3.71.11 \(\int \frac {e^x x^2 (i \pi +\log (2))-2 e^x (i \pi +\log (2)) \log (x)+\log (4 e^{2/x}) (e^x x (i \pi +\log (2))+e^x x^2 (i \pi +\log (2)) \log (x))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ e^x (i \pi +\log (2)) \left (1+\log \left (4 e^{2/x}\right ) \log (x)\right ) \]

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Rubi [A]  time = 0.59, antiderivative size = 37, normalized size of antiderivative = 1.37, number of steps used = 29, number of rules used = 10, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {14, 2194, 2178, 2554, 12, 6485, 2177, 6483, 6475, 2557} \begin {gather*} e^x (\log (2)+i \pi ) \log \left (4 e^{2/x}\right ) \log (x)+e^x (\log (2)+i \pi ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*x^2*(I*Pi + Log[2]) - 2*E^x*(I*Pi + Log[2])*Log[x] + Log[4*E^(2/x)]*(E^x*x*(I*Pi + Log[2]) + E^x*x^2*
(I*Pi + Log[2])*Log[x]))/x^2,x]

[Out]

E^x*(I*Pi + Log[2]) + E^x*(I*Pi + Log[2])*Log[4*E^(2/x)]*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 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 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 6475

Int[ExpIntegralE[1, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)], x
] + (-Simp[EulerGamma*Log[x], x] - Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]

Rule 6483

Int[ExpIntegralEi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -(b*x)]), x
] - Int[ExpIntegralE[1, -(b*x)]/x, x] /; FreeQ[b, x]

Rule 6485

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*ExpInte
gralEi[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*E^(a + b*x))/(a + b*x), x], x] /
; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x (i \pi +\log (2))+\frac {e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right )}{x}-\frac {2 i e^x (\pi -i \log (2)) \log (x)}{x^2}+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)\right ) \, dx\\ &=-\left ((2 i (\pi -i \log (2))) \int \frac {e^x \log (x)}{x^2} \, dx\right )+(i \pi +\log (2)) \int e^x \, dx+(i \pi +\log (2)) \int \frac {e^x \log \left (4 e^{2/x}\right )}{x} \, dx+(i \pi +\log (2)) \int e^x \log \left (4 e^{2/x}\right ) \log (x) \, dx\\ &=e^x (i \pi +\log (2))+\text {Ei}(x) (i \pi +\log (2)) \log \left (4 e^{2/x}\right )+\frac {2 e^x (i \pi +\log (2)) \log (x)}{x}-2 \text {Ei}(x) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(-i \pi -\log (2)) \int -\frac {2 \text {Ei}(x)}{x^2} \, dx+(-i \pi -\log (2)) \int \frac {e^x \log \left (4 e^{2/x}\right )}{x} \, dx+(-i \pi -\log (2)) \int -\frac {2 e^x \log (x)}{x^2} \, dx+(2 i (\pi -i \log (2))) \int \frac {-e^x+x \text {Ei}(x)}{x^2} \, dx\\ &=e^x (i \pi +\log (2))+\frac {2 e^x (i \pi +\log (2)) \log (x)}{x}-2 \text {Ei}(x) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(2 i (\pi -i \log (2))) \int \left (-\frac {e^x}{x^2}+\frac {\text {Ei}(x)}{x}\right ) \, dx+(i \pi +\log (2)) \int -\frac {2 \text {Ei}(x)}{x^2} \, dx+(2 (i \pi +\log (2))) \int \frac {\text {Ei}(x)}{x^2} \, dx+(2 (i \pi +\log (2))) \int \frac {e^x \log (x)}{x^2} \, dx\\ &=e^x (i \pi +\log (2))-\frac {2 \text {Ei}(x) (i \pi +\log (2))}{x}+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)-(2 i (\pi -i \log (2))) \int \frac {e^x}{x^2} \, dx+(2 i (\pi -i \log (2))) \int \frac {\text {Ei}(x)}{x} \, dx+(2 (i \pi +\log (2))) \int \frac {e^x}{x^2} \, dx-(2 (i \pi +\log (2))) \int \frac {\text {Ei}(x)}{x^2} \, dx-(2 (i \pi +\log (2))) \int \frac {-e^x+x \text {Ei}(x)}{x^2} \, dx\\ &=e^x (i \pi +\log (2))+2 (E_1(-x)+\text {Ei}(x)) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)-(2 i (\pi -i \log (2))) \int \frac {e^x}{x} \, dx-(2 i (\pi -i \log (2))) \int \frac {E_1(-x)}{x} \, dx-(2 (i \pi +\log (2))) \int \frac {e^x}{x^2} \, dx+(2 (i \pi +\log (2))) \int \frac {e^x}{x} \, dx-(2 (i \pi +\log (2))) \int \left (-\frac {e^x}{x^2}+\frac {\text {Ei}(x)}{x}\right ) \, dx\\ &=e^x (i \pi +\log (2))+\frac {2 e^x (i \pi +\log (2))}{x}+2 x \, _3F_3(1,1,1;2,2,2;x) (i \pi +\log (2))+(i \pi +\log (2)) \log ^2(-x)+2 \gamma (i \pi +\log (2)) \log (x)+2 (E_1(-x)+\text {Ei}(x)) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(2 (i \pi +\log (2))) \int \frac {e^x}{x^2} \, dx-(2 (i \pi +\log (2))) \int \frac {e^x}{x} \, dx-(2 (i \pi +\log (2))) \int \frac {\text {Ei}(x)}{x} \, dx\\ &=e^x (i \pi +\log (2))-2 \text {Ei}(x) (i \pi +\log (2))+2 x \, _3F_3(1,1,1;2,2,2;x) (i \pi +\log (2))+(i \pi +\log (2)) \log ^2(-x)+2 \gamma (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(2 (i \pi +\log (2))) \int \frac {e^x}{x} \, dx+(2 (i \pi +\log (2))) \int \frac {E_1(-x)}{x} \, dx\\ &=e^x (i \pi +\log (2))+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 27, normalized size = 1.00 \begin {gather*} e^x (i \pi +\log (2)) \left (1+\log \left (4 e^{2/x}\right ) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x^2*(I*Pi + Log[2]) - 2*E^x*(I*Pi + Log[2])*Log[x] + Log[4*E^(2/x)]*(E^x*x*(I*Pi + Log[2]) + E^
x*x^2*(I*Pi + Log[2])*Log[x]))/x^2,x]

[Out]

E^x*(I*Pi + Log[2])*(1 + Log[4*E^(2/x)]*Log[x])

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fricas [A]  time = 1.08, size = 45, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (-i \, \pi - x \log \relax (2)^{2} + {\left (-i \, \pi x - 1\right )} \log \relax (2)\right )} e^{x} \log \relax (x) - {\left (i \, \pi x + x \log \relax (2)\right )} e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(-I*pi - x*log(2)^2 + (-I*pi*x - 1)*log(2))*e^x*log(x) - (I*pi*x + x*log(2))*e^x)/x

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giac [B]  time = 0.21, size = 53, normalized size = 1.96 \begin {gather*} \frac {2 i \, \pi x e^{x} \log \relax (2) \log \relax (x) + 2 \, x e^{x} \log \relax (2)^{2} \log \relax (x) + i \, \pi x e^{x} + x e^{x} \log \relax (2) + 2 i \, \pi e^{x} \log \relax (x) + 2 \, e^{x} \log \relax (2) \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.10, size = 52, normalized size = 1.93

method	result	size
risch	\(\left (\ln \relax (2)+i \pi \right ) {\mathrm e}^{x} \ln \relax (x ) \ln \left ({\mathrm e}^{\frac {2}{x}}\right )+2 i \pi \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )+2 \ln \relax (2)^{2} {\mathrm e}^{x} \ln \relax (x )+i \pi \,{\mathrm e}^{x}+{\mathrm e}^{x} \ln \relax (2)\)	\(52\)
default	\(\frac {x \left (\ln \relax (2)+i \pi \right ) {\mathrm e}^{x}+\left (2 \ln \relax (2)+2 i \pi \right ) {\mathrm e}^{x} \ln \relax (x )+\left (\ln \relax (2) \left (\ln \left (4 \,{\mathrm e}^{\frac {2}{x}}\right )-\frac {2}{x}\right )+i \pi \left (\ln \left (4 \,{\mathrm e}^{\frac {2}{x}}\right )-\frac {2}{x}\right )\right ) x \,{\mathrm e}^{x} \ln \relax (x )}{x}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2*(ln(2)+I*Pi)*exp(x)*ln(x)+x*(ln(2)+I*Pi)*exp(x))*ln(4*exp(2/x))-2*(ln(2)+I*Pi)*exp(x)*ln(x)+x^2*(ln(
2)+I*Pi)*exp(x))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.53, size = 42, normalized size = 1.56 \begin {gather*} i \, \pi e^{x} + e^{x} \log \relax (2) - \frac {2 \, {\left (-i \, \pi + {\left (-i \, \pi \log \relax (2) - \log \relax (2)^{2}\right )} x - \log \relax (2)\right )} e^{x} \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.50, size = 26, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {e}}^x\,\left (\ln \relax (2)+\Pi \,1{}\mathrm {i}\right )\,\left (x+2\,\ln \relax (x)+2\,x\,\ln \relax (2)\,\ln \relax (x)\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4*exp(2/x))*(x*exp(x)*(Pi*1i + log(2)) + x^2*exp(x)*log(x)*(Pi*1i + log(2))) + x^2*exp(x)*(Pi*1i + lo
g(2)) - 2*exp(x)*log(x)*(Pi*1i + log(2)))/x^2,x)

[Out]

(exp(x)*(Pi*1i + log(2))*(x + 2*log(x) + 2*x*log(2)*log(x)))/x

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sympy [B]  time = 0.64, size = 60, normalized size = 2.22 \begin {gather*} - \frac {\left (- 2 x \log {\relax (2 )}^{2} \log {\relax (x )} - 2 i \pi x \log {\relax (2 )} \log {\relax (x )} - x \log {\relax (2 )} - i \pi x - 2 \log {\relax (2 )} \log {\relax (x )} - 2 i \pi \log {\relax (x )}\right ) e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2*(ln(2)+I*pi)*exp(x)*ln(x)+x*(ln(2)+I*pi)*exp(x))*ln(4*exp(2/x))-2*(ln(2)+I*pi)*exp(x)*ln(x)+x
**2*(ln(2)+I*pi)*exp(x))/x**2,x)

[Out]

-(-2*x*log(2)**2*log(x) - 2*I*pi*x*log(2)*log(x) - x*log(2) - I*pi*x - 2*log(2)*log(x) - 2*I*pi*log(x))*exp(x)
/x

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