3.52.89 \(\int e^{1-e^{18 x^7 \log ^8(e^5 x)}+2 e^{9 x^7 \log ^8(e^5 x)} x-x^2} (-2 x+e^{18 x^7 \log ^8(e^5 x)} (-144 x^6 \log ^7(e^5 x)-126 x^6 \log ^8(e^5 x))+e^{9 x^7 \log ^8(e^5 x)} (2+144 x^7 \log ^7(e^5 x)+126 x^7 \log ^8(e^5 x))) \, dx\)

Optimal. Leaf size=27 \[ e^{1-\left (e^{9 x^7 \log ^8\left (e^5 x\right )}-x\right )^2} \]

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Rubi [A]  time = 8.36, antiderivative size = 44, normalized size of antiderivative = 1.63, number of steps used = 1, number of rules used = 1, integrand size = 136, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {6706} \begin {gather*} \exp \left (2 x e^{9 x^7 \log ^8\left (e^5 x\right )}-e^{18 x^7 \log ^8\left (e^5 x\right )}-x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(1 - E^(18*x^7*Log[E^5*x]^8) + 2*E^(9*x^7*Log[E^5*x]^8)*x - x^2)*(-2*x + E^(18*x^7*Log[E^5*x]^8)*(-144*x
^6*Log[E^5*x]^7 - 126*x^6*Log[E^5*x]^8) + E^(9*x^7*Log[E^5*x]^8)*(2 + 144*x^7*Log[E^5*x]^7 + 126*x^7*Log[E^5*x
]^8)),x]

[Out]

E^(1 - E^(18*x^7*Log[E^5*x]^8) + 2*E^(9*x^7*Log[E^5*x]^8)*x - x^2)

Rule 6706

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 {gather*} \begin {aligned} \text {integral} &=\exp \left (1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.52, size = 83, normalized size = 3.07 \begin {gather*} e^{1+2 e^{9 x^7 (5+\log (x))^8} x-x^2-e^{18 x^7 \left (390625+437500 \log ^2(x)+175000 \log ^3(x)+43750 \log ^4(x)+7000 \log ^5(x)+700 \log ^6(x)+40 \log ^7(x)+\log ^8(x)\right )} x^{11250000 x^7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(1 - E^(18*x^7*Log[E^5*x]^8) + 2*E^(9*x^7*Log[E^5*x]^8)*x - x^2)*(-2*x + E^(18*x^7*Log[E^5*x]^8)*(
-144*x^6*Log[E^5*x]^7 - 126*x^6*Log[E^5*x]^8) + E^(9*x^7*Log[E^5*x]^8)*(2 + 144*x^7*Log[E^5*x]^7 + 126*x^7*Log
[E^5*x]^8)),x]

[Out]

E^(1 + 2*E^(9*x^7*(5 + Log[x])^8)*x - x^2 - E^(18*x^7*(390625 + 437500*Log[x]^2 + 175000*Log[x]^3 + 43750*Log[
x]^4 + 7000*Log[x]^5 + 700*Log[x]^6 + 40*Log[x]^7 + Log[x]^8))*x^(11250000*x^7))

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fricas [A]  time = 0.47, size = 39, normalized size = 1.44 \begin {gather*} e^{\left (-x^{2} + 2 \, x e^{\left (9 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} - e^{\left (18 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*log(x*exp(5))^8)^2+(126*x^7*log(x*exp(
5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9*x^7*log(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x
^7*log(x*exp(5))^8)-x^2+1),x, algorithm="fricas")

[Out]

e^(-x^2 + 2*x*e^(9*x^7*log(x*e^5)^8) - e^(18*x^7*log(x*e^5)^8) + 1)

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giac [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(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*log(x*exp(5))^8)^2+(126*x^7*log(x*exp(
5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9*x^7*log(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x
^7*log(x*exp(5))^8)-x^2+1),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.47, size = 40, normalized size = 1.48

method	result	size
risch	\({\mathrm e}^{-{\mathrm e}^{18 x^{7} \ln \left (x \,{\mathrm e}^{5}\right )^{8}}+2 x \,{\mathrm e}^{9 x^{7} \ln \left (x \,{\mathrm e}^{5}\right )^{8}}-x^{2}+1}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-126*x^6*ln(x*exp(5))^8-144*x^6*ln(x*exp(5))^7)*exp(9*x^7*ln(x*exp(5))^8)^2+(126*x^7*ln(x*exp(5))^8+144*
x^7*ln(x*exp(5))^7+2)*exp(9*x^7*ln(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*ln(x*exp(5))^8)^2+2*x*exp(9*x^7*ln(x*exp(5
))^8)-x^2+1),x,method=_RETURNVERBOSE)

[Out]

exp(-exp(18*x^7*ln(x*exp(5))^8)+2*x*exp(9*x^7*ln(x*exp(5))^8)-x^2+1)

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maxima [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(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*log(x*exp(5))^8)^2+(126*x^7*log(x*exp(
5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9*x^7*log(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x
^7*log(x*exp(5))^8)-x^2+1),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 3.52, size = 182, normalized size = 6.74 \begin {gather*} {\mathrm {e}}^{2\,x\,x^{5625000\,x^7}\,{\mathrm {e}}^{3515625\,x^7}\,{\mathrm {e}}^{9\,x^7\,{\ln \relax (x)}^8}\,{\mathrm {e}}^{360\,x^7\,{\ln \relax (x)}^7}\,{\mathrm {e}}^{6300\,x^7\,{\ln \relax (x)}^6}\,{\mathrm {e}}^{63000\,x^7\,{\ln \relax (x)}^5}\,{\mathrm {e}}^{393750\,x^7\,{\ln \relax (x)}^4}\,{\mathrm {e}}^{1575000\,x^7\,{\ln \relax (x)}^3}\,{\mathrm {e}}^{3937500\,x^7\,{\ln \relax (x)}^2}}\,\mathrm {e}\,{\mathrm {e}}^{-x^{11250000\,x^7}\,{\mathrm {e}}^{7031250\,x^7}\,{\mathrm {e}}^{18\,x^7\,{\ln \relax (x)}^8}\,{\mathrm {e}}^{720\,x^7\,{\ln \relax (x)}^7}\,{\mathrm {e}}^{12600\,x^7\,{\ln \relax (x)}^6}\,{\mathrm {e}}^{126000\,x^7\,{\ln \relax (x)}^5}\,{\mathrm {e}}^{787500\,x^7\,{\ln \relax (x)}^4}\,{\mathrm {e}}^{3150000\,x^7\,{\ln \relax (x)}^3}\,{\mathrm {e}}^{7875000\,x^7\,{\ln \relax (x)}^2}}\,{\mathrm {e}}^{-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(2*x*exp(9*x^7*log(x*exp(5))^8) - exp(18*x^7*log(x*exp(5))^8) - x^2 + 1)*(2*x - exp(9*x^7*log(x*exp(5)
)^8)*(144*x^7*log(x*exp(5))^7 + 126*x^7*log(x*exp(5))^8 + 2) + exp(18*x^7*log(x*exp(5))^8)*(144*x^6*log(x*exp(
5))^7 + 126*x^6*log(x*exp(5))^8)),x)

[Out]

exp(2*x*x^(5625000*x^7)*exp(3515625*x^7)*exp(9*x^7*log(x)^8)*exp(360*x^7*log(x)^7)*exp(6300*x^7*log(x)^6)*exp(
63000*x^7*log(x)^5)*exp(393750*x^7*log(x)^4)*exp(1575000*x^7*log(x)^3)*exp(3937500*x^7*log(x)^2))*exp(1)*exp(-
x^(11250000*x^7)*exp(7031250*x^7)*exp(18*x^7*log(x)^8)*exp(720*x^7*log(x)^7)*exp(12600*x^7*log(x)^6)*exp(12600
0*x^7*log(x)^5)*exp(787500*x^7*log(x)^4)*exp(3150000*x^7*log(x)^3)*exp(7875000*x^7*log(x)^2))*exp(-x^2)

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sympy [A]  time = 1.58, size = 39, normalized size = 1.44 \begin {gather*} e^{- x^{2} + 2 x e^{9 x^{7} \log {\left (x e^{5} \right )}^{8}} - e^{18 x^{7} \log {\left (x e^{5} \right )}^{8}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-126*x**6*ln(x*exp(5))**8-144*x**6*ln(x*exp(5))**7)*exp(9*x**7*ln(x*exp(5))**8)**2+(126*x**7*ln(x*
exp(5))**8+144*x**7*ln(x*exp(5))**7+2)*exp(9*x**7*ln(x*exp(5))**8)-2*x)*exp(-exp(9*x**7*ln(x*exp(5))**8)**2+2*
x*exp(9*x**7*ln(x*exp(5))**8)-x**2+1),x)

[Out]

exp(-x**2 + 2*x*exp(9*x**7*log(x*exp(5))**8) - exp(18*x**7*log(x*exp(5))**8) + 1)

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