3.102.22 \(\int \frac {(10-4 x) (i \pi +\log (2))^2+e^{e^{3+x}+e^3 x} (-2 e^{3+x} x (i \pi +\log (2))^2+(-2-2 e^3 x) (i \pi +\log (2))^2)}{-125 x^3+e^{3 e^{3+x}+3 e^3 x} x^3+75 x^4-15 x^5+x^6+e^{2 e^{3+x}+2 e^3 x} (-15 x^3+3 x^4)+e^{e^{3+x}+e^3 x} (75 x^3-30 x^4+3 x^5)} \, dx\)

Optimal. Leaf size=30 \[ \frac {(i \pi +\log (2))^2}{x^2 \left (-5+e^{e^3 \left (e^x+x\right )}+x\right )^2} \]

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Rubi [A]  time = 1.25, antiderivative size = 35, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 3, integrand size = 169, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6688, 12, 6687} \begin {gather*} -\frac {(\pi -i \log (2))^2}{\left (-x-e^{e^3 \left (x+e^x\right )}+5\right )^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((10 - 4*x)*(I*Pi + Log[2])^2 + E^(E^(3 + x) + E^3*x)*(-2*E^(3 + x)*x*(I*Pi + Log[2])^2 + (-2 - 2*E^3*x)*(
I*Pi + Log[2])^2))/(-125*x^3 + E^(3*E^(3 + x) + 3*E^3*x)*x^3 + 75*x^4 - 15*x^5 + x^6 + E^(2*E^(3 + x) + 2*E^3*
x)*(-15*x^3 + 3*x^4) + E^(E^(3 + x) + E^3*x)*(75*x^3 - 30*x^4 + 3*x^5)),x]

[Out]

-((Pi - I*Log[2])^2/((5 - E^(E^3*(E^x + x)) - x)^2*x^2))

Rule 12

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

Rule 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (5-e^{e^3 \left (e^x+x\right )}-2 x-e^{3+e^{3+x}+e^3 x} x-e^{3+e^{3+x}+x+e^3 x} x\right ) (\pi -i \log (2))^2}{\left (5-e^{e^3 \left (e^x+x\right )}-x\right )^3 x^3} \, dx\\ &=\left (2 (\pi -i \log (2))^2\right ) \int \frac {5-e^{e^3 \left (e^x+x\right )}-2 x-e^{3+e^{3+x}+e^3 x} x-e^{3+e^{3+x}+x+e^3 x} x}{\left (5-e^{e^3 \left (e^x+x\right )}-x\right )^3 x^3} \, dx\\ &=-\frac {(\pi -i \log (2))^2}{\left (5-e^{e^3 \left (e^x+x\right )}-x\right )^2 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.85, size = 33, normalized size = 1.10 \begin {gather*} -\frac {(\pi -i \log (2))^2}{x^2 \left (-5+e^{e^{3+x}+e^3 x}+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((10 - 4*x)*(I*Pi + Log[2])^2 + E^(E^(3 + x) + E^3*x)*(-2*E^(3 + x)*x*(I*Pi + Log[2])^2 + (-2 - 2*E^
3*x)*(I*Pi + Log[2])^2))/(-125*x^3 + E^(3*E^(3 + x) + 3*E^3*x)*x^3 + 75*x^4 - 15*x^5 + x^6 + E^(2*E^(3 + x) +
2*E^3*x)*(-15*x^3 + 3*x^4) + E^(E^(3 + x) + E^3*x)*(75*x^3 - 30*x^4 + 3*x^5)),x]

[Out]

-((Pi - I*Log[2])^2/(x^2*(-5 + E^(E^(3 + x) + E^3*x) + x)^2))

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fricas [B]  time = 0.59, size = 71, normalized size = 2.37 \begin {gather*} -\frac {\pi ^{2} - 2 i \, \pi \log \relax (2) - \log \relax (2)^{2}}{x^{4} - 10 \, x^{3} + x^{2} e^{\left (2 \, x e^{3} + 2 \, e^{\left (x + 3\right )}\right )} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(3)*(log(2)+I*pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*pi)^2)*exp(exp(x)*exp(3)+x*exp(3))+(1
0-4*x)*(log(2)+I*pi)^2)/(x^3*exp(exp(x)*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5
-30*x^4+75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x, algorithm="fricas")

[Out]

-(pi^2 - 2*I*pi*log(2) - log(2)^2)/(x^4 - 10*x^3 + x^2*e^(2*x*e^3 + 2*e^(x + 3)) + 25*x^2 + 2*(x^3 - 5*x^2)*e^
(x*e^3 + e^(x + 3)))

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giac [B]  time = 0.49, size = 80, normalized size = 2.67 \begin {gather*} -\frac {\pi ^{2} - 2 i \, \pi \log \relax (2) - \log \relax (2)^{2}}{x^{4} + 2 \, x^{3} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )} - 10 \, x^{3} + x^{2} e^{\left (2 \, x e^{3} + 2 \, e^{\left (x + 3\right )}\right )} - 10 \, x^{2} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )} + 25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(3)*(log(2)+I*pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*pi)^2)*exp(exp(x)*exp(3)+x*exp(3))+(1
0-4*x)*(log(2)+I*pi)^2)/(x^3*exp(exp(x)*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5
-30*x^4+75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x, algorithm="giac")

[Out]

-(pi^2 - 2*I*pi*log(2) - log(2)^2)/(x^4 + 2*x^3*e^(x*e^3 + e^(x + 3)) - 10*x^3 + x^2*e^(2*x*e^3 + 2*e^(x + 3))
- 10*x^2*e^(x*e^3 + e^(x + 3)) + 25*x^2)

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maple [A]  time = 0.16, size = 37, normalized size = 1.23

method	result	size
risch	\(-\frac {-2 i \pi \ln \relax (2)+\pi ^{2}-\ln \relax (2)^{2}}{x^{2} \left ({\mathrm e}^{{\mathrm e}^{3+x}+x \,{\mathrm e}^{3}}+x -5\right )^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*exp(3)*(ln(2)+I*Pi)^2*exp(x)+(-2*x*exp(3)-2)*(ln(2)+I*Pi)^2)*exp(exp(x)*exp(3)+x*exp(3))+(10-4*x)*(
ln(2)+I*Pi)^2)/(x^3*exp(exp(x)*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5-30*x^4+7
5*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x,method=_RETURNVERBOSE)

[Out]

-(-2*I*Pi*ln(2)+Pi^2-ln(2)^2)/x^2/(exp(exp(3+x)+x*exp(3))+x-5)^2

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maxima [B]  time = 4.19, size = 71, normalized size = 2.37 \begin {gather*} -\frac {\pi ^{2} - 2 i \, \pi \log \relax (2) - \log \relax (2)^{2}}{x^{4} - 10 \, x^{3} + x^{2} e^{\left (2 \, x e^{3} + 2 \, e^{\left (x + 3\right )}\right )} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (x e^{3} + e^{\left (x + 3\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(3)*(log(2)+I*pi)^2*exp(x)+(-2*x*exp(3)-2)*(log(2)+I*pi)^2)*exp(exp(x)*exp(3)+x*exp(3))+(1
0-4*x)*(log(2)+I*pi)^2)/(x^3*exp(exp(x)*exp(3)+x*exp(3))^3+(3*x^4-15*x^3)*exp(exp(x)*exp(3)+x*exp(3))^2+(3*x^5
-30*x^4+75*x^3)*exp(exp(x)*exp(3)+x*exp(3))+x^6-15*x^5+75*x^4-125*x^3),x, algorithm="maxima")

[Out]

-(pi^2 - 2*I*pi*log(2) - log(2)^2)/(x^4 - 10*x^3 + x^2*e^(2*x*e^3 + 2*e^(x + 3)) + 25*x^2 + 2*(x^3 - 5*x^2)*e^
(x*e^3 + e^(x + 3)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^3+{\mathrm {e}}^3\,{\mathrm {e}}^x}\,\left ({\left (\ln \relax (2)+\Pi \,1{}\mathrm {i}\right )}^2\,\left (2\,x\,{\mathrm {e}}^3+2\right )+2\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\,{\left (\ln \relax (2)+\Pi \,1{}\mathrm {i}\right )}^2\right )+\left (4\,x-10\right )\,{\left (\ln \relax (2)+\Pi \,1{}\mathrm {i}\right )}^2}{{\mathrm {e}}^{x\,{\mathrm {e}}^3+{\mathrm {e}}^3\,{\mathrm {e}}^x}\,\left (3\,x^5-30\,x^4+75\,x^3\right )-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^3+2\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,\left (15\,x^3-3\,x^4\right )-125\,x^3+75\,x^4-15\,x^5+x^6+x^3\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^3+3\,{\mathrm {e}}^3\,{\mathrm {e}}^x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*exp(3) + exp(3)*exp(x))*((Pi*1i + log(2))^2*(2*x*exp(3) + 2) + 2*x*exp(3)*exp(x)*(Pi*1i + log(2))^
2) + (4*x - 10)*(Pi*1i + log(2))^2)/(exp(x*exp(3) + exp(3)*exp(x))*(75*x^3 - 30*x^4 + 3*x^5) - exp(2*x*exp(3)
+ 2*exp(3)*exp(x))*(15*x^3 - 3*x^4) - 125*x^3 + 75*x^4 - 15*x^5 + x^6 + x^3*exp(3*x*exp(3) + 3*exp(3)*exp(x)))
,x)

[Out]

int(-(exp(x*exp(3) + exp(3)*exp(x))*((Pi*1i + log(2))^2*(2*x*exp(3) + 2) + 2*x*exp(3)*exp(x)*(Pi*1i + log(2))^
2) + (4*x - 10)*(Pi*1i + log(2))^2)/(exp(x*exp(3) + exp(3)*exp(x))*(75*x^3 - 30*x^4 + 3*x^5) - exp(2*x*exp(3)
+ 2*exp(3)*exp(x))*(15*x^3 - 3*x^4) - 125*x^3 + 75*x^4 - 15*x^5 + x^6 + x^3*exp(3*x*exp(3) + 3*exp(3)*exp(x)))
, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(3)*(ln(2)+I*pi)**2*exp(x)+(-2*x*exp(3)-2)*(ln(2)+I*pi)**2)*exp(exp(x)*exp(3)+x*exp(3))+(1
0-4*x)*(ln(2)+I*pi)**2)/(x**3*exp(exp(x)*exp(3)+x*exp(3))**3+(3*x**4-15*x**3)*exp(exp(x)*exp(3)+x*exp(3))**2+(
3*x**5-30*x**4+75*x**3)*exp(exp(x)*exp(3)+x*exp(3))+x**6-15*x**5+75*x**4-125*x**3),x)

[Out]

Timed out

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