-76*exp(exp(5))+9*x^2+380)*log(2*x)+76*exp(exp(5))-3*x^2-380)/log(2*x)^2,x, algorithm="maxima
")

[Out]

-1/2*Ei(log(2*x))*e^(e^5) + 1/2*e^(e^5)*gamma(-1, -log(2*x)) + 9/608*Ei(3*log(2*x)) + 5/2*Ei(log(2*x)) - 5/2*g
amma(-1, -log(2*x)) - 9/608*gamma(-1, -3*log(2*x))

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mupad [B]  time = 1.77, size = 26, normalized size = 1.08 \begin {gather*} -\frac {x^2\,\left ({\mathrm {e}}^{{\mathrm {e}}^5}-5\right )-\frac {3\,x^4}{76}}{x\,\ln \left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(5)) + (log(2*x)*(9*x^2 - 76*exp(exp(5)) + 380))/76 - (3*x^2)/76 - 5)/log(2*x)^2,x)

[Out]

-(x^2*(exp(exp(5)) - 5) - (3*x^4)/76)/(x*log(2*x))

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sympy [A]  time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} \frac {3 x^{3} - 76 x e^{e^{5}} + 380 x}{76 \log {\left (2 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/76*((-76*exp(exp(5))+9*x**2+380)*ln(2*x)+76*exp(exp(5))-3*x**2-380)/ln(2*x)**2,x)

[Out]

(3*x**3 - 76*x*exp(exp(5)) + 380*x)/(76*log(2*x))

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