3.31.50 \(\int \frac {e^{\frac {-3 x^2+(15 x+9 x^2) \log (\log (5-\log (x)))}{(5+3 x) \log (\log (5-\log (x)))}} (75 x^2+45 x^3+(750 x^2+225 x^3+(-150 x^2-45 x^3) \log (x)) \log (5-\log (x)) \log (\log (5-\log (x)))+(-625-2625 x-2475 x^2-675 x^3+(125+525 x+495 x^2+135 x^3) \log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x))))}{(-125-150 x-45 x^2+(25+30 x+9 x^2) \log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx\)

Optimal. Leaf size=31 \[ 5 e^{3 \left (x-\frac {x^2}{(5+3 x) \log (\log (5-\log (x)))}\right )} x \]

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Rubi [F]  time = 9.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {-3 x^2+\left (15 x+9 x^2\right ) \log (\log (5-\log (x)))}{(5+3 x) \log (\log (5-\log (x)))}\right ) \left (75 x^2+45 x^3+\left (750 x^2+225 x^3+\left (-150 x^2-45 x^3\right ) \log (x)\right ) \log (5-\log (x)) \log (\log (5-\log (x)))+\left (-625-2625 x-2475 x^2-675 x^3+\left (125+525 x+495 x^2+135 x^3\right ) \log (x)\right ) \log (5-\log (x)) \log ^2(\log (5-\log (x)))\right )}{\left (-125-150 x-45 x^2+\left (25+30 x+9 x^2\right ) \log (x)\right ) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-3*x^2 + (15*x + 9*x^2)*Log[Log[5 - Log[x]]])/((5 + 3*x)*Log[Log[5 - Log[x]]]))*(75*x^2 + 45*x^3 + (7
50*x^2 + 225*x^3 + (-150*x^2 - 45*x^3)*Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]] + (-625 - 2625*x - 2475*x^
2 - 675*x^3 + (125 + 525*x + 495*x^2 + 135*x^3)*Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]]^2))/((-125 - 150*
x - 45*x^2 + (25 + 30*x + 9*x^2)*Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]]^2),x]

[Out]

5*Defer[Int][E^(3*x*(1 - x/((5 + 3*x)*Log[Log[5 - Log[x]]]))), x] + 15*Defer[Int][E^(3*x*(1 - x/((5 + 3*x)*Log
[Log[5 - Log[x]]])))*x, x] - (25*Defer[Int][E^(3*x*(1 - x/((5 + 3*x)*Log[Log[5 - Log[x]]])))/((-5 + Log[x])*Lo
g[5 - Log[x]]*Log[Log[5 - Log[x]]]^2), x])/3 + 5*Defer[Int][(E^(3*x*(1 - x/((5 + 3*x)*Log[Log[5 - Log[x]]])))*
x)/((-5 + Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]]^2), x] + (125*Defer[Int][E^(3*x*(1 - x/((5 + 3*x)*Log[L
og[5 - Log[x]]])))/((5 + 3*x)*(-5 + Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]]^2), x])/3 - 5*Defer[Int][(E^(
3*x*(1 - x/((5 + 3*x)*Log[Log[5 - Log[x]]])))*x)/Log[Log[5 - Log[x]]], x] - (625*Defer[Int][E^(3*x*(1 - x/((5
+ 3*x)*Log[Log[5 - Log[x]]])))/((5 + 3*x)^2*Log[Log[5 - Log[x]]]), x])/3 + (125*Defer[Int][E^(3*x*(1 - x/((5 +
3*x)*Log[Log[5 - Log[x]]])))/((5 + 3*x)*Log[Log[5 - Log[x]]]), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) \left (-3 x^2 (5+3 x)-(-5+\log (x)) \log (5-\log (x)) \log (\log (5-\log (x))) \left (-3 x^2 (10+3 x)+(1+3 x) (5+3 x)^2 \log (\log (5-\log (x)))\right )\right )}{(5+3 x)^2 (5-\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx\\ &=5 \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) \left (-3 x^2 (5+3 x)-(-5+\log (x)) \log (5-\log (x)) \log (\log (5-\log (x))) \left (-3 x^2 (10+3 x)+(1+3 x) (5+3 x)^2 \log (\log (5-\log (x)))\right )\right )}{(5+3 x)^2 (5-\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx\\ &=5 \int \left (\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )+3 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x+\frac {3 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x^2}{(5+3 x) (-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))}-\frac {3 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x^2 (10+3 x)}{(5+3 x)^2 \log (\log (5-\log (x)))}\right ) \, dx\\ &=5 \int \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) \, dx+15 \int \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x \, dx+15 \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x^2}{(5+3 x) (-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx-15 \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x^2 (10+3 x)}{(5+3 x)^2 \log (\log (5-\log (x)))} \, dx\\ &=5 \int \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) \, dx+15 \int \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x \, dx+15 \int \left (-\frac {5 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{9 (-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))}+\frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x}{3 (-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))}+\frac {25 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{9 (5+3 x) (-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))}\right ) \, dx-15 \int \left (\frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x}{3 \log (\log (5-\log (x)))}+\frac {125 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{9 (5+3 x)^2 \log (\log (5-\log (x)))}-\frac {25 \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{9 (5+3 x) \log (\log (5-\log (x)))}\right ) \, dx\\ &=5 \int \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) \, dx+5 \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x}{(-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx-5 \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x}{\log (\log (5-\log (x)))} \, dx-\frac {25}{3} \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{(-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx+15 \int \exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right ) x \, dx+\frac {125}{3} \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{(5+3 x) (-5+\log (x)) \log (5-\log (x)) \log ^2(\log (5-\log (x)))} \, dx+\frac {125}{3} \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{(5+3 x) \log (\log (5-\log (x)))} \, dx-\frac {625}{3} \int \frac {\exp \left (3 x \left (1-\frac {x}{(5+3 x) \log (\log (5-\log (x)))}\right )\right )}{(5+3 x)^2 \log (\log (5-\log (x)))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 31, normalized size = 1.00 \begin {gather*} 5 e^{3 x-\frac {3 x^2}{(5+3 x) \log (\log (5-\log (x)))}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-3*x^2 + (15*x + 9*x^2)*Log[Log[5 - Log[x]]])/((5 + 3*x)*Log[Log[5 - Log[x]]]))*(75*x^2 + 45*x^
3 + (750*x^2 + 225*x^3 + (-150*x^2 - 45*x^3)*Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]] + (-625 - 2625*x - 2
475*x^2 - 675*x^3 + (125 + 525*x + 495*x^2 + 135*x^3)*Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]]^2))/((-125
- 150*x - 45*x^2 + (25 + 30*x + 9*x^2)*Log[x])*Log[5 - Log[x]]*Log[Log[5 - Log[x]]]^2),x]

[Out]

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

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fricas [A]  time = 0.55, size = 46, normalized size = 1.48 \begin {gather*} 5 \, x e^{\left (-\frac {3 \, {\left (x^{2} - {\left (3 \, x^{2} + 5 \, x\right )} \log \left (\log \left (-\log \relax (x) + 5\right )\right )\right )}}{{\left (3 \, x + 5\right )} \log \left (\log \left (-\log \relax (x) + 5\right )\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((135*x^3+495*x^2+525*x+125)*log(x)-675*x^3-2475*x^2-2625*x-625)*log(5-log(x))*log(log(5-log(x)))^2
+((-45*x^3-150*x^2)*log(x)+225*x^3+750*x^2)*log(5-log(x))*log(log(5-log(x)))+45*x^3+75*x^2)*exp(((9*x^2+15*x)*
log(log(5-log(x)))-3*x^2)/(3*x+5)/log(log(5-log(x))))/((9*x^2+30*x+25)*log(x)-45*x^2-150*x-125)/log(5-log(x))/
log(log(5-log(x)))^2,x, algorithm="fricas")

[Out]

5*x*e^(-3*(x^2 - (3*x^2 + 5*x)*log(log(-log(x) + 5)))/((3*x + 5)*log(log(-log(x) + 5))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((135*x^3+495*x^2+525*x+125)*log(x)-675*x^3-2475*x^2-2625*x-625)*log(5-log(x))*log(log(5-log(x)))^2
+((-45*x^3-150*x^2)*log(x)+225*x^3+750*x^2)*log(5-log(x))*log(log(5-log(x)))+45*x^3+75*x^2)*exp(((9*x^2+15*x)*
log(log(5-log(x)))-3*x^2)/(3*x+5)/log(log(5-log(x))))/((9*x^2+30*x+25)*log(x)-45*x^2-150*x-125)/log(5-log(x))/
log(log(5-log(x)))^2,x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.05, size = 50, normalized size = 1.61

method	result	size
risch	\(5 x \,{\mathrm e}^{\frac {3 x \left (3 \ln \left (\ln \left (5-\ln \relax (x )\right )\right ) x +5 \ln \left (\ln \left (5-\ln \relax (x )\right )\right )-x \right )}{\left (3 x +5\right ) \ln \left (\ln \left (5-\ln \relax (x )\right )\right )}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((135*x^3+495*x^2+525*x+125)*ln(x)-675*x^3-2475*x^2-2625*x-625)*ln(5-ln(x))*ln(ln(5-ln(x)))^2+((-45*x^3-1
50*x^2)*ln(x)+225*x^3+750*x^2)*ln(5-ln(x))*ln(ln(5-ln(x)))+45*x^3+75*x^2)*exp(((9*x^2+15*x)*ln(ln(5-ln(x)))-3*
x^2)/(3*x+5)/ln(ln(5-ln(x))))/((9*x^2+30*x+25)*ln(x)-45*x^2-150*x-125)/ln(5-ln(x))/ln(ln(5-ln(x)))^2,x,method=
_RETURNVERBOSE)

[Out]

5*x*exp(3*x*(3*ln(ln(5-ln(x)))*x+5*ln(ln(5-ln(x)))-x)/(3*x+5)/ln(ln(5-ln(x))))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((135*x^3+495*x^2+525*x+125)*log(x)-675*x^3-2475*x^2-2625*x-625)*log(5-log(x))*log(log(5-log(x)))^2
+((-45*x^3-150*x^2)*log(x)+225*x^3+750*x^2)*log(5-log(x))*log(log(5-log(x)))+45*x^3+75*x^2)*exp(((9*x^2+15*x)*
log(log(5-log(x)))-3*x^2)/(3*x+5)/log(log(5-log(x))))/((9*x^2+30*x+25)*log(x)-45*x^2-150*x-125)/log(5-log(x))/
log(log(5-log(x)))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [B]  time = 3.35, size = 37, normalized size = 1.19 \begin {gather*} 5\,x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-\frac {3\,x^2}{5\,\ln \left (\ln \left (5-\ln \relax (x)\right )\right )+3\,x\,\ln \left (\ln \left (5-\ln \relax (x)\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((log(log(5 - log(x)))*(15*x + 9*x^2) - 3*x^2)/(log(log(5 - log(x)))*(3*x + 5)))*(75*x^2 + 45*x^3 + l
og(5 - log(x))*log(log(5 - log(x)))*(750*x^2 - log(x)*(150*x^2 + 45*x^3) + 225*x^3) - log(5 - log(x))*log(log(
5 - log(x)))^2*(2625*x + 2475*x^2 + 675*x^3 - log(x)*(525*x + 495*x^2 + 135*x^3 + 125) + 625)))/(log(5 - log(x
))*log(log(5 - log(x)))^2*(150*x - log(x)*(30*x + 9*x^2 + 25) + 45*x^2 + 125)),x)

[Out]

5*x*exp(3*x)*exp(-(3*x^2)/(5*log(log(5 - log(x))) + 3*x*log(log(5 - log(x)))))

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sympy [A]  time = 55.61, size = 39, normalized size = 1.26 \begin {gather*} 5 x e^{\frac {- 3 x^{2} + \left (9 x^{2} + 15 x\right ) \log {\left (\log {\left (5 - \log {\relax (x )} \right )} \right )}}{\left (3 x + 5\right ) \log {\left (\log {\left (5 - \log {\relax (x )} \right )} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((135*x**3+495*x**2+525*x+125)*ln(x)-675*x**3-2475*x**2-2625*x-625)*ln(5-ln(x))*ln(ln(5-ln(x)))**2+
((-45*x**3-150*x**2)*ln(x)+225*x**3+750*x**2)*ln(5-ln(x))*ln(ln(5-ln(x)))+45*x**3+75*x**2)*exp(((9*x**2+15*x)*
ln(ln(5-ln(x)))-3*x**2)/(3*x+5)/ln(ln(5-ln(x))))/((9*x**2+30*x+25)*ln(x)-45*x**2-150*x-125)/ln(5-ln(x))/ln(ln(
5-ln(x)))**2,x)

[Out]

5*x*exp((-3*x**2 + (9*x**2 + 15*x)*log(log(5 - log(x))))/((3*x + 5)*log(log(5 - log(x)))))

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