3.80.89 \(\int \frac {-15-5 e^x+(-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} (12288 x^4+384 x^5+3 x^6)+e^x (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6)) \log (x)+(-15+3456 x^4+54 x^5+e^{2 x} (384 x^4+6 x^5)+e^x (-5-5 x+2304 x^4+36 x^5)) \log (x) \log (\log (x))+(27 x^4+18 e^x x^4+3 e^{2 x} x^4) \log (x) \log ^2(\log (x))}{(36864 x^2+1152 x^3+9 x^4+e^{2 x} (4096 x^2+128 x^3+x^4)+e^x (24576 x^2+768 x^3+6 x^4)) \log (x)+(1152 x^2+18 x^3+e^{2 x} (128 x^2+2 x^3)+e^x (768 x^2+12 x^3)) \log (x) \log (\log (x))+(9 x^2+6 e^x x^2+e^{2 x} x^2) \log (x) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=25 \[ -2+x^3+\frac {5}{\left (3+e^x\right ) x (64+x+\log (\log (x)))} \]

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Rubi [F]  time = 5.64, 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 {-15-5 e^x+\left (-960-30 x+110592 x^4+3456 x^5+27 x^6+e^{2 x} \left (12288 x^4+384 x^5+3 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )\right ) \log (x)+\left (-15+3456 x^4+54 x^5+e^{2 x} \left (384 x^4+6 x^5\right )+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (x) \log (\log (x))+\left (27 x^4+18 e^x x^4+3 e^{2 x} x^4\right ) \log (x) \log ^2(\log (x))}{\left (36864 x^2+1152 x^3+9 x^4+e^{2 x} \left (4096 x^2+128 x^3+x^4\right )+e^x \left (24576 x^2+768 x^3+6 x^4\right )\right ) \log (x)+\left (1152 x^2+18 x^3+e^{2 x} \left (128 x^2+2 x^3\right )+e^x \left (768 x^2+12 x^3\right )\right ) \log (x) \log (\log (x))+\left (9 x^2+6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15 - 5*E^x + (-960 - 30*x + 110592*x^4 + 3456*x^5 + 27*x^6 + E^(2*x)*(12288*x^4 + 384*x^5 + 3*x^6) + E^x
*(-320 - 330*x - 5*x^2 + 73728*x^4 + 2304*x^5 + 18*x^6))*Log[x] + (-15 + 3456*x^4 + 54*x^5 + E^(2*x)*(384*x^4
+ 6*x^5) + E^x*(-5 - 5*x + 2304*x^4 + 36*x^5))*Log[x]*Log[Log[x]] + (27*x^4 + 18*E^x*x^4 + 3*E^(2*x)*x^4)*Log[
x]*Log[Log[x]]^2)/((36864*x^2 + 1152*x^3 + 9*x^4 + E^(2*x)*(4096*x^2 + 128*x^3 + x^4) + E^x*(24576*x^2 + 768*x
^3 + 6*x^4))*Log[x] + (1152*x^2 + 18*x^3 + E^(2*x)*(128*x^2 + 2*x^3) + E^x*(768*x^2 + 12*x^3))*Log[x]*Log[Log[
x]] + (9*x^2 + 6*E^x*x^2 + E^(2*x)*x^2)*Log[x]*Log[Log[x]]^2),x]

[Out]

x^3 - 5*Defer[Int][1/((3 + E^x)*(64 + x + Log[Log[x]])^2), x] - 320*Defer[Int][1/((3 + E^x)*x^2*(64 + x + Log[
Log[x]])^2), x] - 330*Defer[Int][1/((3 + E^x)*x*(64 + x + Log[Log[x]])^2), x] - 5*Defer[Int][1/((3 + E^x)*x^2*
Log[x]*(64 + x + Log[Log[x]])^2), x] - 5*Defer[Int][Log[Log[x]]/((3 + E^x)*x^2*(64 + x + Log[Log[x]])^2), x] -
5*Defer[Int][Log[Log[x]]/((3 + E^x)*x*(64 + x + Log[Log[x]])^2), x] + 15*Defer[Int][1/((3 + E^x)^2*x*(64 + x
+ Log[Log[x]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 \left (3+e^x\right )+\log (x) \left (3 e^{2 x} x^4 (64+x)^2+3 \left (-320-10 x+36864 x^4+1152 x^5+9 x^6\right )+e^x \left (-320-330 x-5 x^2+73728 x^4+2304 x^5+18 x^6\right )+\left (-15+3456 x^4+54 x^5+6 e^{2 x} x^4 (64+x)+e^x \left (-5-5 x+2304 x^4+36 x^5\right )\right ) \log (\log (x))+3 \left (3+e^x\right )^2 x^4 \log ^2(\log (x))\right )}{\left (3+e^x\right )^2 x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx\\ &=\int \left (3 x^2+\frac {15}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))}-\frac {5 \left (1+64 \log (x)+66 x \log (x)+x^2 \log (x)+\log (x) \log (\log (x))+x \log (x) \log (\log (x))\right )}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2}\right ) \, dx\\ &=x^3-5 \int \frac {1+64 \log (x)+66 x \log (x)+x^2 \log (x)+\log (x) \log (\log (x))+x \log (x) \log (\log (x))}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx\\ &=x^3-5 \int \frac {1+\log (x) \left (64+66 x+x^2+(1+x) \log (\log (x))\right )}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx\\ &=x^3-5 \int \left (\frac {1}{\left (3+e^x\right ) (64+x+\log (\log (x)))^2}+\frac {64}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2}+\frac {66}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2}+\frac {1}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2}+\frac {\log (\log (x))}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2}+\frac {\log (\log (x))}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2}\right ) \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx\\ &=x^3-5 \int \frac {1}{\left (3+e^x\right ) (64+x+\log (\log (x)))^2} \, dx-5 \int \frac {1}{\left (3+e^x\right ) x^2 \log (x) (64+x+\log (\log (x)))^2} \, dx-5 \int \frac {\log (\log (x))}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2} \, dx-5 \int \frac {\log (\log (x))}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2} \, dx+15 \int \frac {1}{\left (3+e^x\right )^2 x (64+x+\log (\log (x)))} \, dx-320 \int \frac {1}{\left (3+e^x\right ) x^2 (64+x+\log (\log (x)))^2} \, dx-330 \int \frac {1}{\left (3+e^x\right ) x (64+x+\log (\log (x)))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 24, normalized size = 0.96 \begin {gather*} x^3+\frac {5}{\left (3+e^x\right ) x (64+x+\log (\log (x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 - 5*E^x + (-960 - 30*x + 110592*x^4 + 3456*x^5 + 27*x^6 + E^(2*x)*(12288*x^4 + 384*x^5 + 3*x^6)
+ E^x*(-320 - 330*x - 5*x^2 + 73728*x^4 + 2304*x^5 + 18*x^6))*Log[x] + (-15 + 3456*x^4 + 54*x^5 + E^(2*x)*(38
4*x^4 + 6*x^5) + E^x*(-5 - 5*x + 2304*x^4 + 36*x^5))*Log[x]*Log[Log[x]] + (27*x^4 + 18*E^x*x^4 + 3*E^(2*x)*x^4
)*Log[x]*Log[Log[x]]^2)/((36864*x^2 + 1152*x^3 + 9*x^4 + E^(2*x)*(4096*x^2 + 128*x^3 + x^4) + E^x*(24576*x^2 +
768*x^3 + 6*x^4))*Log[x] + (1152*x^2 + 18*x^3 + E^(2*x)*(128*x^2 + 2*x^3) + E^x*(768*x^2 + 12*x^3))*Log[x]*Lo
g[Log[x]] + (9*x^2 + 6*E^x*x^2 + E^(2*x)*x^2)*Log[x]*Log[Log[x]]^2),x]

[Out]

x^3 + 5/((3 + E^x)*x*(64 + x + Log[Log[x]]))

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fricas [B]  time = 0.66, size = 74, normalized size = 2.96 \begin {gather*} \frac {3 \, x^{5} + 192 \, x^{4} + {\left (x^{5} + 64 \, x^{4}\right )} e^{x} + {\left (x^{4} e^{x} + 3 \, x^{4}\right )} \log \left (\log \relax (x)\right ) + 5}{3 \, x^{2} + {\left (x^{2} + 64 \, x\right )} e^{x} + {\left (x e^{x} + 3 \, x\right )} \log \left (\log \relax (x)\right ) + 192 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)^2*x^4+18*exp(x)*x^4+27*x^4)*log(x)*log(log(x))^2+((6*x^5+384*x^4)*exp(x)^2+(36*x^5+2304*x
^4-5*x-5)*exp(x)+54*x^5+3456*x^4-15)*log(x)*log(log(x))+((3*x^6+384*x^5+12288*x^4)*exp(x)^2+(18*x^6+2304*x^5+7
3728*x^4-5*x^2-330*x-320)*exp(x)+27*x^6+3456*x^5+110592*x^4-30*x-960)*log(x)-5*exp(x)-15)/((exp(x)^2*x^2+6*exp
(x)*x^2+9*x^2)*log(x)*log(log(x))^2+((2*x^3+128*x^2)*exp(x)^2+(12*x^3+768*x^2)*exp(x)+18*x^3+1152*x^2)*log(x)*
log(log(x))+((x^4+128*x^3+4096*x^2)*exp(x)^2+(6*x^4+768*x^3+24576*x^2)*exp(x)+9*x^4+1152*x^3+36864*x^2)*log(x)
),x, algorithm="fricas")

[Out]

(3*x^5 + 192*x^4 + (x^5 + 64*x^4)*e^x + (x^4*e^x + 3*x^4)*log(log(x)) + 5)/(3*x^2 + (x^2 + 64*x)*e^x + (x*e^x
+ 3*x)*log(log(x)) + 192*x)

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giac [B]  time = 0.25, size = 78, normalized size = 3.12 \begin {gather*} \frac {x^{5} e^{x} + x^{4} e^{x} \log \left (\log \relax (x)\right ) + 3 \, x^{5} + 64 \, x^{4} e^{x} + 3 \, x^{4} \log \left (\log \relax (x)\right ) + 192 \, x^{4} + 5}{x^{2} e^{x} + x e^{x} \log \left (\log \relax (x)\right ) + 3 \, x^{2} + 64 \, x e^{x} + 3 \, x \log \left (\log \relax (x)\right ) + 192 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)^2*x^4+18*exp(x)*x^4+27*x^4)*log(x)*log(log(x))^2+((6*x^5+384*x^4)*exp(x)^2+(36*x^5+2304*x
^4-5*x-5)*exp(x)+54*x^5+3456*x^4-15)*log(x)*log(log(x))+((3*x^6+384*x^5+12288*x^4)*exp(x)^2+(18*x^6+2304*x^5+7
3728*x^4-5*x^2-330*x-320)*exp(x)+27*x^6+3456*x^5+110592*x^4-30*x-960)*log(x)-5*exp(x)-15)/((exp(x)^2*x^2+6*exp
(x)*x^2+9*x^2)*log(x)*log(log(x))^2+((2*x^3+128*x^2)*exp(x)^2+(12*x^3+768*x^2)*exp(x)+18*x^3+1152*x^2)*log(x)*
log(log(x))+((x^4+128*x^3+4096*x^2)*exp(x)^2+(6*x^4+768*x^3+24576*x^2)*exp(x)+9*x^4+1152*x^3+36864*x^2)*log(x)
),x, algorithm="giac")

[Out]

(x^5*e^x + x^4*e^x*log(log(x)) + 3*x^5 + 64*x^4*e^x + 3*x^4*log(log(x)) + 192*x^4 + 5)/(x^2*e^x + x*e^x*log(lo
g(x)) + 3*x^2 + 64*x*e^x + 3*x*log(log(x)) + 192*x)

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maple [A]  time = 0.06, size = 24, normalized size = 0.96

method	result	size
risch	\(x^{3}+\frac {5}{x \left (\ln \left (\ln \relax (x )\right )+64+x \right ) \left (3+{\mathrm e}^{x}\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*exp(x)^2*x^4+18*exp(x)*x^4+27*x^4)*ln(x)*ln(ln(x))^2+((6*x^5+384*x^4)*exp(x)^2+(36*x^5+2304*x^4-5*x-5)
*exp(x)+54*x^5+3456*x^4-15)*ln(x)*ln(ln(x))+((3*x^6+384*x^5+12288*x^4)*exp(x)^2+(18*x^6+2304*x^5+73728*x^4-5*x
^2-330*x-320)*exp(x)+27*x^6+3456*x^5+110592*x^4-30*x-960)*ln(x)-5*exp(x)-15)/((exp(x)^2*x^2+6*exp(x)*x^2+9*x^2
)*ln(x)*ln(ln(x))^2+((2*x^3+128*x^2)*exp(x)^2+(12*x^3+768*x^2)*exp(x)+18*x^3+1152*x^2)*ln(x)*ln(ln(x))+((x^4+1
28*x^3+4096*x^2)*exp(x)^2+(6*x^4+768*x^3+24576*x^2)*exp(x)+9*x^4+1152*x^3+36864*x^2)*ln(x)),x,method=_RETURNVE
RBOSE)

[Out]

x^3+5/x/(ln(ln(x))+64+x)/(3+exp(x))

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maxima [B]  time = 0.54, size = 74, normalized size = 2.96 \begin {gather*} \frac {3 \, x^{5} + 192 \, x^{4} + {\left (x^{5} + 64 \, x^{4}\right )} e^{x} + {\left (x^{4} e^{x} + 3 \, x^{4}\right )} \log \left (\log \relax (x)\right ) + 5}{3 \, x^{2} + {\left (x^{2} + 64 \, x\right )} e^{x} + {\left (x e^{x} + 3 \, x\right )} \log \left (\log \relax (x)\right ) + 192 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)^2*x^4+18*exp(x)*x^4+27*x^4)*log(x)*log(log(x))^2+((6*x^5+384*x^4)*exp(x)^2+(36*x^5+2304*x
^4-5*x-5)*exp(x)+54*x^5+3456*x^4-15)*log(x)*log(log(x))+((3*x^6+384*x^5+12288*x^4)*exp(x)^2+(18*x^6+2304*x^5+7
3728*x^4-5*x^2-330*x-320)*exp(x)+27*x^6+3456*x^5+110592*x^4-30*x-960)*log(x)-5*exp(x)-15)/((exp(x)^2*x^2+6*exp
(x)*x^2+9*x^2)*log(x)*log(log(x))^2+((2*x^3+128*x^2)*exp(x)^2+(12*x^3+768*x^2)*exp(x)+18*x^3+1152*x^2)*log(x)*
log(log(x))+((x^4+128*x^3+4096*x^2)*exp(x)^2+(6*x^4+768*x^3+24576*x^2)*exp(x)+9*x^4+1152*x^3+36864*x^2)*log(x)
),x, algorithm="maxima")

[Out]

(3*x^5 + 192*x^4 + (x^5 + 64*x^4)*e^x + (x^4*e^x + 3*x^4)*log(log(x)) + 5)/(3*x^2 + (x^2 + 64*x)*e^x + (x*e^x
+ 3*x)*log(log(x)) + 192*x)

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mupad [B]  time = 6.03, size = 191, normalized size = 7.64 \begin {gather*} \frac {15}{x\,\left ({\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^x+9\right )}+x^3+\frac {\frac {5\,\left ({\mathrm {e}}^x+192\,\ln \relax (x)+64\,{\mathrm {e}}^x\,\ln \relax (x)+6\,x\,\ln \relax (x)+66\,x\,{\mathrm {e}}^x\,\ln \relax (x)+x^2\,{\mathrm {e}}^x\,\ln \relax (x)+3\right )}{x\,\left (x\,\ln \relax (x)+1\right )\,{\left ({\mathrm {e}}^x+3\right )}^2}+\frac {5\,\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left ({\mathrm {e}}^x+x\,{\mathrm {e}}^x+3\right )}{x\,\left (x\,\ln \relax (x)+1\right )\,{\left ({\mathrm {e}}^x+3\right )}^2}}{x+\ln \left (\ln \relax (x)\right )+64}-\frac {5\,\left (x^2+x\right )}{x^3\,\left ({\mathrm {e}}^x+3\right )}+\frac {5\,\left (9\,x-{\mathrm {e}}^{2\,x}-6\,{\mathrm {e}}^x+3\,x^2\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^x-9\right )}{x^2\,\left (x\,\ln \relax (x)+1\right )\,{\left ({\mathrm {e}}^x+3\right )}^3\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(exp(2*x)*(12288*x^4 + 384*x^5 + 3*x^6) - exp(x)*(330*x + 5*x^2 - 73728*x^4 - 2304*x^5 - 18*x^6 +
320) - 30*x + 110592*x^4 + 3456*x^5 + 27*x^6 - 960) - 5*exp(x) + log(log(x))^2*log(x)*(18*x^4*exp(x) + 3*x^4*e
xp(2*x) + 27*x^4) + log(log(x))*log(x)*(exp(2*x)*(384*x^4 + 6*x^5) + 3456*x^4 + 54*x^5 - exp(x)*(5*x - 2304*x^
4 - 36*x^5 + 5) - 15) - 15)/(log(x)*(exp(2*x)*(4096*x^2 + 128*x^3 + x^4) + exp(x)*(24576*x^2 + 768*x^3 + 6*x^4
) + 36864*x^2 + 1152*x^3 + 9*x^4) + log(log(x))^2*log(x)*(6*x^2*exp(x) + x^2*exp(2*x) + 9*x^2) + log(log(x))*l
og(x)*(exp(x)*(768*x^2 + 12*x^3) + exp(2*x)*(128*x^2 + 2*x^3) + 1152*x^2 + 18*x^3)),x)

[Out]

15/(x*(exp(2*x) + 6*exp(x) + 9)) + x^3 + ((5*(exp(x) + 192*log(x) + 64*exp(x)*log(x) + 6*x*log(x) + 66*x*exp(x
)*log(x) + x^2*exp(x)*log(x) + 3))/(x*(x*log(x) + 1)*(exp(x) + 3)^2) + (5*log(log(x))*log(x)*(exp(x) + x*exp(x
) + 3))/(x*(x*log(x) + 1)*(exp(x) + 3)^2))/(x + log(log(x)) + 64) - (5*(x + x^2))/(x^3*(exp(x) + 3)) + (5*(9*x
- exp(2*x) - 6*exp(x) + 3*x^2*exp(x) + x^2*exp(2*x) + 3*x*exp(x) - 9))/(x^2*(x*log(x) + 1)*(exp(x) + 3)^3*(x
- 1))

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sympy [A]  time = 0.52, size = 37, normalized size = 1.48 \begin {gather*} x^{3} + \frac {5}{3 x^{2} + 3 x \log {\left (\log {\relax (x )} \right )} + 192 x + \left (x^{2} + x \log {\left (\log {\relax (x )} \right )} + 64 x\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(x)**2*x**4+18*exp(x)*x**4+27*x**4)*ln(x)*ln(ln(x))**2+((6*x**5+384*x**4)*exp(x)**2+(36*x**5+
2304*x**4-5*x-5)*exp(x)+54*x**5+3456*x**4-15)*ln(x)*ln(ln(x))+((3*x**6+384*x**5+12288*x**4)*exp(x)**2+(18*x**6
+2304*x**5+73728*x**4-5*x**2-330*x-320)*exp(x)+27*x**6+3456*x**5+110592*x**4-30*x-960)*ln(x)-5*exp(x)-15)/((ex
p(x)**2*x**2+6*exp(x)*x**2+9*x**2)*ln(x)*ln(ln(x))**2+((2*x**3+128*x**2)*exp(x)**2+(12*x**3+768*x**2)*exp(x)+1
8*x**3+1152*x**2)*ln(x)*ln(ln(x))+((x**4+128*x**3+4096*x**2)*exp(x)**2+(6*x**4+768*x**3+24576*x**2)*exp(x)+9*x
**4+1152*x**3+36864*x**2)*ln(x)),x)

[Out]

x**3 + 5/(3*x**2 + 3*x*log(log(x)) + 192*x + (x**2 + x*log(log(x)) + 64*x)*exp(x))

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