3.5.12 \(\int \frac {e^{\frac {-1+\log (5)}{\log (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x)}} (-9 x+9 x \log (5)+(-6 x+6 x \log (5)) \log (x)+(-x+x \log (5)) \log ^2(x)+e^{\frac {75+28 x+x \log (x)}{3+\log (x)}} (-75+59 x+(75-59 x) \log (5)+(31 x-31 x \log (5)) \log (x)+(x-x \log (5)) \log ^2(x)))}{(-9 x^2-6 x^2 \log (x)-x^2 \log ^2(x)+e^{\frac {75+28 x+x \log (x)}{3+\log (x)}} (9 x+6 x \log (x)+x \log ^2(x))) \log ^2(-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x)} \, dx\)

Optimal. Leaf size=29 \[ e^{\frac {-1+\log (5)}{\log \left (-e^{x+\frac {25 (3+x)}{3+\log (x)}}+x\right )}} \]

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Rubi [F]  time = 66.67, 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 {-1+\log (5)}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}\right ) \left (-9 x+9 x \log (5)+(-6 x+6 x \log (5)) \log (x)+(-x+x \log (5)) \log ^2(x)+e^{\frac {75+28 x+x \log (x)}{3+\log (x)}} \left (-75+59 x+(75-59 x) \log (5)+(31 x-31 x \log (5)) \log (x)+(x-x \log (5)) \log ^2(x)\right )\right )}{\left (-9 x^2-6 x^2 \log (x)-x^2 \log ^2(x)+e^{\frac {75+28 x+x \log (x)}{3+\log (x)}} \left (9 x+6 x \log (x)+x \log ^2(x)\right )\right ) \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-1 + Log[5])/Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x])*(-9*x + 9*x*Log[5] + (-6*x + 6*x*Log[
5])*Log[x] + (-x + x*Log[5])*Log[x]^2 + E^((75 + 28*x + x*Log[x])/(3 + Log[x]))*(-75 + 59*x + (75 - 59*x)*Log[
5] + (31*x - 31*x*Log[5])*Log[x] + (x - x*Log[5])*Log[x]^2)))/((-9*x^2 - 6*x^2*Log[x] - x^2*Log[x]^2 + E^((75
+ 28*x + x*Log[x])/(3 + Log[x]))*(9*x + 6*x*Log[x] + x*Log[x]^2))*Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x]))
+ x]^2),x]

[Out]

59*(1 - Log[5])*Defer[Int][(5/E)^Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1)/((3 + Log[x])^2*Log[-E
^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] - 3*(1 - Log[5])*Defer[Int][5^(2 + Log[-E^((75 + 28*x + x*L
og[x])/(3 + Log[x])) + x]^(-1))/(E^Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1)*x*(3 + Log[x])^2*Log
[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] - 84*(1 - Log[5])*Defer[Int][(5/E)^Log[-E^((75 + 28*x +
x*Log[x])/(3 + Log[x])) + x]^(-1)/((-x + E^(75/(3 + Log[x]) + (28*x)/(3 + Log[x]))*x^(x/(3 + Log[x])))*(3 + Lo
g[x])^2*Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] + 59*(1 - Log[5])*Defer[Int][((5/E)^Log[-E^((
75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1)*x)/((-x + E^(75/(3 + Log[x]) + (28*x)/(3 + Log[x]))*x^(x/(3 + Lo
g[x])))*(3 + Log[x])^2*Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] + 31*(1 - Log[5])*Defer[Int][(
(5/E)^Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1)*Log[x])/((3 + Log[x])^2*Log[-E^((75 + 28*x + x*Lo
g[x])/(3 + Log[x])) + x]^2), x] - 6*(1 - Log[5])*Defer[Int][((5/E)^Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])
) + x]^(-1)*Log[x])/((-x + E^(75/(3 + Log[x]) + (28*x)/(3 + Log[x]))*x^(x/(3 + Log[x])))*(3 + Log[x])^2*Log[-E
^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] + 31*(1 - Log[5])*Defer[Int][((5/E)^Log[-E^((75 + 28*x + x*
Log[x])/(3 + Log[x])) + x]^(-1)*x*Log[x])/((-x + E^(75/(3 + Log[x]) + (28*x)/(3 + Log[x]))*x^(x/(3 + Log[x])))
*(3 + Log[x])^2*Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] + (1 - Log[5])*Defer[Int][((5/E)^Log[
-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1)*Log[x]^2)/((3 + Log[x])^2*Log[-E^((75 + 28*x + x*Log[x])/(3
+ Log[x])) + x]^2), x] - (1 - Log[5])*Defer[Int][((5/E)^Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1
)*Log[x]^2)/((-x + E^(75/(3 + Log[x]) + (28*x)/(3 + Log[x]))*x^(x/(3 + Log[x])))*(3 + Log[x])^2*Log[-E^((75 +
28*x + x*Log[x])/(3 + Log[x])) + x]^2), x] + (1 - Log[5])*Defer[Int][((5/E)^Log[-E^((75 + 28*x + x*Log[x])/(3
+ Log[x])) + x]^(-1)*x*Log[x]^2)/((-x + E^(75/(3 + Log[x]) + (28*x)/(3 + Log[x]))*x^(x/(3 + Log[x])))*(3 + Log
[x])^2*Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-1+\log (5)}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}\right ) \left (x (-9+9 \log (5))+(-6 x+6 x \log (5)) \log (x)+(-x+x \log (5)) \log ^2(x)+e^{\frac {75+28 x+x \log (x)}{3+\log (x)}} \left (-75+59 x+(75-59 x) \log (5)+(31 x-31 x \log (5)) \log (x)+(x-x \log (5)) \log ^2(x)\right )\right )}{\left (-9 x^2-6 x^2 \log (x)-x^2 \log ^2(x)+e^{\frac {75+28 x+x \log (x)}{3+\log (x)}} \left (9 x+6 x \log (x)+x \log ^2(x)\right )\right ) \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )} \, dx\\ &=\int \frac {\left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} (1-\log (5)) \left (9 x+6 x \log (x)+x \log ^2(x)-e^{\frac {75+28 x}{3+\log (x)}} x^{\frac {x}{3+\log (x)}} \left (-75+59 x+31 x \log (x)+x \log ^2(x)\right )\right )}{x \left (x-e^{\frac {75+28 x}{3+\log (x)}} x^{\frac {x}{3+\log (x)}}\right ) (3+\log (x))^2 \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )} \, dx\\ &=(1-\log (5)) \int \frac {\left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} \left (9 x+6 x \log (x)+x \log ^2(x)-e^{\frac {75+28 x}{3+\log (x)}} x^{\frac {x}{3+\log (x)}} \left (-75+59 x+31 x \log (x)+x \log ^2(x)\right )\right )}{x \left (x-e^{\frac {75+28 x}{3+\log (x)}} x^{\frac {x}{3+\log (x)}}\right ) (3+\log (x))^2 \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )} \, dx\\ &=(1-\log (5)) \int \left (\frac {\left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} \left (-75+59 x+31 x \log (x)+x \log ^2(x)\right )}{x (3+\log (x))^2 \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}+\frac {\left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} \left (-84+59 x-6 \log (x)+31 x \log (x)-\log ^2(x)+x \log ^2(x)\right )}{\left (-x+e^{\frac {75}{3+\log (x)}+\frac {28 x}{3+\log (x)}} x^{\frac {x}{3+\log (x)}}\right ) (3+\log (x))^2 \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}\right ) \, dx\\ &=(1-\log (5)) \int \frac {\left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} \left (-75+59 x+31 x \log (x)+x \log ^2(x)\right )}{x (3+\log (x))^2 \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )} \, dx+(1-\log (5)) \int \frac {\left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} \left (-84+59 x-6 \log (x)+31 x \log (x)-\log ^2(x)+x \log ^2(x)\right )}{\left (-x+e^{\frac {75}{3+\log (x)}+\frac {28 x}{3+\log (x)}} x^{\frac {x}{3+\log (x)}}\right ) (3+\log (x))^2 \log ^2\left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 31, normalized size = 1.07 \begin {gather*} \left (\frac {5}{e}\right )^{\frac {1}{\log \left (-e^{\frac {75+28 x+x \log (x)}{3+\log (x)}}+x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-1 + Log[5])/Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x])*(-9*x + 9*x*Log[5] + (-6*x + 6*
x*Log[5])*Log[x] + (-x + x*Log[5])*Log[x]^2 + E^((75 + 28*x + x*Log[x])/(3 + Log[x]))*(-75 + 59*x + (75 - 59*x
)*Log[5] + (31*x - 31*x*Log[5])*Log[x] + (x - x*Log[5])*Log[x]^2)))/((-9*x^2 - 6*x^2*Log[x] - x^2*Log[x]^2 + E
^((75 + 28*x + x*Log[x])/(3 + Log[x]))*(9*x + 6*x*Log[x] + x*Log[x]^2))*Log[-E^((75 + 28*x + x*Log[x])/(3 + Lo
g[x])) + x]^2),x]

[Out]

(5/E)^Log[-E^((75 + 28*x + x*Log[x])/(3 + Log[x])) + x]^(-1)

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fricas [A]  time = 0.54, size = 30, normalized size = 1.03 \begin {gather*} e^{\left (\frac {\log \relax (5) - 1}{\log \left (x - e^{\left (\frac {x \log \relax (x) + 28 \, x + 75}{\log \relax (x) + 3}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x*log(5)+x)*log(x)^2+(-31*x*log(5)+31*x)*log(x)+(-59*x+75)*log(5)+59*x-75)*exp((x*log(x)+28*x+75
)/(3+log(x)))+(x*log(5)-x)*log(x)^2+(6*x*log(5)-6*x)*log(x)+9*x*log(5)-9*x)*exp((log(5)-1)/log(-exp((x*log(x)+
28*x+75)/(3+log(x)))+x))/((x*log(x)^2+6*x*log(x)+9*x)*exp((x*log(x)+28*x+75)/(3+log(x)))-x^2*log(x)^2-6*x^2*lo
g(x)-9*x^2)/log(-exp((x*log(x)+28*x+75)/(3+log(x)))+x)^2,x, algorithm="fricas")

[Out]

e^((log(5) - 1)/log(x - e^((x*log(x) + 28*x + 75)/(log(x) + 3))))

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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((((-x*log(5)+x)*log(x)^2+(-31*x*log(5)+31*x)*log(x)+(-59*x+75)*log(5)+59*x-75)*exp((x*log(x)+28*x+75
)/(3+log(x)))+(x*log(5)-x)*log(x)^2+(6*x*log(5)-6*x)*log(x)+9*x*log(5)-9*x)*exp((log(5)-1)/log(-exp((x*log(x)+
28*x+75)/(3+log(x)))+x))/((x*log(x)^2+6*x*log(x)+9*x)*exp((x*log(x)+28*x+75)/(3+log(x)))-x^2*log(x)^2-6*x^2*lo
g(x)-9*x^2)/log(-exp((x*log(x)+28*x+75)/(3+log(x)))+x)^2,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.08, size = 31, normalized size = 1.07

method	result	size
risch	\({\mathrm e}^{\frac {\ln \relax (5)-1}{\ln \left (-{\mathrm e}^{\frac {x \ln \relax (x )+28 x +75}{3+\ln \relax (x )}}+x \right )}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x*ln(5)+x)*ln(x)^2+(-31*x*ln(5)+31*x)*ln(x)+(-59*x+75)*ln(5)+59*x-75)*exp((x*ln(x)+28*x+75)/(3+ln(x)))
+(x*ln(5)-x)*ln(x)^2+(6*x*ln(5)-6*x)*ln(x)+9*x*ln(5)-9*x)*exp((ln(5)-1)/ln(-exp((x*ln(x)+28*x+75)/(3+ln(x)))+x
))/((x*ln(x)^2+6*x*ln(x)+9*x)*exp((x*ln(x)+28*x+75)/(3+ln(x)))-x^2*ln(x)^2-6*x^2*ln(x)-9*x^2)/ln(-exp((x*ln(x)
+28*x+75)/(3+ln(x)))+x)^2,x,method=_RETURNVERBOSE)

[Out]

exp((ln(5)-1)/ln(-exp((x*ln(x)+28*x+75)/(3+ln(x)))+x))

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maxima [B]  time = 1.17, size = 79, normalized size = 2.72 \begin {gather*} e^{\left (\frac {\log \relax (5)}{\log \left (x - e^{\left (\frac {x \log \relax (x)}{\log \relax (x) + 3} + \frac {28 \, x}{\log \relax (x) + 3} + \frac {75}{\log \relax (x) + 3}\right )}\right )} - \frac {1}{\log \left (x - e^{\left (\frac {x \log \relax (x)}{\log \relax (x) + 3} + \frac {28 \, x}{\log \relax (x) + 3} + \frac {75}{\log \relax (x) + 3}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x*log(5)+x)*log(x)^2+(-31*x*log(5)+31*x)*log(x)+(-59*x+75)*log(5)+59*x-75)*exp((x*log(x)+28*x+75
)/(3+log(x)))+(x*log(5)-x)*log(x)^2+(6*x*log(5)-6*x)*log(x)+9*x*log(5)-9*x)*exp((log(5)-1)/log(-exp((x*log(x)+
28*x+75)/(3+log(x)))+x))/((x*log(x)^2+6*x*log(x)+9*x)*exp((x*log(x)+28*x+75)/(3+log(x)))-x^2*log(x)^2-6*x^2*lo
g(x)-9*x^2)/log(-exp((x*log(x)+28*x+75)/(3+log(x)))+x)^2,x, algorithm="maxima")

[Out]

e^(log(5)/log(x - e^(x*log(x)/(log(x) + 3) + 28*x/(log(x) + 3) + 75/(log(x) + 3))) - 1/log(x - e^(x*log(x)/(lo
g(x) + 3) + 28*x/(log(x) + 3) + 75/(log(x) + 3))))

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mupad [B]  time = 1.46, size = 78, normalized size = 2.69 \begin {gather*} 5^{\frac {1}{\ln \left (x-x^{\frac {x}{\ln \relax (x)+3}}\,{\mathrm {e}}^{\frac {75}{\ln \relax (x)+3}}\,{\mathrm {e}}^{\frac {28\,x}{\ln \relax (x)+3}}\right )}}\,{\mathrm {e}}^{-\frac {1}{\ln \left (x-x^{\frac {x}{\ln \relax (x)+3}}\,{\mathrm {e}}^{\frac {75}{\ln \relax (x)+3}}\,{\mathrm {e}}^{\frac {28\,x}{\ln \relax (x)+3}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(5) - 1)/log(x - exp((28*x + x*log(x) + 75)/(log(x) + 3))))*(9*x - 9*x*log(5) - exp((28*x + x*log
(x) + 75)/(log(x) + 3))*(59*x - log(5)*(59*x - 75) + log(x)*(31*x - 31*x*log(5)) + log(x)^2*(x - x*log(5)) - 7
5) + log(x)*(6*x - 6*x*log(5)) + log(x)^2*(x - x*log(5))))/(log(x - exp((28*x + x*log(x) + 75)/(log(x) + 3)))^
2*(6*x^2*log(x) + x^2*log(x)^2 - exp((28*x + x*log(x) + 75)/(log(x) + 3))*(9*x + x*log(x)^2 + 6*x*log(x)) + 9*
x^2)),x)

[Out]

5^(1/log(x - x^(x/(log(x) + 3))*exp(75/(log(x) + 3))*exp((28*x)/(log(x) + 3))))*exp(-1/log(x - x^(x/(log(x) +
3))*exp(75/(log(x) + 3))*exp((28*x)/(log(x) + 3))))

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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((((-x*ln(5)+x)*ln(x)**2+(-31*x*ln(5)+31*x)*ln(x)+(-59*x+75)*ln(5)+59*x-75)*exp((x*ln(x)+28*x+75)/(3+
ln(x)))+(x*ln(5)-x)*ln(x)**2+(6*x*ln(5)-6*x)*ln(x)+9*x*ln(5)-9*x)*exp((ln(5)-1)/ln(-exp((x*ln(x)+28*x+75)/(3+l
n(x)))+x))/((x*ln(x)**2+6*x*ln(x)+9*x)*exp((x*ln(x)+28*x+75)/(3+ln(x)))-x**2*ln(x)**2-6*x**2*ln(x)-9*x**2)/ln(
-exp((x*ln(x)+28*x+75)/(3+ln(x)))+x)**2,x)

[Out]

Timed out

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