3.29.67 \(\int \frac {e^{x+\frac {3 x}{\log (x)}} (6+24 x) \log ^2(x)+e^{x+\frac {2 x}{\log (x)}} (72+288 x) \log ^2(x)+(20-20 x+e^x (384+1536 x)) \log ^2(x)+e^{\frac {x}{\log (x)}} (10 x-10 x \log (x)+(5-5 x+e^x (288+1152 x)) \log ^2(x))}{64 e^x \log ^2(x)+48 e^{x+\frac {x}{\log (x)}} \log ^2(x)+12 e^{x+\frac {2 x}{\log (x)}} \log ^2(x)+e^{x+\frac {3 x}{\log (x)}} \log ^2(x)} \, dx\) [2867]

Optimal. Leaf size=26 \[ x \left (6+\frac {5 e^{-x}}{\left (4+e^{\frac {x}{\log (x)}}\right )^2}+12 x\right ) \]

[Out]

(6+5/exp(x)/(exp(x/ln(x))+4)^2+12*x)*x

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Rubi [F]
time = 2.84, 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 {e^{x+\frac {3 x}{\log (x)}} (6+24 x) \log ^2(x)+e^{x+\frac {2 x}{\log (x)}} (72+288 x) \log ^2(x)+\left (20-20 x+e^x (384+1536 x)\right ) \log ^2(x)+e^{\frac {x}{\log (x)}} \left (10 x-10 x \log (x)+\left (5-5 x+e^x (288+1152 x)\right ) \log ^2(x)\right )}{64 e^x \log ^2(x)+48 e^{x+\frac {x}{\log (x)}} \log ^2(x)+12 e^{x+\frac {2 x}{\log (x)}} \log ^2(x)+e^{x+\frac {3 x}{\log (x)}} \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x + (3*x)/Log[x])*(6 + 24*x)*Log[x]^2 + E^(x + (2*x)/Log[x])*(72 + 288*x)*Log[x]^2 + (20 - 20*x + E^x*
(384 + 1536*x))*Log[x]^2 + E^(x/Log[x])*(10*x - 10*x*Log[x] + (5 - 5*x + E^x*(288 + 1152*x))*Log[x]^2))/(64*E^
x*Log[x]^2 + 48*E^(x + x/Log[x])*Log[x]^2 + 12*E^(x + (2*x)/Log[x])*Log[x]^2 + E^(x + (3*x)/Log[x])*Log[x]^2),
x]

[Out]

(3*(1 + 4*x)^2)/4 + 5*Defer[Int][1/(E^x*(4 + E^(x/Log[x]))^2), x] - 5*Defer[Int][x/(E^x*(4 + E^(x/Log[x]))^2),
x] - 40*Defer[Int][x/(E^x*(4 + E^(x/Log[x]))^3*Log[x]^2), x] + 10*Defer[Int][x/(E^x*(4 + E^(x/Log[x]))^2*Log[
x]^2), x] + 40*Defer[Int][x/(E^x*(4 + E^(x/Log[x]))^3*Log[x]), x] - 10*Defer[Int][x/(E^x*(4 + E^(x/Log[x]))^2*
Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (10 e^{\frac {x}{\log (x)}} x-10 e^{\frac {x}{\log (x)}} x \log (x)+\left (4+e^{\frac {x}{\log (x)}}\right ) \left (5-5 x+96 e^x (1+4 x)+48 e^{x+\frac {x}{\log (x)}} (1+4 x)+6 e^{x+\frac {2 x}{\log (x)}} (1+4 x)\right ) \log ^2(x)\right )}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log ^2(x)} \, dx\\ &=\int \left (6 (1+4 x)+\frac {40 e^{-x} x (-1+\log (x))}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log ^2(x)}-\frac {5 e^{-x} \left (-2 x+2 x \log (x)-\log ^2(x)+x \log ^2(x)\right )}{\left (4+e^{\frac {x}{\log (x)}}\right )^2 \log ^2(x)}\right ) \, dx\\ &=\frac {3}{4} (1+4 x)^2-5 \int \frac {e^{-x} \left (-2 x+2 x \log (x)-\log ^2(x)+x \log ^2(x)\right )}{\left (4+e^{\frac {x}{\log (x)}}\right )^2 \log ^2(x)} \, dx+40 \int \frac {e^{-x} x (-1+\log (x))}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log ^2(x)} \, dx\\ &=\frac {3}{4} (1+4 x)^2-5 \int \left (-\frac {e^{-x}}{\left (4+e^{\frac {x}{\log (x)}}\right )^2}+\frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^2}-\frac {2 e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^2 \log ^2(x)}+\frac {2 e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^2 \log (x)}\right ) \, dx+40 \int \left (-\frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log ^2(x)}+\frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log (x)}\right ) \, dx\\ &=\frac {3}{4} (1+4 x)^2+5 \int \frac {e^{-x}}{\left (4+e^{\frac {x}{\log (x)}}\right )^2} \, dx-5 \int \frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^2} \, dx+10 \int \frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^2 \log ^2(x)} \, dx-10 \int \frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^2 \log (x)} \, dx-40 \int \frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log ^2(x)} \, dx+40 \int \frac {e^{-x} x}{\left (4+e^{\frac {x}{\log (x)}}\right )^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.14, size = 26, normalized size = 1.00 \begin {gather*} x \left (6+\frac {5 e^{-x}}{\left (4+e^{\frac {x}{\log (x)}}\right )^2}+12 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x + (3*x)/Log[x])*(6 + 24*x)*Log[x]^2 + E^(x + (2*x)/Log[x])*(72 + 288*x)*Log[x]^2 + (20 - 20*x
+ E^x*(384 + 1536*x))*Log[x]^2 + E^(x/Log[x])*(10*x - 10*x*Log[x] + (5 - 5*x + E^x*(288 + 1152*x))*Log[x]^2))/
(64*E^x*Log[x]^2 + 48*E^(x + x/Log[x])*Log[x]^2 + 12*E^(x + (2*x)/Log[x])*Log[x]^2 + E^(x + (3*x)/Log[x])*Log[
x]^2),x]

[Out]

x*(6 + 5/(E^x*(4 + E^(x/Log[x]))^2) + 12*x)

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Maple [A]
time = 0.13, size = 28, normalized size = 1.08

method	result	size
risch	\(12 x^{2}+6 x +\frac {5 x \,{\mathrm e}^{-x}}{\left ({\mathrm e}^{\frac {x}{\ln \left (x \right )}}+4\right )^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x+6)*exp(x)*ln(x)^2*exp(x/ln(x))^3+(288*x+72)*exp(x)*ln(x)^2*exp(x/ln(x))^2+(((1152*x+288)*exp(x)-5*x
+5)*ln(x)^2-10*x*ln(x)+10*x)*exp(x/ln(x))+((1536*x+384)*exp(x)-20*x+20)*ln(x)^2)/(exp(x)*ln(x)^2*exp(x/ln(x))^
3+12*exp(x)*ln(x)^2*exp(x/ln(x))^2+48*exp(x)*ln(x)^2*exp(x/ln(x))+64*exp(x)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

12*x^2+6*x+5*x*exp(-x)/(exp(x/ln(x))+4)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (24) = 48\).
time = 0.33, size = 81, normalized size = 3.12 \begin {gather*} \frac {6 \, {\left (2 \, x^{2} + x\right )} e^{\left (x + \frac {2 \, x}{\log \left (x\right )}\right )} + 48 \, {\left (2 \, x^{2} + x\right )} e^{\left (x + \frac {x}{\log \left (x\right )}\right )} + 96 \, {\left (2 \, x^{2} + x\right )} e^{x} + 5 \, x}{e^{\left (x + \frac {2 \, x}{\log \left (x\right )}\right )} + 8 \, e^{\left (x + \frac {x}{\log \left (x\right )}\right )} + 16 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*exp(x)*log(x)^2*exp(x/log(x))^3+(288*x+72)*exp(x)*log(x)^2*exp(x/log(x))^2+(((1152*x+288)*
exp(x)-5*x+5)*log(x)^2-10*x*log(x)+10*x)*exp(x/log(x))+((1536*x+384)*exp(x)-20*x+20)*log(x)^2)/(exp(x)*log(x)^
2*exp(x/log(x))^3+12*exp(x)*log(x)^2*exp(x/log(x))^2+48*exp(x)*log(x)^2*exp(x/log(x))+64*exp(x)*log(x)^2),x, a
lgorithm="maxima")

[Out]

(6*(2*x^2 + x)*e^(x + 2*x/log(x)) + 48*(2*x^2 + x)*e^(x + x/log(x)) + 96*(2*x^2 + x)*e^x + 5*x)/(e^(x + 2*x/lo
g(x)) + 8*e^(x + x/log(x)) + 16*e^x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (24) = 48\).
time = 0.39, size = 103, normalized size = 3.96 \begin {gather*} \frac {96 \, {\left (2 \, x^{2} + x\right )} e^{\left (2 \, x\right )} + 48 \, {\left (2 \, x^{2} + x\right )} e^{\left (x + \frac {x \log \left (x\right ) + x}{\log \left (x\right )}\right )} + 5 \, x e^{x} + 6 \, {\left (2 \, x^{2} + x\right )} e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + x\right )}}{\log \left (x\right )}\right )}}{16 \, e^{\left (2 \, x\right )} + 8 \, e^{\left (x + \frac {x \log \left (x\right ) + x}{\log \left (x\right )}\right )} + e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + x\right )}}{\log \left (x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*exp(x)*log(x)^2*exp(x/log(x))^3+(288*x+72)*exp(x)*log(x)^2*exp(x/log(x))^2+(((1152*x+288)*
exp(x)-5*x+5)*log(x)^2-10*x*log(x)+10*x)*exp(x/log(x))+((1536*x+384)*exp(x)-20*x+20)*log(x)^2)/(exp(x)*log(x)^
2*exp(x/log(x))^3+12*exp(x)*log(x)^2*exp(x/log(x))^2+48*exp(x)*log(x)^2*exp(x/log(x))+64*exp(x)*log(x)^2),x, a
lgorithm="fricas")

[Out]

(96*(2*x^2 + x)*e^(2*x) + 48*(2*x^2 + x)*e^(x + (x*log(x) + x)/log(x)) + 5*x*e^x + 6*(2*x^2 + x)*e^(2*(x*log(x
) + x)/log(x)))/(16*e^(2*x) + 8*e^(x + (x*log(x) + x)/log(x)) + e^(2*(x*log(x) + x)/log(x)))

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Sympy [A]
time = 0.13, size = 39, normalized size = 1.50 \begin {gather*} 12 x^{2} + 6 x + \frac {5 x}{e^{x} e^{\frac {2 x}{\log {\left (x \right )}}} + 8 e^{x} e^{\frac {x}{\log {\left (x \right )}}} + 16 e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*exp(x)*ln(x)**2*exp(x/ln(x))**3+(288*x+72)*exp(x)*ln(x)**2*exp(x/ln(x))**2+(((1152*x+288)*
exp(x)-5*x+5)*ln(x)**2-10*x*ln(x)+10*x)*exp(x/ln(x))+((1536*x+384)*exp(x)-20*x+20)*ln(x)**2)/(exp(x)*ln(x)**2*
exp(x/ln(x))**3+12*exp(x)*ln(x)**2*exp(x/ln(x))**2+48*exp(x)*ln(x)**2*exp(x/ln(x))+64*exp(x)*ln(x)**2),x)

[Out]

12*x**2 + 6*x + 5*x/(exp(x)*exp(2*x/log(x)) + 8*exp(x)*exp(x/log(x)) + 16*exp(x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (24) = 48\).
time = 0.42, size = 212, normalized size = 8.15 \begin {gather*} \frac {12 \, x^{2} e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + 2 \, x\right )}}{\log \left (x\right )}\right )} + 96 \, x^{2} e^{\left (\frac {x \log \left (x\right ) + 2 \, x}{\log \left (x\right )} + \frac {x \log \left (x\right ) + x}{\log \left (x\right )}\right )} + 192 \, x^{2} e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + x\right )}}{\log \left (x\right )}\right )} + 6 \, x e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + 2 \, x\right )}}{\log \left (x\right )}\right )} + 48 \, x e^{\left (\frac {x \log \left (x\right ) + 2 \, x}{\log \left (x\right )} + \frac {x \log \left (x\right ) + x}{\log \left (x\right )}\right )} + 5 \, x e^{\left (\frac {x \log \left (x\right ) + 2 \, x}{\log \left (x\right )}\right )} + 96 \, x e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + x\right )}}{\log \left (x\right )}\right )}}{e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + 2 \, x\right )}}{\log \left (x\right )}\right )} + 8 \, e^{\left (\frac {x \log \left (x\right ) + 2 \, x}{\log \left (x\right )} + \frac {x \log \left (x\right ) + x}{\log \left (x\right )}\right )} + 16 \, e^{\left (\frac {2 \, {\left (x \log \left (x\right ) + x\right )}}{\log \left (x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*exp(x)*log(x)^2*exp(x/log(x))^3+(288*x+72)*exp(x)*log(x)^2*exp(x/log(x))^2+(((1152*x+288)*
exp(x)-5*x+5)*log(x)^2-10*x*log(x)+10*x)*exp(x/log(x))+((1536*x+384)*exp(x)-20*x+20)*log(x)^2)/(exp(x)*log(x)^
2*exp(x/log(x))^3+12*exp(x)*log(x)^2*exp(x/log(x))^2+48*exp(x)*log(x)^2*exp(x/log(x))+64*exp(x)*log(x)^2),x, a
lgorithm="giac")

[Out]

(12*x^2*e^(2*(x*log(x) + 2*x)/log(x)) + 96*x^2*e^((x*log(x) + 2*x)/log(x) + (x*log(x) + x)/log(x)) + 192*x^2*e
^(2*(x*log(x) + x)/log(x)) + 6*x*e^(2*(x*log(x) + 2*x)/log(x)) + 48*x*e^((x*log(x) + 2*x)/log(x) + (x*log(x) +
x)/log(x)) + 5*x*e^((x*log(x) + 2*x)/log(x)) + 96*x*e^(2*(x*log(x) + x)/log(x)))/(e^(2*(x*log(x) + 2*x)/log(x
)) + 8*e^((x*log(x) + 2*x)/log(x) + (x*log(x) + x)/log(x)) + 16*e^(2*(x*log(x) + x)/log(x)))

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Mupad [B]
time = 1.98, size = 49, normalized size = 1.88 \begin {gather*} 6\,x+12\,x^2-\frac {5\,{\mathrm {e}}^{-x}\,\left (x-x\,\ln \left (x\right )\right )}{\left (\ln \left (x\right )-1\right )\,\left (8\,{\mathrm {e}}^{\frac {x}{\ln \left (x\right )}}+{\mathrm {e}}^{\frac {2\,x}{\ln \left (x\right )}}+16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x/log(x))*(10*x + log(x)^2*(exp(x)*(1152*x + 288) - 5*x + 5) - 10*x*log(x)) + log(x)^2*(exp(x)*(1536*
x + 384) - 20*x + 20) + exp((3*x)/log(x))*exp(x)*log(x)^2*(24*x + 6) + exp((2*x)/log(x))*exp(x)*log(x)^2*(288*
x + 72))/(64*exp(x)*log(x)^2 + 48*exp(x/log(x))*exp(x)*log(x)^2 + 12*exp((2*x)/log(x))*exp(x)*log(x)^2 + exp((
3*x)/log(x))*exp(x)*log(x)^2),x)

[Out]

6*x + 12*x^2 - (5*exp(-x)*(x - x*log(x)))/((log(x) - 1)*(8*exp(x/log(x)) + exp((2*x)/log(x)) + 16))

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