3.2.60 \(\int \frac {e^{\frac {2 (3 x-3 x^3+(x-x^3) \log (4) \log (x^2))}{8+3 x+x \log (4) \log (x^2)}} (48-144 x^2-36 x^3+(32-32 x^2) \log (4)+(16-48 x^2-24 x^3) \log (4) \log (x^2)-4 x^3 \log ^2(4) \log ^2(x^2))}{64+48 x+9 x^2+(16 x+6 x^2) \log (4) \log (x^2)+x^2 \log ^2(4) \log ^2(x^2)} \, dx\) [160]

Optimal. Leaf size=28 \[ e^{\frac {2 \left (x-x^3\right )}{x+\frac {8}{3+\log (4) \log \left (x^2\right )}}} \]

[Out]

exp((-x^3+x)/(8/(2*ln(2)*ln(x^2)+3)+x))^2

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Rubi [F]
time = 4.73, 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 {2 \left (3 x-3 x^3+\left (x-x^3\right ) \log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \left (48-144 x^2-36 x^3+\left (32-32 x^2\right ) \log (4)+\left (16-48 x^2-24 x^3\right ) \log (4) \log \left (x^2\right )-4 x^3 \log ^2(4) \log ^2\left (x^2\right )\right )}{64+48 x+9 x^2+\left (16 x+6 x^2\right ) \log (4) \log \left (x^2\right )+x^2 \log ^2(4) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(3*x - 3*x^3 + (x - x^3)*Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*(48 - 144*x^2 - 36*x^3 +
(32 - 32*x^2)*Log[4] + (16 - 48*x^2 - 24*x^3)*Log[4]*Log[x^2] - 4*x^3*Log[4]^2*Log[x^2]^2))/(64 + 48*x + 9*x^2
+ (16*x + 6*x^2)*Log[4]*Log[x^2] + x^2*Log[4]^2*Log[x^2]^2),x]

[Out]

-4*Defer[Int][x/E^((2*x*(-1 + x^2)*(3 + Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2])), x] + 32*Log[4]*Defer
[Int][1/(E^((2*x*(-1 + x^2)*(3 + Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*(8 + 3*x + x*Log[4]*Log[x^2]
)^2), x] - 128*Defer[Int][1/(E^((2*x*(-1 + x^2)*(3 + Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*x*(8 + 3
*x + x*Log[4]*Log[x^2])^2), x] + 128*Defer[Int][x/(E^((2*x*(-1 + x^2)*(3 + Log[4]*Log[x^2]))/(8 + 3*x + x*Log[
4]*Log[x^2]))*(8 + 3*x + x*Log[4]*Log[x^2])^2), x] - 32*Log[4]*Defer[Int][x^2/(E^((2*x*(-1 + x^2)*(3 + Log[4]*
Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*(8 + 3*x + x*Log[4]*Log[x^2])^2), x] + 16*Defer[Int][1/(E^((2*x*(-1
+ x^2)*(3 + Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*x*(8 + 3*x + x*Log[4]*Log[x^2])), x] + 16*Defer[I
nt][x/(E^((2*x*(-1 + x^2)*(3 + Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*(8 + 3*x + x*Log[4]*Log[x^2]))
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \left (48-144 x^2-36 x^3+\left (32-32 x^2\right ) \log (4)+\left (16-48 x^2-24 x^3\right ) \log (4) \log \left (x^2\right )-4 x^3 \log ^2(4) \log ^2\left (x^2\right )\right )}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (-4 \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x-\frac {32 \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \left (-1+x^2\right ) (-4+x \log (4))}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2}+\frac {16 \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \left (1+x^2\right )}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )}\right ) \, dx\\ &=-\left (4 \int \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x \, dx\right )+16 \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \left (1+x^2\right )}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )} \, dx-32 \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \left (-1+x^2\right ) (-4+x \log (4))}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2} \, dx\\ &=-\left (4 \int \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x \, dx\right )+16 \int \left (\frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right )}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )}+\frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \, dx-32 \int \left (\frac {4 \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right )}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2}-\frac {4 \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2}-\frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) \log (4)}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2}+\frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x^2 \log (4)}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x \, dx\right )+16 \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right )}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )} \, dx+16 \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x}{8+3 x+x \log (4) \log \left (x^2\right )} \, dx-128 \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right )}{x \left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2} \, dx+128 \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2} \, dx+(32 \log (4)) \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right )}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2} \, dx-(32 \log (4)) \int \frac {\exp \left (-\frac {2 x \left (-1+x^2\right ) \left (3+\log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}\right ) x^2}{\left (8+3 x+x \log (4) \log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]
time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {2 \left (3 x-3 x^3+\left (x-x^3\right ) \log (4) \log \left (x^2\right )\right )}{8+3 x+x \log (4) \log \left (x^2\right )}} \left (48-144 x^2-36 x^3+\left (32-32 x^2\right ) \log (4)+\left (16-48 x^2-24 x^3\right ) \log (4) \log \left (x^2\right )-4 x^3 \log ^2(4) \log ^2\left (x^2\right )\right )}{64+48 x+9 x^2+\left (16 x+6 x^2\right ) \log (4) \log \left (x^2\right )+x^2 \log ^2(4) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((2*(3*x - 3*x^3 + (x - x^3)*Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*(48 - 144*x^2 - 36*
x^3 + (32 - 32*x^2)*Log[4] + (16 - 48*x^2 - 24*x^3)*Log[4]*Log[x^2] - 4*x^3*Log[4]^2*Log[x^2]^2))/(64 + 48*x +
9*x^2 + (16*x + 6*x^2)*Log[4]*Log[x^2] + x^2*Log[4]^2*Log[x^2]^2),x]

[Out]

Integrate[(E^((2*(3*x - 3*x^3 + (x - x^3)*Log[4]*Log[x^2]))/(8 + 3*x + x*Log[4]*Log[x^2]))*(48 - 144*x^2 - 36*
x^3 + (32 - 32*x^2)*Log[4] + (16 - 48*x^2 - 24*x^3)*Log[4]*Log[x^2] - 4*x^3*Log[4]^2*Log[x^2]^2))/(64 + 48*x +
9*x^2 + (16*x + 6*x^2)*Log[4]*Log[x^2] + x^2*Log[4]^2*Log[x^2]^2), x]

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Maple [A]
time = 3.00, size = 37, normalized size = 1.32

method	result	size
risch	\({\mathrm e}^{-\frac {2 x \left (x -1\right ) \left (x +1\right ) \left (2 \ln \left (2\right ) \ln \left (x^{2}\right )+3\right )}{2 x \ln \left (2\right ) \ln \left (x^{2}\right )+3 x +8}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x^3*ln(2)^2*ln(x^2)^2+2*(-24*x^3-48*x^2+16)*ln(2)*ln(x^2)+2*(-32*x^2+32)*ln(2)-36*x^3-144*x^2+48)*exp
((2*(-x^3+x)*ln(2)*ln(x^2)-3*x^3+3*x)/(2*x*ln(2)*ln(x^2)+3*x+8))^2/(4*x^2*ln(2)^2*ln(x^2)^2+2*(6*x^2+16*x)*ln(
2)*ln(x^2)+9*x^2+48*x+64),x,method=_RETURNVERBOSE)

[Out]

exp(-2*x*(x-1)*(x+1)*(2*ln(2)*ln(x^2)+3)/(2*x*ln(2)*ln(x^2)+3*x+8))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (28) = 56\).
time = 13.97, size = 591, normalized size = 21.11 \begin {gather*} e^{\left (-\frac {128 \, x^{2} \log \left (2\right )^{3} \log \left (x\right )^{3}}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} - \frac {288 \, x^{2} \log \left (2\right )^{2} \log \left (x\right )^{2}}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} + \frac {256 \, x \log \left (2\right )^{2} \log \left (x\right )^{2}}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} - \frac {216 \, x^{2} \log \left (2\right ) \log \left (x\right )}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} + \frac {384 \, x \log \left (2\right ) \log \left (x\right )}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} - \frac {54 \, x^{2}}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} + \frac {4096 \, \log \left (2\right ) \log \left (x\right )}{256 \, x \log \left (2\right )^{4} \log \left (x\right )^{4} + 256 \, {\left (3 \, x \log \left (2\right )^{3} + 2 \, \log \left (2\right )^{3}\right )} \log \left (x\right )^{3} + 288 \, {\left (3 \, x \log \left (2\right )^{2} + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} + 432 \, {\left (x \log \left (2\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right ) + 81 \, x + 216} - \frac {512 \, \log \left (2\right ) \log \left (x\right )}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} - \frac {64 \, \log \left (2\right ) \log \left (x\right )}{16 \, x \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, {\left (3 \, x \log \left (2\right ) + 4 \, \log \left (2\right )\right )} \log \left (x\right ) + 9 \, x + 24} + \frac {8 \, \log \left (2\right ) \log \left (x\right )}{4 \, \log \left (2\right ) \log \left (x\right ) + 3} + \frac {144 \, x}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} + \frac {3072}{256 \, x \log \left (2\right )^{4} \log \left (x\right )^{4} + 256 \, {\left (3 \, x \log \left (2\right )^{3} + 2 \, \log \left (2\right )^{3}\right )} \log \left (x\right )^{3} + 288 \, {\left (3 \, x \log \left (2\right )^{2} + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} + 432 \, {\left (x \log \left (2\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right ) + 81 \, x + 216} - \frac {384}{64 \, \log \left (2\right )^{3} \log \left (x\right )^{3} + 144 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (2\right ) \log \left (x\right ) + 27} - \frac {48}{16 \, x \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, {\left (3 \, x \log \left (2\right ) + 4 \, \log \left (2\right )\right )} \log \left (x\right ) + 9 \, x + 24} + \frac {6}{4 \, \log \left (2\right ) \log \left (x\right ) + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^3*log(2)^2*log(x^2)^2+2*(-24*x^3-48*x^2+16)*log(2)*log(x^2)+2*(-32*x^2+32)*log(2)-36*x^3-144*
x^2+48)*exp((2*(-x^3+x)*log(2)*log(x^2)-3*x^3+3*x)/(2*x*log(2)*log(x^2)+3*x+8))^2/(4*x^2*log(2)^2*log(x^2)^2+2
*(6*x^2+16*x)*log(2)*log(x^2)+9*x^2+48*x+64),x, algorithm="maxima")

[Out]

e^(-128*x^2*log(2)^3*log(x)^3/(64*log(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) - 288*x^
2*log(2)^2*log(x)^2/(64*log(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) + 256*x*log(2)^2*l
og(x)^2/(64*log(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) - 216*x^2*log(2)*log(x)/(64*lo
g(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) + 384*x*log(2)*log(x)/(64*log(2)^3*log(x)^3
+ 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) - 54*x^2/(64*log(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108
*log(2)*log(x) + 27) + 4096*log(2)*log(x)/(256*x*log(2)^4*log(x)^4 + 256*(3*x*log(2)^3 + 2*log(2)^3)*log(x)^3
+ 288*(3*x*log(2)^2 + 4*log(2)^2)*log(x)^2 + 432*(x*log(2) + 2*log(2))*log(x) + 81*x + 216) - 512*log(2)*log(x
)/(64*log(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) - 64*log(2)*log(x)/(16*x*log(2)^2*lo
g(x)^2 + 8*(3*x*log(2) + 4*log(2))*log(x) + 9*x + 24) + 8*log(2)*log(x)/(4*log(2)*log(x) + 3) + 144*x/(64*log(
2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) + 3072/(256*x*log(2)^4*log(x)^4 + 256*(3*x*log
(2)^3 + 2*log(2)^3)*log(x)^3 + 288*(3*x*log(2)^2 + 4*log(2)^2)*log(x)^2 + 432*(x*log(2) + 2*log(2))*log(x) + 8
1*x + 216) - 384/(64*log(2)^3*log(x)^3 + 144*log(2)^2*log(x)^2 + 108*log(2)*log(x) + 27) - 48/(16*x*log(2)^2*l
og(x)^2 + 8*(3*x*log(2) + 4*log(2))*log(x) + 9*x + 24) + 6/(4*log(2)*log(x) + 3))

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Fricas [A]
time = 0.31, size = 43, normalized size = 1.54 \begin {gather*} e^{\left (-\frac {2 \, {\left (3 \, x^{3} + 2 \, {\left (x^{3} - x\right )} \log \left (2\right ) \log \left (x^{2}\right ) - 3 \, x\right )}}{2 \, x \log \left (2\right ) \log \left (x^{2}\right ) + 3 \, x + 8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^3*log(2)^2*log(x^2)^2+2*(-24*x^3-48*x^2+16)*log(2)*log(x^2)+2*(-32*x^2+32)*log(2)-36*x^3-144*
x^2+48)*exp((2*(-x^3+x)*log(2)*log(x^2)-3*x^3+3*x)/(2*x*log(2)*log(x^2)+3*x+8))^2/(4*x^2*log(2)^2*log(x^2)^2+2
*(6*x^2+16*x)*log(2)*log(x^2)+9*x^2+48*x+64),x, algorithm="fricas")

[Out]

e^(-2*(3*x^3 + 2*(x^3 - x)*log(2)*log(x^2) - 3*x)/(2*x*log(2)*log(x^2) + 3*x + 8))

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Sympy [A]
time = 0.37, size = 44, normalized size = 1.57 \begin {gather*} e^{\frac {2 \left (- 3 x^{3} + 3 x + \left (- 2 x^{3} + 2 x\right ) \log {\left (2 \right )} \log {\left (x^{2} \right )}\right )}{2 x \log {\left (2 \right )} \log {\left (x^{2} \right )} + 3 x + 8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**3*ln(2)**2*ln(x**2)**2+2*(-24*x**3-48*x**2+16)*ln(2)*ln(x**2)+2*(-32*x**2+32)*ln(2)-36*x**3-
144*x**2+48)*exp((2*(-x**3+x)*ln(2)*ln(x**2)-3*x**3+3*x)/(2*x*ln(2)*ln(x**2)+3*x+8))**2/(4*x**2*ln(2)**2*ln(x*
*2)**2+2*(6*x**2+16*x)*ln(2)*ln(x**2)+9*x**2+48*x+64),x)

[Out]

exp(2*(-3*x**3 + 3*x + (-2*x**3 + 2*x)*log(2)*log(x**2))/(2*x*log(2)*log(x**2) + 3*x + 8))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (28) = 56\).
time = 0.59, size = 94, normalized size = 3.36 \begin {gather*} e^{\left (-\frac {4 \, x^{3} \log \left (2\right ) \log \left (x^{2}\right )}{2 \, x \log \left (2\right ) \log \left (x^{2}\right ) + 3 \, x + 8} - \frac {6 \, x^{3}}{2 \, x \log \left (2\right ) \log \left (x^{2}\right ) + 3 \, x + 8} + \frac {4 \, x \log \left (2\right ) \log \left (x^{2}\right )}{2 \, x \log \left (2\right ) \log \left (x^{2}\right ) + 3 \, x + 8} + \frac {6 \, x}{2 \, x \log \left (2\right ) \log \left (x^{2}\right ) + 3 \, x + 8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^3*log(2)^2*log(x^2)^2+2*(-24*x^3-48*x^2+16)*log(2)*log(x^2)+2*(-32*x^2+32)*log(2)-36*x^3-144*
x^2+48)*exp((2*(-x^3+x)*log(2)*log(x^2)-3*x^3+3*x)/(2*x*log(2)*log(x^2)+3*x+8))^2/(4*x^2*log(2)^2*log(x^2)^2+2
*(6*x^2+16*x)*log(2)*log(x^2)+9*x^2+48*x+64),x, algorithm="giac")

[Out]

e^(-4*x^3*log(2)*log(x^2)/(2*x*log(2)*log(x^2) + 3*x + 8) - 6*x^3/(2*x*log(2)*log(x^2) + 3*x + 8) + 4*x*log(2)
*log(x^2)/(2*x*log(2)*log(x^2) + 3*x + 8) + 6*x/(2*x*log(2)*log(x^2) + 3*x + 8))

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Mupad [B]
time = 0.88, size = 76, normalized size = 2.71 \begin {gather*} {\mathrm {e}}^{\frac {6\,x}{3\,x+2\,x\,\ln \left (x^2\right )\,\ln \left (2\right )+8}}\,{\mathrm {e}}^{-\frac {6\,x^3}{3\,x+2\,x\,\ln \left (x^2\right )\,\ln \left (2\right )+8}}\,{\left (x^8\right )}^{\frac {x\,\ln \left (2\right )-x^3\,\ln \left (2\right )}{3\,x+2\,x\,\ln \left (x^2\right )\,\ln \left (2\right )+8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((2*(3*x - 3*x^3 + 2*log(x^2)*log(2)*(x - x^3)))/(3*x + 2*x*log(x^2)*log(2) + 8))*(2*log(2)*(32*x^2 -
32) + 144*x^2 + 36*x^3 + 16*x^3*log(x^2)^2*log(2)^2 + 2*log(x^2)*log(2)*(48*x^2 + 24*x^3 - 16) - 48))/(48*x +
9*x^2 + 2*log(x^2)*log(2)*(16*x + 6*x^2) + 4*x^2*log(x^2)^2*log(2)^2 + 64),x)

[Out]

exp((6*x)/(3*x + 2*x*log(x^2)*log(2) + 8))*exp(-(6*x^3)/(3*x + 2*x*log(x^2)*log(2) + 8))*(x^8)^((x*log(2) - x^
3*log(2))/(3*x + 2*x*log(x^2)*log(2) + 8))

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