\(\int e^{x^2+(-6 e^2 x+24 e x^2-24 x^3) \log (5)+(9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4) \log ^2(5)} (2 x+(-6 e^2+48 e x-72 x^2) \log (5)+(-72 e^3+432 e^2 x-864 e x^2+576 x^3) \log ^2(5)) \, dx\) [9018]

Optimal result

Integrand size = 114, antiderivative size = 21 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=e^{\left (-x+3 (-e+2 x)^2 \log (5)\right )^2} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(21)=42\).

Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {6838} \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=5^{-24 x^3+24 e x^2-6 e^2 x} \exp \left (x^2+9 \left (16 x^4-32 e x^3+24 e^2 x^2-8 e^3 x+e^4\right ) \log ^2(5)\right ) \]

[In]

Int[E^(x^2 + (-6*E^2*x + 24*E*x^2 - 24*x^3)*Log[5] + (9*E^4 - 72*E^3*x + 216*E^2*x^2 - 288*E*x^3 + 144*x^4)*Lo
g[5]^2)*(2*x + (-6*E^2 + 48*E*x - 72*x^2)*Log[5] + (-72*E^3 + 432*E^2*x - 864*E*x^2 + 576*x^3)*Log[5]^2),x]

[Out]

5^(-6*E^2*x + 24*E*x^2 - 24*x^3)*E^(x^2 + 9*(E^4 - 8*E^3*x + 24*E^2*x^2 - 32*E*x^3 + 16*x^4)*Log[5]^2)

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = 5^{-6 e^2 x+24 e x^2-24 x^3} \exp \left (x^2+9 \left (e^4-8 e^3 x+24 e^2 x^2-32 e x^3+16 x^4\right ) \log ^2(5)\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(21)=42\).

Time = 2.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=5^{-6 (e-2 x)^2 x} e^{\frac {1}{4} \left (e^2-2 e (e-2 x)+(e-2 x)^2+36 (e-2 x)^4 \log ^2(5)\right )} \]

[In]

Integrate[E^(x^2 + (-6*E^2*x + 24*E*x^2 - 24*x^3)*Log[5] + (9*E^4 - 72*E^3*x + 216*E^2*x^2 - 288*E*x^3 + 144*x
^4)*Log[5]^2)*(2*x + (-6*E^2 + 48*E*x - 72*x^2)*Log[5] + (-72*E^3 + 432*E^2*x - 864*E*x^2 + 576*x^3)*Log[5]^2)
,x]

[Out]

E^((E^2 - 2*E*(E - 2*x) + (E - 2*x)^2 + 36*(E - 2*x)^4*Log[5]^2)/4)/5^(6*(E - 2*x)^2*x)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(21)=42\).

Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.29

method	result	size
derivativedivides	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \left (5\right )^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \left (5\right )+x^{2}}\)	\(69\)
default	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \left (5\right )^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \left (5\right )+x^{2}}\)	\(69\)
norman	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \left (5\right )^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \left (5\right )+x^{2}}\)	\(69\)
parallelrisch	\({\mathrm e}^{\left (9 \,{\mathrm e}^{4}-72 x \,{\mathrm e}^{3}+216 x^{2} {\mathrm e}^{2}-288 x^{3} {\mathrm e}+144 x^{4}\right ) \ln \left (5\right )^{2}+\left (-6 \,{\mathrm e}^{2} x +24 x^{2} {\mathrm e}-24 x^{3}\right ) \ln \left (5\right )+x^{2}}\)	\(69\)
risch	\(15625^{-\left (-4 x \,{\mathrm e}+4 x^{2}+{\mathrm e}^{2}\right ) x} {\mathrm e}^{-288 \ln \left (5\right )^{2} {\mathrm e} x^{3}+144 x^{4} \ln \left (5\right )^{2}+216 \ln \left (5\right )^{2} {\mathrm e}^{2} x^{2}-72 \ln \left (5\right )^{2} {\mathrm e}^{3} x +9 \ln \left (5\right )^{2} {\mathrm e}^{4}+x^{2}}\)	\(73\)
gosper	\({\mathrm e}^{-288 \ln \left (5\right )^{2} {\mathrm e} x^{3}+144 x^{4} \ln \left (5\right )^{2}+216 \ln \left (5\right )^{2} {\mathrm e}^{2} x^{2}-72 \ln \left (5\right )^{2} {\mathrm e}^{3} x +24 x^{2} {\mathrm e} \ln \left (5\right )-24 x^{3} \ln \left (5\right )+9 \ln \left (5\right )^{2} {\mathrm e}^{4}-6 x \,{\mathrm e}^{2} \ln \left (5\right )+x^{2}}\)	\(85\)

[In]

int(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*ln(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*ln(5)+2*x)*
exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*ln(5)^2+(-6*x*exp(1)^2+24*x^2*exp(1)-24
*x^3)*ln(5)+x^2),x,method=_RETURNVERBOSE)

[Out]

exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*ln(5)^2+(-6*x*exp(1)^2+24*x^2*exp(1)-24
*x^3)*ln(5)+x^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (21) = 42\).

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=e^{\left (9 \, {\left (16 \, x^{4} - 32 \, x^{3} e + 24 \, x^{2} e^{2} - 8 \, x e^{3} + e^{4}\right )} \log \left (5\right )^{2} + x^{2} - 6 \, {\left (4 \, x^{3} - 4 \, x^{2} e + x e^{2}\right )} \log \left (5\right )\right )} \]

[In]

integrate(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*log(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*log(
5)+2*x)*exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*log(5)^2+(-6*x*exp(1)^2+24*x^2*
exp(1)-24*x^3)*log(5)+x^2),x, algorithm="fricas")

[Out]

e^(9*(16*x^4 - 32*x^3*e + 24*x^2*e^2 - 8*x*e^3 + e^4)*log(5)^2 + x^2 - 6*(4*x^3 - 4*x^2*e + x*e^2)*log(5))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (17) = 34\).

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.14 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=e^{x^{2} + \left (- 24 x^{3} + 24 e x^{2} - 6 x e^{2}\right ) \log {\left (5 \right )} + \left (144 x^{4} - 288 e x^{3} + 216 x^{2} e^{2} - 72 x e^{3} + 9 e^{4}\right ) \log {\left (5 \right )}^{2}} \]

[In]

integrate(((-72*exp(1)**3+432*x*exp(1)**2-864*x**2*exp(1)+576*x**3)*ln(5)**2+(-6*exp(1)**2+48*x*exp(1)-72*x**2
)*ln(5)+2*x)*exp((9*exp(1)**4-72*x*exp(1)**3+216*x**2*exp(1)**2-288*x**3*exp(1)+144*x**4)*ln(5)**2+(-6*x*exp(1
)**2+24*x**2*exp(1)-24*x**3)*ln(5)+x**2),x)

[Out]

exp(x**2 + (-24*x**3 + 24*E*x**2 - 6*x*exp(2))*log(5) + (144*x**4 - 288*E*x**3 + 216*x**2*exp(2) - 72*x*exp(3)
+ 9*exp(4))*log(5)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (21) = 42\).

Time = 0.49 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.62 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=e^{\left (144 \, x^{4} \log \left (5\right )^{2} - 288 \, x^{3} e \log \left (5\right )^{2} + 216 \, x^{2} e^{2} \log \left (5\right )^{2} - 24 \, x^{3} \log \left (5\right ) + 24 \, x^{2} e \log \left (5\right ) - 72 \, x e^{3} \log \left (5\right )^{2} - 6 \, x e^{2} \log \left (5\right ) + 9 \, e^{4} \log \left (5\right )^{2} + x^{2}\right )} \]

[In]

integrate(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*log(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*log(
5)+2*x)*exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*log(5)^2+(-6*x*exp(1)^2+24*x^2*
exp(1)-24*x^3)*log(5)+x^2),x, algorithm="maxima")

[Out]

e^(144*x^4*log(5)^2 - 288*x^3*e*log(5)^2 + 216*x^2*e^2*log(5)^2 - 24*x^3*log(5) + 24*x^2*e*log(5) - 72*x*e^3*l
og(5)^2 - 6*x*e^2*log(5) + 9*e^4*log(5)^2 + x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (21) = 42\).

Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.62 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=e^{\left (144 \, x^{4} \log \left (5\right )^{2} - 288 \, x^{3} e \log \left (5\right )^{2} + 216 \, x^{2} e^{2} \log \left (5\right )^{2} - 24 \, x^{3} \log \left (5\right ) + 24 \, x^{2} e \log \left (5\right ) - 72 \, x e^{3} \log \left (5\right )^{2} - 6 \, x e^{2} \log \left (5\right ) + 9 \, e^{4} \log \left (5\right )^{2} + x^{2}\right )} \]

[In]

integrate(((-72*exp(1)^3+432*x*exp(1)^2-864*x^2*exp(1)+576*x^3)*log(5)^2+(-6*exp(1)^2+48*x*exp(1)-72*x^2)*log(
5)+2*x)*exp((9*exp(1)^4-72*x*exp(1)^3+216*x^2*exp(1)^2-288*x^3*exp(1)+144*x^4)*log(5)^2+(-6*x*exp(1)^2+24*x^2*
exp(1)-24*x^3)*log(5)+x^2),x, algorithm="giac")

[Out]

e^(144*x^4*log(5)^2 - 288*x^3*e*log(5)^2 + 216*x^2*e^2*log(5)^2 - 24*x^3*log(5) + 24*x^2*e*log(5) - 72*x*e^3*l
og(5)^2 - 6*x*e^2*log(5) + 9*e^4*log(5)^2 + x^2)

Mupad [B] (verification not implemented)

Time = 13.55 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.05 \[ \int e^{x^2+\left (-6 e^2 x+24 e x^2-24 x^3\right ) \log (5)+\left (9 e^4-72 e^3 x+216 e^2 x^2-288 e x^3+144 x^4\right ) \log ^2(5)} \left (2 x+\left (-6 e^2+48 e x-72 x^2\right ) \log (5)+\left (-72 e^3+432 e^2 x-864 e x^2+576 x^3\right ) \log ^2(5)\right ) \, dx=\frac {5^{24\,x^2\,\mathrm {e}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{144\,x^4\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{-72\,x\,{\mathrm {e}}^3\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{216\,x^2\,{\mathrm {e}}^2\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{-288\,x^3\,\mathrm {e}\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{9\,{\mathrm {e}}^4\,{\ln \left (5\right )}^2}}{5^{24\,x^3}\,5^{6\,x\,{\mathrm {e}}^2}} \]

[In]

int(-exp(x^2 - log(5)*(6*x*exp(2) - 24*x^2*exp(1) + 24*x^3) + log(5)^2*(9*exp(4) - 72*x*exp(3) + 216*x^2*exp(2
) - 288*x^3*exp(1) + 144*x^4))*(log(5)^2*(72*exp(3) - 432*x*exp(2) + 864*x^2*exp(1) - 576*x^3) - 2*x + log(5)*
(6*exp(2) - 48*x*exp(1) + 72*x^2)),x)

[Out]

(5^(24*x^2*exp(1))*exp(x^2)*exp(144*x^4*log(5)^2)*exp(-72*x*exp(3)*log(5)^2)*exp(216*x^2*exp(2)*log(5)^2)*exp(
-288*x^3*exp(1)*log(5)^2)*exp(9*exp(4)*log(5)^2))/(5^(24*x^3)*5^(6*x*exp(2)))