3.85.18 $\int \frac{e^{-\frac{3600-2280x+361x^2}{400{\log }^2(x)}}(3600-2280x+361x^2+(1140x-361x^2)\log (x)+200{\log }^2(x)+200{\log }^3(x))}{100{\log }^2(x)}dx$

Optimal. Leaf size=24 $$-5+2e^{-\frac{{(3-\frac{19x}{20})}^2}{{\log }^2(x)}}x\log (x)$$

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Rubi [B]  time = 0.22, antiderivative size = 81, normalized size of antiderivative = 3.38, number of steps used = 2, number of rules used = 2, integrand size = 62, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.032, Rules used = {12, 2288} $$\begin{array}{c}\frac{2e^{-\frac{361x^2-2280x+3600}{400{\log }^2(x)}}(361x^2+19(60x-19x^2)\log (x)-2280x+3600)}{(\frac{361x^2-2280x+3600}{x{\log }^3(x)}+\frac{19(60-19x)}{{\log }^2(x)}){\log }^2(x)}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(3600 - 2280*x + 361*x^2 + (1140*x - 361*x^2)*Log[x] + 200*Log[x]^2 + 200*Log[x]^3)/(100*E^((3600 - 2280*x
+ 361*x^2)/(400*Log[x]^2))*Log[x]^2),x]

[Out]

(2*(3600 - 2280*x + 361*x^2 + 19*(60*x - 19*x^2)*Log[x]))/(E^((3600 - 2280*x + 361*x^2)/(400*Log[x]^2))*((3600
- 2280*x + 361*x^2)/(x*Log[x]^3) + (19*(60 - 19*x))/Log[x]^2)*Log[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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{100}\int \frac{e^{-\frac{3600-2280x+361x^2}{400{\log }^2(x)}}(3600-2280x+361x^2+(1140x-361x^2)\log (x)+200{\log }^2(x)+200{\log }^3(x))}{{\log }^2(x)}dx\\ & =\frac{2e^{-\frac{3600-2280x+361x^2}{400{\log }^2(x)}}(3600-2280x+361x^2+19(60x-19x^2)\log (x))}{(\frac{3600-2280x+361x^2}{x{\log }^3(x)}+\frac{19(60-19x)}{{\log }^2(x)}){\log }^2(x)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.92 $$\begin{array}{c}2e^{-\frac{(60-19x{)}^2}{400{\log }^2(x)}}x\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(3600 - 2280*x + 361*x^2 + (1140*x - 361*x^2)*Log[x] + 200*Log[x]^2 + 200*Log[x]^3)/(100*E^((3600 -
2280*x + 361*x^2)/(400*Log[x]^2))*Log[x]^2),x]

[Out]

(2*x*Log[x])/E^((60 - 19*x)^2/(400*Log[x]^2))

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fricas [A]  time = 0.83, size = 22, normalized size = 0.92 $$\begin{array}{c}2x{e}^{(-\frac{361x^2-2280x+3600}{400\log (x{)}^2})}\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^2-2280*x+3600)/log(x)^2/exp(1/400*(3
61*x^2-2280*x+3600)/log(x)^2),x, algorithm="fricas")

[Out]

2*x*e^(-1/400*(361*x^2 - 2280*x + 3600)/log(x)^2)*log(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\mathit{undef}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^2-2280*x+3600)/log(x)^2/exp(1/400*(3
61*x^2-2280*x+3600)/log(x)^2),x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.03, size = 20, normalized size = 0.83

method	result	size
risch	$2\ln (x)x{\mathrm{e}}^{-\frac{{(19x-60)}^2}{400\ln (x{)}^2}}$	$20$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/100*(200*ln(x)^3+200*ln(x)^2+(-361*x^2+1140*x)*ln(x)+361*x^2-2280*x+3600)/ln(x)^2/exp(1/400*(361*x^2-228
0*x+3600)/ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)*x*exp(-1/400*(19*x-60)^2/ln(x)^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\frac{1}{100}\int \frac{(200\log (x{)}^3+361x^2-19(19x^2-60x)\log (x)+200\log (x{)}^2-2280x+3600)e^{(-\frac{361x^2-2280x+3600}{400\log (x{)}^2})}}{\log (x{)}^2}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^2-2280*x+3600)/log(x)^2/exp(1/400*(3
61*x^2-2280*x+3600)/log(x)^2),x, algorithm="maxima")

[Out]

1/100*integrate((200*log(x)^3 + 361*x^2 - 19*(19*x^2 - 60*x)*log(x) + 200*log(x)^2 - 2280*x + 3600)*e^(-1/400*
(361*x^2 - 2280*x + 3600)/log(x)^2)/log(x)^2, x)

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mupad [B]  time = 5.51, size = 22, normalized size = 0.92 $$\begin{array}{c}2x{\mathrm{e}}^{-\frac{361x^2-2280x+3600}{400{\ln (x)}^2}}\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-((361*x^2)/400 - (57*x)/10 + 9)/log(x)^2)*(2*log(x)^2 - (114*x)/5 + 2*log(x)^3 + (log(x)*(1140*x - 3
61*x^2))/100 + (361*x^2)/100 + 36))/log(x)^2,x)

[Out]

2*x*exp(-(361*x^2 - 2280*x + 3600)/(400*log(x)^2))*log(x)

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sympy [A]  time = 4.01, size = 26, normalized size = 1.08 $$\begin{array}{c}2x{e}^{-\frac{\frac{361x^2}{400}-\frac{57x}{10}+9}{\log {(x)}^2}}\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/100*(200*ln(x)**3+200*ln(x)**2+(-361*x**2+1140*x)*ln(x)+361*x**2-2280*x+3600)/ln(x)**2/exp(1/400*(
361*x**2-2280*x+3600)/ln(x)**2),x)

[Out]

2*x*exp(-(361*x**2/400 - 57*x/10 + 9)/log(x)**2)*log(x)

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