3.45.39 \(\int \frac {e^{(4 x^4-4 x^5+x^6) \log ^2(x)} ((32 x^3-24 x^4+2 x^6) \log (x)+(64 x^3-64 x^4+4 x^5+6 x^6) \log ^2(x))+(-4-2 e^{(4 x^4-4 x^5+x^6) \log ^2(x)}-2 \log (4)) \log (2+e^{(4 x^4-4 x^5+x^6) \log ^2(x)}+\log (4))}{128+96 x+24 x^2+2 x^3+e^{(4 x^4-4 x^5+x^6) \log ^2(x)} (64+48 x+12 x^2+x^3)+(64+48 x+12 x^2+x^3) \log (4)} \, dx\)

Optimal. Leaf size=32 \[ \frac {\log \left (2+e^{x^2 \left (2 x-x^2\right )^2 \log ^2(x)}+\log (4)\right )}{(4+x)^2} \]

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Rubi [F]  time = 37.92, 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^{\left (4 x^4-4 x^5+x^6\right ) \log ^2(x)} \left (\left (32 x^3-24 x^4+2 x^6\right ) \log (x)+\left (64 x^3-64 x^4+4 x^5+6 x^6\right ) \log ^2(x)\right )+\left (-4-2 e^{\left (4 x^4-4 x^5+x^6\right ) \log ^2(x)}-2 \log (4)\right ) \log \left (2+e^{\left (4 x^4-4 x^5+x^6\right ) \log ^2(x)}+\log (4)\right )}{128+96 x+24 x^2+2 x^3+e^{\left (4 x^4-4 x^5+x^6\right ) \log ^2(x)} \left (64+48 x+12 x^2+x^3\right )+\left (64+48 x+12 x^2+x^3\right ) \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2)*((32*x^3 - 24*x^4 + 2*x^6)*Log[x] + (64*x^3 - 64*x^4 + 4*x^5 + 6*x^6)*
Log[x]^2) + (-4 - 2*E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2) - 2*Log[4])*Log[2 + E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2)
+ Log[4]])/(128 + 96*x + 24*x^2 + 2*x^3 + E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2)*(64 + 48*x + 12*x^2 + x^3) + (64
+ 48*x + 12*x^2 + x^3)*Log[4]),x]

[Out]

-4288*x + 74*x^2 - (64*x^3)/27 + x^4/16 + 4288*x*Log[x] - 148*x^2*Log[x] + (64*x^3*Log[x])/9 - (x^4*Log[x])/4
- (1152*x*Log[x])/(4 + x) + 11136*Log[1 + x/4]*Log[x] - 2624*x*Log[x]^2 + 232*x^2*Log[x]^2 - (68*x^3*Log[x]^2)
/3 + (3*x^4*Log[x]^2)/2 - (3072*x*Log[x]^2)/(4 + x) + 13568*Log[1 + x/4]*Log[x]^2 + 1152*Log[4 + x] + 11136*Po
lyLog[2, -1/4*x] + 27136*Log[x]*PolyLog[2, -1/4*x] - 27136*PolyLog[3, -1/4*x] + 336*(1 + Log[2])*Defer[Int][(x
*Log[x])/(-E^((-2 + x)^2*x^4*Log[x]^2) - 2*(1 + Log[2])), x] + 4*(1 + Log[2])*Defer[Int][(x^3*Log[x])/(-E^((-2
+ x)^2*x^4*Log[x]^2) - 2*(1 + Log[2])), x] + 1920*(1 + Log[2])*Defer[Int][Log[x]/(E^((-2 + x)^2*x^4*Log[x]^2)
+ 2*(1 + Log[2])), x] + 9984*(1 + Log[2])*Defer[Int][Log[x]/((-4 - x)*(E^((-2 + x)^2*x^4*Log[x]^2) + 2*(1 + L
og[2]))), x] + 48*(1 + Log[2])*Defer[Int][(x^2*Log[x])/(E^((-2 + x)^2*x^4*Log[x]^2) + 2*(1 + Log[2])), x] + 92
16*(1 + Log[2])*Defer[Int][Log[x]/((4 + x)^2*(E^((-2 + x)^2*x^4*Log[x]^2) + 2*(1 + Log[2]))), x] + 928*(1 + Lo
g[2])*Defer[Int][(x*Log[x]^2)/(-E^((-2 + x)^2*x^4*Log[x]^2) - 2*(1 + Log[2])), x] + 12*(1 + Log[2])*Defer[Int]
[(x^3*Log[x]^2)/(-E^((-2 + x)^2*x^4*Log[x]^2) - 2*(1 + Log[2])), x] + 5248*(1 + Log[2])*Defer[Int][Log[x]^2/(E
^((-2 + x)^2*x^4*Log[x]^2) + 2*(1 + Log[2])), x] + 27136*(1 + Log[2])*Defer[Int][Log[x]^2/((-4 - x)*(E^((-2 +
x)^2*x^4*Log[x]^2) + 2*(1 + Log[2]))), x] + 136*(1 + Log[2])*Defer[Int][(x^2*Log[x]^2)/(E^((-2 + x)^2*x^4*Log[
x]^2) + 2*(1 + Log[2])), x] + 24576*(1 + Log[2])*Defer[Int][Log[x]^2/((4 + x)^2*(E^((-2 + x)^2*x^4*Log[x]^2) +
2*(1 + Log[2]))), x] + 2*Defer[Int][Log[E^((-2 + x)^2*x^4*Log[x]^2) + 2*(1 + Log[2])]/(-4 - x)^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{(-2+x)^2 x^4 \log ^2(x)} (-2+x)^2 x^3 (4+x) \log (x)+2 e^{(-2+x)^2 x^4 \log ^2(x)} x^3 \left (32-32 x+2 x^2+3 x^3\right ) \log ^2(x)-2 \left (2+e^{(-2+x)^2 x^4 \log ^2(x)}+\log (4)\right ) \log \left (2+e^{(-2+x)^2 x^4 \log ^2(x)}+\log (4)\right )}{(4+x)^3 \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )} \, dx\\ &=\int \left (\frac {2 x^3 \log (x) \left (-32 (1+\log (2))+24 x (1+\log (2))-2 x^3 (1+\log (2))-64 (1+\log (2)) \log (x)+64 x (1+\log (2)) \log (x)-4 x^2 (1+\log (2)) \log (x)-6 x^3 (1+\log (2)) \log (x)\right )}{(4+x)^3 \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )}+\frac {2 \left (16 x^3 \log (x)-12 x^4 \log (x)+x^6 \log (x)+32 x^3 \log ^2(x)-32 x^4 \log ^2(x)+2 x^5 \log ^2(x)+3 x^6 \log ^2(x)-\log \left (2+e^{(-2+x)^2 x^4 \log ^2(x)}+\log (4)\right )\right )}{(4+x)^3}\right ) \, dx\\ &=2 \int \frac {x^3 \log (x) \left (-32 (1+\log (2))+24 x (1+\log (2))-2 x^3 (1+\log (2))-64 (1+\log (2)) \log (x)+64 x (1+\log (2)) \log (x)-4 x^2 (1+\log (2)) \log (x)-6 x^3 (1+\log (2)) \log (x)\right )}{(4+x)^3 \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )} \, dx+2 \int \frac {16 x^3 \log (x)-12 x^4 \log (x)+x^6 \log (x)+32 x^3 \log ^2(x)-32 x^4 \log ^2(x)+2 x^5 \log ^2(x)+3 x^6 \log ^2(x)-\log \left (2+e^{(-2+x)^2 x^4 \log ^2(x)}+\log (4)\right )}{(4+x)^3} \, dx\\ &=2 \int \frac {2 (2-x) x^3 (1+\log (2)) \log (x) (-2+x+(-4+3 x) \log (x))}{(4+x)^2 \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )} \, dx+2 \int \left (\frac {16 x^3 \log (x)}{(4+x)^3}-\frac {12 x^4 \log (x)}{(4+x)^3}+\frac {x^6 \log (x)}{(4+x)^3}+\frac {32 x^3 \log ^2(x)}{(4+x)^3}-\frac {32 x^4 \log ^2(x)}{(4+x)^3}+\frac {2 x^5 \log ^2(x)}{(4+x)^3}+\frac {3 x^6 \log ^2(x)}{(4+x)^3}+\frac {\log \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )}{(-4-x)^3}\right ) \, dx\\ &=2 \int \frac {x^6 \log (x)}{(4+x)^3} \, dx+2 \int \frac {\log \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )}{(-4-x)^3} \, dx+4 \int \frac {x^5 \log ^2(x)}{(4+x)^3} \, dx+6 \int \frac {x^6 \log ^2(x)}{(4+x)^3} \, dx-24 \int \frac {x^4 \log (x)}{(4+x)^3} \, dx+32 \int \frac {x^3 \log (x)}{(4+x)^3} \, dx+64 \int \frac {x^3 \log ^2(x)}{(4+x)^3} \, dx-64 \int \frac {x^4 \log ^2(x)}{(4+x)^3} \, dx+(4 (1+\log (2))) \int \frac {(2-x) x^3 \log (x) (-2+x+(-4+3 x) \log (x))}{(4+x)^2 \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )} \, dx\\ &=2 \int \left (-640 \log (x)+96 x \log (x)-12 x^2 \log (x)+x^3 \log (x)+\frac {4096 \log (x)}{(4+x)^3}-\frac {6144 \log (x)}{(4+x)^2}+\frac {3840 \log (x)}{4+x}\right ) \, dx+2 \int \frac {\log \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )}{(-4-x)^3} \, dx+4 \int \left (96 \log ^2(x)-12 x \log ^2(x)+x^2 \log ^2(x)-\frac {1024 \log ^2(x)}{(4+x)^3}+\frac {1280 \log ^2(x)}{(4+x)^2}-\frac {640 \log ^2(x)}{4+x}\right ) \, dx+6 \int \left (-640 \log ^2(x)+96 x \log ^2(x)-12 x^2 \log ^2(x)+x^3 \log ^2(x)+\frac {4096 \log ^2(x)}{(4+x)^3}-\frac {6144 \log ^2(x)}{(4+x)^2}+\frac {3840 \log ^2(x)}{4+x}\right ) \, dx-24 \int \left (-12 \log (x)+x \log (x)+\frac {256 \log (x)}{(4+x)^3}-\frac {256 \log (x)}{(4+x)^2}+\frac {96 \log (x)}{4+x}\right ) \, dx+32 \int \left (\log (x)-\frac {64 \log (x)}{(4+x)^3}+\frac {48 \log (x)}{(4+x)^2}-\frac {12 \log (x)}{4+x}\right ) \, dx+64 \int \left (\log ^2(x)-\frac {64 \log ^2(x)}{(4+x)^3}+\frac {48 \log ^2(x)}{(4+x)^2}-\frac {12 \log ^2(x)}{4+x}\right ) \, dx-64 \int \left (-12 \log ^2(x)+x \log ^2(x)+\frac {256 \log ^2(x)}{(4+x)^3}-\frac {256 \log ^2(x)}{(4+x)^2}+\frac {96 \log ^2(x)}{4+x}\right ) \, dx+(4 (1+\log (2))) \int \left (\frac {64 \log (x) (2-x+4 \log (x)-3 x \log (x))}{e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))}+\frac {x^2 \log (x) (2-x+4 \log (x)-3 x \log (x))}{e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))}+\frac {384 \log (x) (2-x+4 \log (x)-3 x \log (x))}{(4+x)^2 \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )}+\frac {10 x \log (x) (-2+x-4 \log (x)+3 x \log (x))}{e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))}+\frac {352 \log (x) (-2+x-4 \log (x)+3 x \log (x))}{(4+x) \left (e^{(-2+x)^2 x^4 \log ^2(x)}+2 (1+\log (2))\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 26, normalized size = 0.81 \begin {gather*} \frac {\log \left (2+e^{(-2+x)^2 x^4 \log ^2(x)}+\log (4)\right )}{(4+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2)*((32*x^3 - 24*x^4 + 2*x^6)*Log[x] + (64*x^3 - 64*x^4 + 4*x^5 + 6
*x^6)*Log[x]^2) + (-4 - 2*E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2) - 2*Log[4])*Log[2 + E^((4*x^4 - 4*x^5 + x^6)*Log[
x]^2) + Log[4]])/(128 + 96*x + 24*x^2 + 2*x^3 + E^((4*x^4 - 4*x^5 + x^6)*Log[x]^2)*(64 + 48*x + 12*x^2 + x^3)
+ (64 + 48*x + 12*x^2 + x^3)*Log[4]),x]

[Out]

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

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fricas [A]  time = 0.49, size = 38, normalized size = 1.19 \begin {gather*} \frac {\log \left (e^{\left ({\left (x^{6} - 4 \, x^{5} + 4 \, x^{4}\right )} \log \relax (x)^{2}\right )} + 2 \, \log \relax (2) + 2\right )}{x^{2} + 8 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp((x^6-4*x^5+4*x^4)*log(x)^2)-4*log(2)-4)*log(exp((x^6-4*x^5+4*x^4)*log(x)^2)+2+2*log(2))+((6
*x^6+4*x^5-64*x^4+64*x^3)*log(x)^2+(2*x^6-24*x^4+32*x^3)*log(x))*exp((x^6-4*x^5+4*x^4)*log(x)^2))/((x^3+12*x^2
+48*x+64)*exp((x^6-4*x^5+4*x^4)*log(x)^2)+2*(x^3+12*x^2+48*x+64)*log(2)+2*x^3+24*x^2+96*x+128),x, algorithm="f
ricas")

[Out]

log(e^((x^6 - 4*x^5 + 4*x^4)*log(x)^2) + 2*log(2) + 2)/(x^2 + 8*x + 16)

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giac [A]  time = 3.00, size = 46, normalized size = 1.44 \begin {gather*} \frac {\log \left (e^{\left (x^{6} \log \relax (x)^{2} - 4 \, x^{5} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x)^{2}\right )} + 2 \, \log \relax (2) + 2\right )}{x^{2} + 8 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp((x^6-4*x^5+4*x^4)*log(x)^2)-4*log(2)-4)*log(exp((x^6-4*x^5+4*x^4)*log(x)^2)+2+2*log(2))+((6
*x^6+4*x^5-64*x^4+64*x^3)*log(x)^2+(2*x^6-24*x^4+32*x^3)*log(x))*exp((x^6-4*x^5+4*x^4)*log(x)^2))/((x^3+12*x^2
+48*x+64)*exp((x^6-4*x^5+4*x^4)*log(x)^2)+2*(x^3+12*x^2+48*x+64)*log(2)+2*x^3+24*x^2+96*x+128),x, algorithm="g
iac")

[Out]

log(e^(x^6*log(x)^2 - 4*x^5*log(x)^2 + 4*x^4*log(x)^2) + 2*log(2) + 2)/(x^2 + 8*x + 16)

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maple [A]  time = 0.05, size = 33, normalized size = 1.03

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{x^{4} \left (x -2\right )^{2} \ln \relax (x )^{2}}+2+2 \ln \relax (2)\right )}{x^{2}+8 x +16}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp((x^6-4*x^5+4*x^4)*ln(x)^2)-4*ln(2)-4)*ln(exp((x^6-4*x^5+4*x^4)*ln(x)^2)+2+2*ln(2))+((6*x^6+4*x^5-
64*x^4+64*x^3)*ln(x)^2+(2*x^6-24*x^4+32*x^3)*ln(x))*exp((x^6-4*x^5+4*x^4)*ln(x)^2))/((x^3+12*x^2+48*x+64)*exp(
(x^6-4*x^5+4*x^4)*ln(x)^2)+2*(x^3+12*x^2+48*x+64)*ln(2)+2*x^3+24*x^2+96*x+128),x,method=_RETURNVERBOSE)

[Out]

1/(x^2+8*x+16)*ln(exp(x^4*(x-2)^2*ln(x)^2)+2+2*ln(2))

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maxima [A]  time = 0.63, size = 61, normalized size = 1.91 \begin {gather*} \frac {\log \left (2 \, {\left (\log \relax (2) + 1\right )} e^{\left (4 \, x^{5} \log \relax (x)^{2}\right )} + e^{\left (x^{6} \log \relax (x)^{2} + 4 \, x^{4} \log \relax (x)^{2}\right )}\right ) - 4 \, \log \left (e^{\left (x^{5} \log \relax (x)^{2}\right )}\right )}{x^{2} + 8 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp((x^6-4*x^5+4*x^4)*log(x)^2)-4*log(2)-4)*log(exp((x^6-4*x^5+4*x^4)*log(x)^2)+2+2*log(2))+((6
*x^6+4*x^5-64*x^4+64*x^3)*log(x)^2+(2*x^6-24*x^4+32*x^3)*log(x))*exp((x^6-4*x^5+4*x^4)*log(x)^2))/((x^3+12*x^2
+48*x+64)*exp((x^6-4*x^5+4*x^4)*log(x)^2)+2*(x^3+12*x^2+48*x+64)*log(2)+2*x^3+24*x^2+96*x+128),x, algorithm="m
axima")

[Out]

(log(2*(log(2) + 1)*e^(4*x^5*log(x)^2) + e^(x^6*log(x)^2 + 4*x^4*log(x)^2)) - 4*log(e^(x^5*log(x)^2)))/(x^2 +
8*x + 16)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{{\ln \relax (x)}^2\,\left (x^6-4\,x^5+4\,x^4\right )}\,\left (\left (6\,x^6+4\,x^5-64\,x^4+64\,x^3\right )\,{\ln \relax (x)}^2+\left (2\,x^6-24\,x^4+32\,x^3\right )\,\ln \relax (x)\right )-\ln \left (2\,\ln \relax (2)+{\mathrm {e}}^{{\ln \relax (x)}^2\,\left (x^6-4\,x^5+4\,x^4\right )}+2\right )\,\left (4\,\ln \relax (2)+2\,{\mathrm {e}}^{{\ln \relax (x)}^2\,\left (x^6-4\,x^5+4\,x^4\right )}+4\right )}{96\,x+2\,\ln \relax (2)\,\left (x^3+12\,x^2+48\,x+64\right )+{\mathrm {e}}^{{\ln \relax (x)}^2\,\left (x^6-4\,x^5+4\,x^4\right )}\,\left (x^3+12\,x^2+48\,x+64\right )+24\,x^2+2\,x^3+128} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x)^2*(4*x^4 - 4*x^5 + x^6))*(log(x)*(32*x^3 - 24*x^4 + 2*x^6) + log(x)^2*(64*x^3 - 64*x^4 + 4*x^5
+ 6*x^6)) - log(2*log(2) + exp(log(x)^2*(4*x^4 - 4*x^5 + x^6)) + 2)*(4*log(2) + 2*exp(log(x)^2*(4*x^4 - 4*x^5
+ x^6)) + 4))/(96*x + 2*log(2)*(48*x + 12*x^2 + x^3 + 64) + exp(log(x)^2*(4*x^4 - 4*x^5 + x^6))*(48*x + 12*x^
2 + x^3 + 64) + 24*x^2 + 2*x^3 + 128),x)

[Out]

int((exp(log(x)^2*(4*x^4 - 4*x^5 + x^6))*(log(x)*(32*x^3 - 24*x^4 + 2*x^6) + log(x)^2*(64*x^3 - 64*x^4 + 4*x^5
+ 6*x^6)) - log(2*log(2) + exp(log(x)^2*(4*x^4 - 4*x^5 + x^6)) + 2)*(4*log(2) + 2*exp(log(x)^2*(4*x^4 - 4*x^5
+ x^6)) + 4))/(96*x + 2*log(2)*(48*x + 12*x^2 + x^3 + 64) + exp(log(x)^2*(4*x^4 - 4*x^5 + x^6))*(48*x + 12*x^
2 + x^3 + 64) + 24*x^2 + 2*x^3 + 128), x)

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sympy [A]  time = 20.09, size = 36, normalized size = 1.12 \begin {gather*} \frac {\log {\left (e^{\left (x^{6} - 4 x^{5} + 4 x^{4}\right ) \log {\relax (x )}^{2}} + 2 \log {\relax (2 )} + 2 \right )}}{x^{2} + 8 x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp((x**6-4*x**5+4*x**4)*ln(x)**2)-4*ln(2)-4)*ln(exp((x**6-4*x**5+4*x**4)*ln(x)**2)+2+2*ln(2))+
((6*x**6+4*x**5-64*x**4+64*x**3)*ln(x)**2+(2*x**6-24*x**4+32*x**3)*ln(x))*exp((x**6-4*x**5+4*x**4)*ln(x)**2))/
((x**3+12*x**2+48*x+64)*exp((x**6-4*x**5+4*x**4)*ln(x)**2)+2*(x**3+12*x**2+48*x+64)*ln(2)+2*x**3+24*x**2+96*x+
128),x)

[Out]

log(exp((x**6 - 4*x**5 + 4*x**4)*log(x)**2) + 2*log(2) + 2)/(x**2 + 8*x + 16)

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