3.90.86 \(\int \frac {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x (-384-64 x+40 x^2+8 x^3)+(4 e^{3 x}+e^{2 x} (96-112 x-16 x^2+8 x^3)+e^x (-384 x+288 x^2+144 x^3-20 x^4-8 x^5)) \log (x)+(-1536-640 x+96 x^2+72 x^3+8 x^4+(-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x (768-416 x-280 x^2+8 x^4)) \log (x)) \log (\log (x))+(1536+128 x-480 x^2-168 x^3-16 x^4+e^x (192+96 x+12 x^2)) \log (x) \log ^2(\log (x))+(256+192 x+48 x^2+4 x^3) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x (48+24 x+3 x^2) \log (x) \log ^2(\log (x))+(64+48 x+12 x^2+x^3) \log (x) \log ^3(\log (x))} \, dx\)

Optimal. Leaf size=28 \[ 4+(-3+x) \left (2-\frac {2 x}{\frac {e^x}{4+x}+\log (\log (x))}\right )^2 \]

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Rubi [F]  time = 17.36, 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 {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1536*x + 640*x^2 - 96*x^3 - 72*x^4 - 8*x^5 + E^x*(-384 - 64*x + 40*x^2 + 8*x^3) + (4*E^(3*x) + E^(2*x)*(9
6 - 112*x - 16*x^2 + 8*x^3) + E^x*(-384*x + 288*x^2 + 144*x^3 - 20*x^4 - 8*x^5))*Log[x] + (-1536 - 640*x + 96*
x^2 + 72*x^3 + 8*x^4 + (-1536*x - 384*x^2 + 288*x^3 + 120*x^4 + 12*x^5 + E^(2*x)*(48 + 12*x) + E^x*(768 - 416*
x - 280*x^2 + 8*x^4))*Log[x])*Log[Log[x]] + (1536 + 128*x - 480*x^2 - 168*x^3 - 16*x^4 + E^x*(192 + 96*x + 12*
x^2))*Log[x]*Log[Log[x]]^2 + (256 + 192*x + 48*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^3)/(E^(3*x)*Log[x] + E^(2*x)*(1
2 + 3*x)*Log[x]*Log[Log[x]] + E^x*(48 + 24*x + 3*x^2)*Log[x]*Log[Log[x]]^2 + (64 + 48*x + 12*x^2 + x^3)*Log[x]
*Log[Log[x]]^3),x]

[Out]

4*x + 1536*Defer[Int][x/(Log[x]*(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^3), x] + 640*Defer[Int][x^2/(Log[x]*(E^x
+ 4*Log[Log[x]] + x*Log[Log[x]])^3), x] - 96*Defer[Int][x^3/(Log[x]*(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^3),
x] - 72*Defer[Int][x^4/(Log[x]*(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^3), x] - 8*Defer[Int][x^5/(Log[x]*(E^x +
4*Log[Log[x]] + x*Log[Log[x]])^3), x] - 1152*Defer[Int][(x^2*Log[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log[Log[x]
])^3, x] - 576*Defer[Int][(x^3*Log[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^3, x] + 56*Defer[Int][(x^4*L
og[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^3, x] + 64*Defer[Int][(x^5*Log[Log[x]])/(E^x + 4*Log[Log[x]]
+ x*Log[Log[x]])^3, x] + 8*Defer[Int][(x^6*Log[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^3, x] - 384*Def
er[Int][x/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2, x] + 288*Defer[Int][x^2/(E^x + 4*Log[Log[x]] + x*Log[Log[x]
])^2, x] + 144*Defer[Int][x^3/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2, x] - 20*Defer[Int][x^4/(E^x + 4*Log[Log
[x]] + x*Log[Log[x]])^2, x] - 8*Defer[Int][x^5/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2, x] - 384*Defer[Int][1/
(Log[x]*(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2), x] - 64*Defer[Int][x/(Log[x]*(E^x + 4*Log[Log[x]] + x*Log[Lo
g[x]])^2), x] + 40*Defer[Int][x^2/(Log[x]*(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2), x] + 8*Defer[Int][x^3/(Log
[x]*(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2), x] + 288*Defer[Int][(x*Log[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log
[Log[x]])^2, x] + 72*Defer[Int][(x^2*Log[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2, x] - 32*Defer[Int][
(x^3*Log[Log[x]])/(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^2, x] - 8*Defer[Int][(x^4*Log[Log[x]])/(E^x + 4*Log[Lo
g[x]] + x*Log[Log[x]])^2, x] + 96*Defer[Int][(E^x + 4*Log[Log[x]] + x*Log[Log[x]])^(-1), x] - 112*Defer[Int][x
/(E^x + 4*Log[Log[x]] + x*Log[Log[x]]), x] - 16*Defer[Int][x^2/(E^x + 4*Log[Log[x]] + x*Log[Log[x]]), x] + 8*D
efer[Int][x^3/(E^x + 4*Log[Log[x]] + x*Log[Log[x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (e^x-x (4+x)+(4+x) \log (\log (x))\right ) \left (2 (-3+x) (4+x)^2+\log (x) \left (e^x \left (24+e^x-24 x-3 x^2+2 x^3\right )+(4+x) \left (24+2 e^x-6 x-3 x^2\right ) \log (\log (x))+(4+x)^2 \log ^2(\log (x))\right )\right )}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx\\ &=4 \int \frac {\left (e^x-x (4+x)+(4+x) \log (\log (x))\right ) \left (2 (-3+x) (4+x)^2+\log (x) \left (e^x \left (24+e^x-24 x-3 x^2+2 x^3\right )+(4+x) \left (24+2 e^x-6 x-3 x^2\right ) \log (\log (x))+(4+x)^2 \log ^2(\log (x))\right )\right )}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx\\ &=4 \int \left (1+\frac {2 \left (12-14 x-2 x^2+x^3\right )}{e^x+4 \log (\log (x))+x \log (\log (x))}+\frac {2 (-3+x) x (4+x)^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}-\frac {(4+x) \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2}\right ) \, dx\\ &=4 x-4 \int \frac {(4+x) \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2} \, dx+8 \int \frac {12-14 x-2 x^2+x^3}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+8 \int \frac {(-3+x) x (4+x)^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx\\ &=4 x-4 \int \frac {(4+x) \left (-2 \left (-12+x+x^2\right )+x \log (x) \left (24-24 x-3 x^2+2 x^3+2 \left (-9+x^2\right ) \log (\log (x))\right )\right )}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^2} \, dx+8 \int \left (\frac {12}{e^x+4 \log (\log (x))+x \log (\log (x))}-\frac {14 x}{e^x+4 \log (\log (x))+x \log (\log (x))}-\frac {2 x^2}{e^x+4 \log (\log (x))+x \log (\log (x))}+\frac {x^3}{e^x+4 \log (\log (x))+x \log (\log (x))}\right ) \, dx+8 \int \left (-\frac {48 x \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}-\frac {8 x^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}+\frac {5 x^3 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}+\frac {x^4 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}\right ) \, dx\\ &=4 x-4 \int \left (\frac {4 \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2}+\frac {x \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2}\right ) \, dx+8 \int \frac {x^3}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+8 \int \frac {x^4 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx-16 \int \frac {x^2}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+40 \int \frac {x^3 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx-64 \int \frac {x^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx+96 \int \frac {1}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-112 \int \frac {x}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-384 \int \frac {x \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx\\ &=4 x-4 \int \frac {x \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2} \, dx+8 \int \frac {x^3}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+8 \int \frac {x^4 (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx-16 \int \frac {x^2}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-16 \int \frac {24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2} \, dx+40 \int \frac {x^3 (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx-64 \int \frac {x^2 (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx+96 \int \frac {1}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-112 \int \frac {x}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-384 \int \frac {x (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 49, normalized size = 1.75 \begin {gather*} 4 x \left (1+\frac {(-3+x) x (4+x)^2}{\left (e^x+(4+x) \log (\log (x))\right )^2}-\frac {2 (-3+x) (4+x)}{e^x+(4+x) \log (\log (x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1536*x + 640*x^2 - 96*x^3 - 72*x^4 - 8*x^5 + E^x*(-384 - 64*x + 40*x^2 + 8*x^3) + (4*E^(3*x) + E^(2
*x)*(96 - 112*x - 16*x^2 + 8*x^3) + E^x*(-384*x + 288*x^2 + 144*x^3 - 20*x^4 - 8*x^5))*Log[x] + (-1536 - 640*x
+ 96*x^2 + 72*x^3 + 8*x^4 + (-1536*x - 384*x^2 + 288*x^3 + 120*x^4 + 12*x^5 + E^(2*x)*(48 + 12*x) + E^x*(768
- 416*x - 280*x^2 + 8*x^4))*Log[x])*Log[Log[x]] + (1536 + 128*x - 480*x^2 - 168*x^3 - 16*x^4 + E^x*(192 + 96*x
+ 12*x^2))*Log[x]*Log[Log[x]]^2 + (256 + 192*x + 48*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^3)/(E^(3*x)*Log[x] + E^(2
*x)*(12 + 3*x)*Log[x]*Log[Log[x]] + E^x*(48 + 24*x + 3*x^2)*Log[x]*Log[Log[x]]^2 + (64 + 48*x + 12*x^2 + x^3)*
Log[x]*Log[Log[x]]^3),x]

[Out]

4*x*(1 + ((-3 + x)*x*(4 + x)^2)/(E^x + (4 + x)*Log[Log[x]])^2 - (2*(-3 + x)*(4 + x))/(E^x + (4 + x)*Log[Log[x]
]))

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fricas [B]  time = 0.53, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \, {\left (x^{5} + 5 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (\log \relax (x)\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} - 12 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 5 \, x^{3} - 8 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 48 \, x\right )} \log \left (\log \relax (x)\right )\right )}}{2 \, {\left (x + 4\right )} e^{x} \log \left (\log \relax (x)\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (\log \relax (x)\right )^{2} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192)*exp(x)-16*x^4-168*x^3-480*x^2+128*
x+1536)*log(x)*log(log(x))^2+(((12*x+48)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*
x^2-1536*x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+(8*x^3-16*x^2-112*x+96)*exp(x)^2+(-
8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp(x))*log(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+
1536*x)/((x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log(x)*log(log(x))^2+(3*x+12)*exp(x)
^2*log(x)*log(log(x))+exp(x)^3*log(x)),x, algorithm="fricas")

[Out]

4*(x^5 + 5*x^4 - 8*x^3 + (x^3 + 8*x^2 + 16*x)*log(log(x))^2 - 48*x^2 + x*e^(2*x) - 2*(x^3 + x^2 - 12*x)*e^x -
2*(x^4 + 5*x^3 - 8*x^2 - (x^2 + 4*x)*e^x - 48*x)*log(log(x)))/(2*(x + 4)*e^x*log(log(x)) + (x^2 + 8*x + 16)*lo
g(log(x))^2 + e^(2*x))

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giac [B]  time = 0.59, size = 167, normalized size = 5.96 \begin {gather*} \frac {4 \, {\left (x^{5} - 2 \, x^{4} \log \left (\log \relax (x)\right ) + x^{3} \log \left (\log \relax (x)\right )^{2} + 5 \, x^{4} - 2 \, x^{3} e^{x} - 10 \, x^{3} \log \left (\log \relax (x)\right ) + 2 \, x^{2} e^{x} \log \left (\log \relax (x)\right ) + 8 \, x^{2} \log \left (\log \relax (x)\right )^{2} - 8 \, x^{3} - 2 \, x^{2} e^{x} + 16 \, x^{2} \log \left (\log \relax (x)\right ) + 8 \, x e^{x} \log \left (\log \relax (x)\right ) + 16 \, x \log \left (\log \relax (x)\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} + 24 \, x e^{x} + 96 \, x \log \left (\log \relax (x)\right )\right )}}{x^{2} \log \left (\log \relax (x)\right )^{2} + 2 \, x e^{x} \log \left (\log \relax (x)\right ) + 8 \, x \log \left (\log \relax (x)\right )^{2} + 8 \, e^{x} \log \left (\log \relax (x)\right ) + 16 \, \log \left (\log \relax (x)\right )^{2} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192)*exp(x)-16*x^4-168*x^3-480*x^2+128*
x+1536)*log(x)*log(log(x))^2+(((12*x+48)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*
x^2-1536*x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+(8*x^3-16*x^2-112*x+96)*exp(x)^2+(-
8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp(x))*log(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+
1536*x)/((x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log(x)*log(log(x))^2+(3*x+12)*exp(x)
^2*log(x)*log(log(x))+exp(x)^3*log(x)),x, algorithm="giac")

[Out]

4*(x^5 - 2*x^4*log(log(x)) + x^3*log(log(x))^2 + 5*x^4 - 2*x^3*e^x - 10*x^3*log(log(x)) + 2*x^2*e^x*log(log(x)
) + 8*x^2*log(log(x))^2 - 8*x^3 - 2*x^2*e^x + 16*x^2*log(log(x)) + 8*x*e^x*log(log(x)) + 16*x*log(log(x))^2 -
48*x^2 + x*e^(2*x) + 24*x*e^x + 96*x*log(log(x)))/(x^2*log(log(x))^2 + 2*x*e^x*log(log(x)) + 8*x*log(log(x))^2
+ 8*e^x*log(log(x)) + 16*log(log(x))^2 + e^(2*x))

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maple [B]  time = 0.07, size = 64, normalized size = 2.29

method	result	size
risch	\(4 x +\frac {4 \left (x^{3}-2 x^{2} \ln \left (\ln \relax (x )\right )+x^{2}-2 \,{\mathrm e}^{x} x -2 x \ln \left (\ln \relax (x )\right )-12 x +6 \,{\mathrm e}^{x}+24 \ln \left (\ln \relax (x )\right )\right ) \left (4+x \right ) x}{\left (x \ln \left (\ln \relax (x )\right )+{\mathrm e}^{x}+4 \ln \left (\ln \relax (x )\right )\right )^{2}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+48*x^2+192*x+256)*ln(x)*ln(ln(x))^3+((12*x^2+96*x+192)*exp(x)-16*x^4-168*x^3-480*x^2+128*x+1536)*l
n(x)*ln(ln(x))^2+(((12*x+48)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*x^2-1536*x)*
ln(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*ln(ln(x))+(4*exp(x)^3+(8*x^3-16*x^2-112*x+96)*exp(x)^2+(-8*x^5-20*x^4+14
4*x^3+288*x^2-384*x)*exp(x))*ln(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+1536*x)/((x^3+12
*x^2+48*x+64)*ln(x)*ln(ln(x))^3+(3*x^2+24*x+48)*exp(x)*ln(x)*ln(ln(x))^2+(3*x+12)*exp(x)^2*ln(x)*ln(ln(x))+exp
(x)^3*ln(x)),x,method=_RETURNVERBOSE)

[Out]

4*x+4*(x^3-2*x^2*ln(ln(x))+x^2-2*exp(x)*x-2*x*ln(ln(x))-12*x+6*exp(x)+24*ln(ln(x)))*(4+x)*x/(x*ln(ln(x))+exp(x
)+4*ln(ln(x)))^2

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maxima [B]  time = 0.54, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \, {\left (x^{5} + 5 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (\log \relax (x)\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} - 12 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 5 \, x^{3} - 8 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 48 \, x\right )} \log \left (\log \relax (x)\right )\right )}}{2 \, {\left (x + 4\right )} e^{x} \log \left (\log \relax (x)\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (\log \relax (x)\right )^{2} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+48*x^2+192*x+256)*log(x)*log(log(x))^3+((12*x^2+96*x+192)*exp(x)-16*x^4-168*x^3-480*x^2+128*
x+1536)*log(x)*log(log(x))^2+(((12*x+48)*exp(x)^2+(8*x^4-280*x^2-416*x+768)*exp(x)+12*x^5+120*x^4+288*x^3-384*
x^2-1536*x)*log(x)+8*x^4+72*x^3+96*x^2-640*x-1536)*log(log(x))+(4*exp(x)^3+(8*x^3-16*x^2-112*x+96)*exp(x)^2+(-
8*x^5-20*x^4+144*x^3+288*x^2-384*x)*exp(x))*log(x)+(8*x^3+40*x^2-64*x-384)*exp(x)-8*x^5-72*x^4-96*x^3+640*x^2+
1536*x)/((x^3+12*x^2+48*x+64)*log(x)*log(log(x))^3+(3*x^2+24*x+48)*exp(x)*log(x)*log(log(x))^2+(3*x+12)*exp(x)
^2*log(x)*log(log(x))+exp(x)^3*log(x)),x, algorithm="maxima")

[Out]

4*(x^5 + 5*x^4 - 8*x^3 + (x^3 + 8*x^2 + 16*x)*log(log(x))^2 - 48*x^2 + x*e^(2*x) - 2*(x^3 + x^2 - 12*x)*e^x -
2*(x^4 + 5*x^3 - 8*x^2 - (x^2 + 4*x)*e^x - 48*x)*log(log(x)))/(2*(x + 4)*e^x*log(log(x)) + (x^2 + 8*x + 16)*lo
g(log(x))^2 + e^(2*x))

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mupad [B]  time = 8.70, size = 121, normalized size = 4.32 \begin {gather*} \frac {4\,x\,\left ({\mathrm {e}}^{2\,x}-48\,x+96\,\ln \left (\ln \relax (x)\right )+24\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^x+16\,x\,\ln \left (\ln \relax (x)\right )+8\,x\,{\ln \left (\ln \relax (x)\right )}^2-10\,x^2\,\ln \left (\ln \relax (x)\right )-2\,x^3\,\ln \left (\ln \relax (x)\right )+8\,\ln \left (\ln \relax (x)\right )\,{\mathrm {e}}^x+16\,{\ln \left (\ln \relax (x)\right )}^2-2\,x\,{\mathrm {e}}^x-8\,x^2+5\,x^3+x^4+x^2\,{\ln \left (\ln \relax (x)\right )}^2+2\,x\,\ln \left (\ln \relax (x)\right )\,{\mathrm {e}}^x\right )}{{\left (4\,\ln \left (\ln \relax (x)\right )+{\mathrm {e}}^x+x\,\ln \left (\ln \relax (x)\right )\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1536*x + log(log(x))*(log(x)*(exp(2*x)*(12*x + 48) - 1536*x - 384*x^2 + 288*x^3 + 120*x^4 + 12*x^5 - exp(
x)*(416*x + 280*x^2 - 8*x^4 - 768)) - 640*x + 96*x^2 + 72*x^3 + 8*x^4 - 1536) - log(x)*(exp(2*x)*(112*x + 16*x
^2 - 8*x^3 - 96) - 4*exp(3*x) + exp(x)*(384*x - 288*x^2 - 144*x^3 + 20*x^4 + 8*x^5)) + 640*x^2 - 96*x^3 - 72*x
^4 - 8*x^5 - exp(x)*(64*x - 40*x^2 - 8*x^3 + 384) + log(log(x))^2*log(x)*(128*x + exp(x)*(96*x + 12*x^2 + 192)
- 480*x^2 - 168*x^3 - 16*x^4 + 1536) + log(log(x))^3*log(x)*(192*x + 48*x^2 + 4*x^3 + 256))/(exp(3*x)*log(x)
+ log(log(x))^3*log(x)*(48*x + 12*x^2 + x^3 + 64) + log(log(x))*exp(2*x)*log(x)*(3*x + 12) + log(log(x))^2*exp
(x)*log(x)*(24*x + 3*x^2 + 48)),x)

[Out]

(4*x*(exp(2*x) - 48*x + 96*log(log(x)) + 24*exp(x) - 2*x^2*exp(x) + 16*x*log(log(x)) + 8*x*log(log(x))^2 - 10*
x^2*log(log(x)) - 2*x^3*log(log(x)) + 8*log(log(x))*exp(x) + 16*log(log(x))^2 - 2*x*exp(x) - 8*x^2 + 5*x^3 + x
^4 + x^2*log(log(x))^2 + 2*x*log(log(x))*exp(x)))/(4*log(log(x)) + exp(x) + x*log(log(x)))^2

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sympy [B]  time = 0.76, size = 131, normalized size = 4.68 \begin {gather*} 4 x + \frac {4 x^{5} - 8 x^{4} \log {\left (\log {\relax (x )} \right )} + 20 x^{4} - 40 x^{3} \log {\left (\log {\relax (x )} \right )} - 32 x^{3} + 64 x^{2} \log {\left (\log {\relax (x )} \right )} - 192 x^{2} + 384 x \log {\left (\log {\relax (x )} \right )} + \left (- 8 x^{3} - 8 x^{2} + 96 x\right ) e^{x}}{x^{2} \log {\left (\log {\relax (x )} \right )}^{2} + 8 x \log {\left (\log {\relax (x )} \right )}^{2} + \left (2 x \log {\left (\log {\relax (x )} \right )} + 8 \log {\left (\log {\relax (x )} \right )}\right ) e^{x} + e^{2 x} + 16 \log {\left (\log {\relax (x )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+48*x**2+192*x+256)*ln(x)*ln(ln(x))**3+((12*x**2+96*x+192)*exp(x)-16*x**4-168*x**3-480*x**2+
128*x+1536)*ln(x)*ln(ln(x))**2+(((12*x+48)*exp(x)**2+(8*x**4-280*x**2-416*x+768)*exp(x)+12*x**5+120*x**4+288*x
**3-384*x**2-1536*x)*ln(x)+8*x**4+72*x**3+96*x**2-640*x-1536)*ln(ln(x))+(4*exp(x)**3+(8*x**3-16*x**2-112*x+96)
*exp(x)**2+(-8*x**5-20*x**4+144*x**3+288*x**2-384*x)*exp(x))*ln(x)+(8*x**3+40*x**2-64*x-384)*exp(x)-8*x**5-72*
x**4-96*x**3+640*x**2+1536*x)/((x**3+12*x**2+48*x+64)*ln(x)*ln(ln(x))**3+(3*x**2+24*x+48)*exp(x)*ln(x)*ln(ln(x
))**2+(3*x+12)*exp(x)**2*ln(x)*ln(ln(x))+exp(x)**3*ln(x)),x)

[Out]

4*x + (4*x**5 - 8*x**4*log(log(x)) + 20*x**4 - 40*x**3*log(log(x)) - 32*x**3 + 64*x**2*log(log(x)) - 192*x**2
+ 384*x*log(log(x)) + (-8*x**3 - 8*x**2 + 96*x)*exp(x))/(x**2*log(log(x))**2 + 8*x*log(log(x))**2 + (2*x*log(l
og(x)) + 8*log(log(x)))*exp(x) + exp(2*x) + 16*log(log(x))**2)

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