3.87.87 \(\int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+(x^2+4 \log (4)) \log (x^2)}{e^4 (256 x^2-128 x^3+16 x^4)+e^4 (512 x-256 x^2+32 x^3) \log (4)+e^4 (256-128 x+16 x^2) \log ^2(4)+(e^4 (-128 x^2+64 x^3-8 x^4)+e^4 (-256 x+128 x^2-16 x^3) \log (4)+e^4 (-128+64 x-8 x^2) \log ^2(4)) \log (x^2)+(e^4 (16 x^2-8 x^3+x^4)+e^4 (32 x-16 x^2+2 x^3) \log (4)+e^4 (16-8 x+x^2) \log ^2(4)) \log ^2(x^2)} \, dx\)

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

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Rubi [F]  time = 0.79, 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 {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x - 2*x^2 + (-24 + 2*x)*Log[4] + (x^2 + 4*Log[4])*Log[x^2])/(E^4*(256*x^2 - 128*x^3 + 16*x^4) + E^4*(5
12*x - 256*x^2 + 32*x^3)*Log[4] + E^4*(256 - 128*x + 16*x^2)*Log[4]^2 + (E^4*(-128*x^2 + 64*x^3 - 8*x^4) + E^4
*(-256*x + 128*x^2 - 16*x^3)*Log[4] + E^4*(-128 + 64*x - 8*x^2)*Log[4]^2)*Log[x^2] + (E^4*(16*x^2 - 8*x^3 + x^
4) + E^4*(32*x - 16*x^2 + 2*x^3)*Log[4] + E^4*(16 - 8*x + x^2)*Log[4]^2)*Log[x^2]^2),x]

[Out]

(-2*Defer[Int][1/((4 - x)*(x + Log[4])*(4 - Log[x^2])^2), x])/E^4 + Defer[Int][(x^2 + Log[256])/((-4 + x)^2*(x
+ Log[4])^2*(-4 + Log[x^2])), x]/E^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^2+2 x (-4+\log (4))-24 \log (4)+\left (x^2+\log (256)\right ) \log \left (x^2\right )}{e^4 (4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx\\ &=\frac {\int \frac {-2 x^2+2 x (-4+\log (4))-24 \log (4)+\left (x^2+\log (256)\right ) \log \left (x^2\right )}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4}\\ &=\frac {\int \left (\frac {2 \left (x^2-x (4-\log (4))-\log (256)\right )}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2}+\frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )}\right ) \, dx}{e^4}\\ &=\frac {\int \frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \, dx}{e^4}+\frac {2 \int \frac {x^2-x (4-\log (4))-\log (256)}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4}\\ &=\frac {\int \frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \, dx}{e^4}+\frac {2 \int \frac {-x-\log (4)}{(4-x) (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4}\\ &=\frac {\int \frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \, dx}{e^4}-\frac {2 \int \frac {1}{(4-x) (x+\log (4)) \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 39, normalized size = 1.50 \begin {gather*} -\frac {x \left (x^2+x (-4+\log (4))-\log (256)\right )}{e^4 (-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x - 2*x^2 + (-24 + 2*x)*Log[4] + (x^2 + 4*Log[4])*Log[x^2])/(E^4*(256*x^2 - 128*x^3 + 16*x^4) +
E^4*(512*x - 256*x^2 + 32*x^3)*Log[4] + E^4*(256 - 128*x + 16*x^2)*Log[4]^2 + (E^4*(-128*x^2 + 64*x^3 - 8*x^4)
+ E^4*(-256*x + 128*x^2 - 16*x^3)*Log[4] + E^4*(-128 + 64*x - 8*x^2)*Log[4]^2)*Log[x^2] + (E^4*(16*x^2 - 8*x^
3 + x^4) + E^4*(32*x - 16*x^2 + 2*x^3)*Log[4] + E^4*(16 - 8*x + x^2)*Log[4]^2)*Log[x^2]^2),x]

[Out]

-((x*(x^2 + x*(-4 + Log[4]) - Log[256]))/(E^4*(-4 + x)^2*(x + Log[4])^2*(-4 + Log[x^2])))

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fricas [A]  time = 0.73, size = 51, normalized size = 1.96 \begin {gather*} \frac {x}{8 \, {\left (x - 4\right )} e^{4} \log \relax (2) + 4 \, {\left (x^{2} - 4 \, x\right )} e^{4} - {\left (2 \, {\left (x - 4\right )} e^{4} \log \relax (2) + {\left (x^{2} - 4 \, x\right )} e^{4}\right )} \log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(2)+x^2)*log(x^2)+2*(2*x-24)*log(2)-2*x^2-8*x)/((4*(x^2-8*x+16)*exp(4)*log(2)^2+2*(2*x^3-16*x
^2+32*x)*exp(4)*log(2)+(x^4-8*x^3+16*x^2)*exp(4))*log(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*log(2)^2+2*(-16*x^3+1
28*x^2-256*x)*exp(4)*log(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*log(x^2)+4*(16*x^2-128*x+256)*exp(4)*log(2)^2+2*(3
2*x^3-256*x^2+512*x)*exp(4)*log(2)+(16*x^4-128*x^3+256*x^2)*exp(4)),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.36, size = 71, normalized size = 2.73 \begin {gather*} -\frac {x}{x^{2} e^{4} \log \left (x^{2}\right ) + 2 \, x e^{4} \log \relax (2) \log \left (x^{2}\right ) - 4 \, x^{2} e^{4} - 8 \, x e^{4} \log \relax (2) - 4 \, x e^{4} \log \left (x^{2}\right ) - 8 \, e^{4} \log \relax (2) \log \left (x^{2}\right ) + 16 \, x e^{4} + 32 \, e^{4} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(2)+x^2)*log(x^2)+2*(2*x-24)*log(2)-2*x^2-8*x)/((4*(x^2-8*x+16)*exp(4)*log(2)^2+2*(2*x^3-16*x
^2+32*x)*exp(4)*log(2)+(x^4-8*x^3+16*x^2)*exp(4))*log(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*log(2)^2+2*(-16*x^3+1
28*x^2-256*x)*exp(4)*log(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*log(x^2)+4*(16*x^2-128*x+256)*exp(4)*log(2)^2+2*(3
2*x^3-256*x^2+512*x)*exp(4)*log(2)+(16*x^4-128*x^3+256*x^2)*exp(4)),x, algorithm="giac")

[Out]

-x/(x^2*e^4*log(x^2) + 2*x*e^4*log(2)*log(x^2) - 4*x^2*e^4 - 8*x*e^4*log(2) - 4*x*e^4*log(x^2) - 8*e^4*log(2)*
log(x^2) + 16*x*e^4 + 32*e^4*log(2))

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maple [A]  time = 0.39, size = 29, normalized size = 1.12

method	result	size
norman	\(-\frac {x \,{\mathrm e}^{-4}}{\left (\ln \left (x^{2}\right )-4\right ) \left (x -4\right ) \left (x +2 \ln \relax (2)\right )}\)	\(29\)
risch	\(-\frac {x \,{\mathrm e}^{-4}}{\left (2 x \ln \relax (2)+x^{2}-8 \ln \relax (2)-4 x \right ) \left (\ln \left (x^{2}\right )-4\right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*ln(2)+x^2)*ln(x^2)+2*(2*x-24)*ln(2)-2*x^2-8*x)/((4*(x^2-8*x+16)*exp(4)*ln(2)^2+2*(2*x^3-16*x^2+32*x)*e
xp(4)*ln(2)+(x^4-8*x^3+16*x^2)*exp(4))*ln(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*ln(2)^2+2*(-16*x^3+128*x^2-256*x)
*exp(4)*ln(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*ln(x^2)+4*(16*x^2-128*x+256)*exp(4)*ln(2)^2+2*(32*x^3-256*x^2+51
2*x)*exp(4)*ln(2)+(16*x^4-128*x^3+256*x^2)*exp(4)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.51, size = 54, normalized size = 2.08 \begin {gather*} \frac {x}{2 \, {\left (2 \, x^{2} e^{4} + 4 \, x {\left (\log \relax (2) - 2\right )} e^{4} - 16 \, e^{4} \log \relax (2) - {\left (x^{2} e^{4} + 2 \, x {\left (\log \relax (2) - 2\right )} e^{4} - 8 \, e^{4} \log \relax (2)\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(2)+x^2)*log(x^2)+2*(2*x-24)*log(2)-2*x^2-8*x)/((4*(x^2-8*x+16)*exp(4)*log(2)^2+2*(2*x^3-16*x
^2+32*x)*exp(4)*log(2)+(x^4-8*x^3+16*x^2)*exp(4))*log(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*log(2)^2+2*(-16*x^3+1
28*x^2-256*x)*exp(4)*log(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*log(x^2)+4*(16*x^2-128*x+256)*exp(4)*log(2)^2+2*(3
2*x^3-256*x^2+512*x)*exp(4)*log(2)+(16*x^4-128*x^3+256*x^2)*exp(4)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.42, size = 562, normalized size = 21.62 \begin {gather*} \frac {\frac {x\,\left (4\,x+24\,\ln \relax (2)-2\,x\,\ln \relax (2)+x^2\right )}{64\,{\mathrm {e}}^4\,{\ln \relax (2)}^2+16\,x^2\,{\mathrm {e}}^4-8\,x^3\,{\mathrm {e}}^4+x^4\,{\mathrm {e}}^4-32\,x\,{\mathrm {e}}^4\,{\ln \relax (2)}^2-32\,x^2\,{\mathrm {e}}^4\,\ln \relax (2)+4\,x^3\,{\mathrm {e}}^4\,\ln \relax (2)+4\,x^2\,{\mathrm {e}}^4\,{\ln \relax (2)}^2+64\,x\,{\mathrm {e}}^4\,\ln \relax (2)}-\frac {x\,\ln \left (x^2\right )\,\left (x^2+\ln \left (256\right )\right )}{2\,\left (64\,{\mathrm {e}}^4\,{\ln \relax (2)}^2+16\,x^2\,{\mathrm {e}}^4-8\,x^3\,{\mathrm {e}}^4+x^4\,{\mathrm {e}}^4-32\,x\,{\mathrm {e}}^4\,{\ln \relax (2)}^2-32\,x^2\,{\mathrm {e}}^4\,\ln \relax (2)+4\,x^3\,{\mathrm {e}}^4\,\ln \relax (2)+4\,x^2\,{\mathrm {e}}^4\,{\ln \relax (2)}^2+64\,x\,{\mathrm {e}}^4\,\ln \relax (2)\right )}}{\ln \left (x^2\right )-4}+\frac {\frac {{\mathrm {e}}^{-4}\,\left (32\,\ln \relax (2)-6\,\ln \relax (2)\,\ln \relax (4)+3\,{\ln \relax (2)}^2\,\ln \relax (4)+36\,{\ln \relax (2)}^2+2\,{\ln \relax (2)}^3+{\ln \relax (2)}^4+16\right )\,x^3}{2\,\left (32\,\ln \relax (2)+24\,{\ln \relax (2)}^2+8\,{\ln \relax (2)}^3+{\ln \relax (2)}^4+16\right )}+\frac {{\mathrm {e}}^{-4}\,\left (32\,\ln \relax (2)-16\,\ln \relax (4)+40\,\ln \relax (2)\,\ln \relax (4)-96\,{\ln \relax (2)}^2\,\ln \relax (4)+10\,{\ln \relax (2)}^3\,\ln \relax (4)-{\ln \relax (2)}^4\,\ln \relax (4)-80\,{\ln \relax (2)}^2+192\,{\ln \relax (2)}^3-20\,{\ln \relax (2)}^4+2\,{\ln \relax (2)}^5\right )\,x^2}{4\,\left (32\,\ln \relax (2)+24\,{\ln \relax (2)}^2+8\,{\ln \relax (2)}^3+{\ln \relax (2)}^4+16\right )}+\frac {2\,{\mathrm {e}}^{-4}\,\left (32\,\ln \relax (2)-8\,\ln \relax (2)\,\ln \relax (4)+32\,{\ln \relax (2)}^2\,\ln \relax (4)-16\,{\ln \relax (2)}^3\,\ln \relax (4)+{\ln \relax (2)}^4\,\ln \relax (4)+80\,{\ln \relax (2)}^2-16\,{\ln \relax (2)}^3+48\,{\ln \relax (2)}^4\right )\,x}{32\,\ln \relax (2)+24\,{\ln \relax (2)}^2+8\,{\ln \relax (2)}^3+{\ln \relax (2)}^4+16}-\frac {4\,{\mathrm {e}}^{-4}\,\left (4\,{\ln \relax (2)}^2\,\ln \relax (4)-20\,{\ln \relax (2)}^3\,\ln \relax (4)+{\ln \relax (2)}^4\,\ln \relax (4)-8\,{\ln \relax (2)}^3+40\,{\ln \relax (2)}^4-2\,{\ln \relax (2)}^5\right )}{32\,\ln \relax (2)+24\,{\ln \relax (2)}^2+8\,{\ln \relax (2)}^3+{\ln \relax (2)}^4+16}}{x^4+\left (4\,\ln \relax (2)-8\right )\,x^3+\left (4\,{\ln \relax (2)}^2-32\,\ln \relax (2)+16\right )\,x^2+\left (64\,\ln \relax (2)-32\,{\ln \relax (2)}^2\right )\,x+64\,{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x - 2*log(2)*(2*x - 24) - log(x^2)*(8*log(2) + x^2) + 2*x^2)/(log(x^2)^2*(exp(4)*(16*x^2 - 8*x^3 + x^4
) + 4*exp(4)*log(2)^2*(x^2 - 8*x + 16) + 2*exp(4)*log(2)*(32*x - 16*x^2 + 2*x^3)) - log(x^2)*(exp(4)*(128*x^2
- 64*x^3 + 8*x^4) + 2*exp(4)*log(2)*(256*x - 128*x^2 + 16*x^3) + 4*exp(4)*log(2)^2*(8*x^2 - 64*x + 128)) + exp
(4)*(256*x^2 - 128*x^3 + 16*x^4) + 2*exp(4)*log(2)*(512*x - 256*x^2 + 32*x^3) + 4*exp(4)*log(2)^2*(16*x^2 - 12
8*x + 256)),x)

[Out]

((x*(4*x + 24*log(2) - 2*x*log(2) + x^2))/(64*exp(4)*log(2)^2 + 16*x^2*exp(4) - 8*x^3*exp(4) + x^4*exp(4) - 32
*x*exp(4)*log(2)^2 - 32*x^2*exp(4)*log(2) + 4*x^3*exp(4)*log(2) + 4*x^2*exp(4)*log(2)^2 + 64*x*exp(4)*log(2))
- (x*log(x^2)*(log(256) + x^2))/(2*(64*exp(4)*log(2)^2 + 16*x^2*exp(4) - 8*x^3*exp(4) + x^4*exp(4) - 32*x*exp(
4)*log(2)^2 - 32*x^2*exp(4)*log(2) + 4*x^3*exp(4)*log(2) + 4*x^2*exp(4)*log(2)^2 + 64*x*exp(4)*log(2))))/(log(
x^2) - 4) + ((x^3*exp(-4)*(32*log(2) - 6*log(2)*log(4) + 3*log(2)^2*log(4) + 36*log(2)^2 + 2*log(2)^3 + log(2)
^4 + 16))/(2*(32*log(2) + 24*log(2)^2 + 8*log(2)^3 + log(2)^4 + 16)) - (4*exp(-4)*(4*log(2)^2*log(4) - 20*log(
2)^3*log(4) + log(2)^4*log(4) - 8*log(2)^3 + 40*log(2)^4 - 2*log(2)^5))/(32*log(2) + 24*log(2)^2 + 8*log(2)^3
+ log(2)^4 + 16) + (2*x*exp(-4)*(32*log(2) - 8*log(2)*log(4) + 32*log(2)^2*log(4) - 16*log(2)^3*log(4) + log(2
)^4*log(4) + 80*log(2)^2 - 16*log(2)^3 + 48*log(2)^4))/(32*log(2) + 24*log(2)^2 + 8*log(2)^3 + log(2)^4 + 16)
+ (x^2*exp(-4)*(32*log(2) - 16*log(4) + 40*log(2)*log(4) - 96*log(2)^2*log(4) + 10*log(2)^3*log(4) - log(2)^4*
log(4) - 80*log(2)^2 + 192*log(2)^3 - 20*log(2)^4 + 2*log(2)^5))/(4*(32*log(2) + 24*log(2)^2 + 8*log(2)^3 + lo
g(2)^4 + 16)))/(x*(64*log(2) - 32*log(2)^2) + x^3*(4*log(2) - 8) + x^2*(4*log(2)^2 - 32*log(2) + 16) + 64*log(
2)^2 + x^4)

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sympy [B]  time = 0.25, size = 73, normalized size = 2.81 \begin {gather*} - \frac {x}{- 4 x^{2} e^{4} - 8 x e^{4} \log {\relax (2 )} + 16 x e^{4} + \left (x^{2} e^{4} - 4 x e^{4} + 2 x e^{4} \log {\relax (2 )} - 8 e^{4} \log {\relax (2 )}\right ) \log {\left (x^{2} \right )} + 32 e^{4} \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*ln(2)+x**2)*ln(x**2)+2*(2*x-24)*ln(2)-2*x**2-8*x)/((4*(x**2-8*x+16)*exp(4)*ln(2)**2+2*(2*x**3-16
*x**2+32*x)*exp(4)*ln(2)+(x**4-8*x**3+16*x**2)*exp(4))*ln(x**2)**2+(4*(-8*x**2+64*x-128)*exp(4)*ln(2)**2+2*(-1
6*x**3+128*x**2-256*x)*exp(4)*ln(2)+(-8*x**4+64*x**3-128*x**2)*exp(4))*ln(x**2)+4*(16*x**2-128*x+256)*exp(4)*l
n(2)**2+2*(32*x**3-256*x**2+512*x)*exp(4)*ln(2)+(16*x**4-128*x**3+256*x**2)*exp(4)),x)

[Out]

-x/(-4*x**2*exp(4) - 8*x*exp(4)*log(2) + 16*x*exp(4) + (x**2*exp(4) - 4*x*exp(4) + 2*x*exp(4)*log(2) - 8*exp(4
)*log(2))*log(x**2) + 32*exp(4)*log(2))

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