3.102.25 \(\int \frac {-128-384 x-256 x^2+e^{2 x} (-128 x^2+640 x^4+512 x^5)+e^{4 x} (-32 x^4+96 x^5+128 x^6)+(-256-256 x+e^{2 x} (128 x^2+1152 x^3+1024 x^4)+e^{4 x} (128 x^4+128 x^5)) \log (x)+e^{2 x} (512 x^2+512 x^3) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ \left (-16+\frac {4+\frac {8 e^{-2 x}}{x^2}}{x+\log (x)}\right )^2 \]

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Rubi [F]  time = 11.67, 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 {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-128 - 384*x - 256*x^2 + E^(2*x)*(-128*x^2 + 640*x^4 + 512*x^5) + E^(4*x)*(-32*x^4 + 96*x^5 + 128*x^6) +
(-256 - 256*x + E^(2*x)*(128*x^2 + 1152*x^3 + 1024*x^4) + E^(4*x)*(128*x^4 + 128*x^5))*Log[x] + E^(2*x)*(512*x
^2 + 512*x^3)*Log[x]^2)/(E^(4*x)*x^8 + 3*E^(4*x)*x^7*Log[x] + 3*E^(4*x)*x^6*Log[x]^2 + E^(4*x)*x^5*Log[x]^3),x
]

[Out]

16/(x + Log[x])^2 - 128/(x + Log[x]) - 128*Defer[Int][1/(E^(4*x)*x^5*(x + Log[x])^3), x] - 128*Defer[Int][1/(E
^(4*x)*x^4*(x + Log[x])^3), x] - 128*Defer[Int][1/(E^(2*x)*x^3*(x + Log[x])^3), x] - 128*Defer[Int][1/(E^(2*x)
*x^2*(x + Log[x])^3), x] - 256*Defer[Int][1/(E^(4*x)*x^5*(x + Log[x])^2), x] - 256*Defer[Int][1/(E^(4*x)*x^4*(
x + Log[x])^2), x] + 128*Defer[Int][1/(E^(2*x)*x^3*(x + Log[x])^2), x] + 128*Defer[Int][1/(E^(2*x)*x^2*(x + Lo
g[x])^2), x] + 512*Defer[Int][1/(E^(2*x)*x^3*(x + Log[x])), x] + 512*Defer[Int][1/(E^(2*x)*x^2*(x + Log[x])),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 e^{-4 x} (1+x) \left (-4-8 x-4 e^{2 x} x^2+4 e^{2 x} x^3-e^{2 x} \left (-16+e^{2 x}\right ) x^4+4 e^{4 x} x^5+4 \left (-2+e^{4 x} x^4+e^{2 x} x^2 (1+8 x)\right ) \log (x)+16 e^{2 x} x^2 \log ^2(x)\right )}{x^5 (x+\log (x))^3} \, dx\\ &=32 \int \frac {e^{-4 x} (1+x) \left (-4-8 x-4 e^{2 x} x^2+4 e^{2 x} x^3-e^{2 x} \left (-16+e^{2 x}\right ) x^4+4 e^{4 x} x^5+4 \left (-2+e^{4 x} x^4+e^{2 x} x^2 (1+8 x)\right ) \log (x)+16 e^{2 x} x^2 \log ^2(x)\right )}{x^5 (x+\log (x))^3} \, dx\\ &=32 \int \left (-\frac {4 e^{-4 x} (1+x)}{x^5 (x+\log (x))^3}-\frac {8 e^{-4 x} (1+x)}{x^4 (x+\log (x))^3}-\frac {8 e^{-4 x} (1+x) \log (x)}{x^5 (x+\log (x))^3}+\frac {(1+x) (-1+4 x+4 \log (x))}{x (x+\log (x))^3}+\frac {4 e^{-2 x} (1+x) \left (-1+x+4 x^2+\log (x)+8 x \log (x)+4 \log ^2(x)\right )}{x^3 (x+\log (x))^3}\right ) \, dx\\ &=32 \int \frac {(1+x) (-1+4 x+4 \log (x))}{x (x+\log (x))^3} \, dx-128 \int \frac {e^{-4 x} (1+x)}{x^5 (x+\log (x))^3} \, dx+128 \int \frac {e^{-2 x} (1+x) \left (-1+x+4 x^2+\log (x)+8 x \log (x)+4 \log ^2(x)\right )}{x^3 (x+\log (x))^3} \, dx-256 \int \frac {e^{-4 x} (1+x)}{x^4 (x+\log (x))^3} \, dx-256 \int \frac {e^{-4 x} (1+x) \log (x)}{x^5 (x+\log (x))^3} \, dx\\ &=32 \int \left (\frac {-1-x}{x (x+\log (x))^3}+\frac {4 (1+x)}{x (x+\log (x))^2}\right ) \, dx-128 \int \left (\frac {e^{-4 x}}{x^5 (x+\log (x))^3}+\frac {e^{-4 x}}{x^4 (x+\log (x))^3}\right ) \, dx+128 \int \left (\frac {e^{-2 x} (-1-x)}{x^3 (x+\log (x))^3}+\frac {e^{-2 x} (1+x)}{x^3 (x+\log (x))^2}+\frac {4 e^{-2 x} (1+x)}{x^3 (x+\log (x))}\right ) \, dx-256 \int \left (\frac {e^{-4 x}}{x^4 (x+\log (x))^3}+\frac {e^{-4 x}}{x^3 (x+\log (x))^3}\right ) \, dx-256 \int \left (\frac {e^{-4 x} (-1-x)}{x^4 (x+\log (x))^3}+\frac {e^{-4 x} (1+x)}{x^5 (x+\log (x))^2}\right ) \, dx\\ &=32 \int \frac {-1-x}{x (x+\log (x))^3} \, dx-128 \int \frac {e^{-4 x}}{x^5 (x+\log (x))^3} \, dx-128 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^3} \, dx+128 \int \frac {e^{-2 x} (-1-x)}{x^3 (x+\log (x))^3} \, dx+128 \int \frac {e^{-2 x} (1+x)}{x^3 (x+\log (x))^2} \, dx+128 \int \frac {1+x}{x (x+\log (x))^2} \, dx-256 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^3} \, dx-256 \int \frac {e^{-4 x} (-1-x)}{x^4 (x+\log (x))^3} \, dx-256 \int \frac {e^{-4 x}}{x^3 (x+\log (x))^3} \, dx-256 \int \frac {e^{-4 x} (1+x)}{x^5 (x+\log (x))^2} \, dx+512 \int \frac {e^{-2 x} (1+x)}{x^3 (x+\log (x))} \, dx\\ &=\frac {16}{(x+\log (x))^2}-\frac {128}{x+\log (x)}-128 \int \frac {e^{-4 x}}{x^5 (x+\log (x))^3} \, dx-128 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^3} \, dx+128 \int \left (-\frac {e^{-2 x}}{x^3 (x+\log (x))^3}-\frac {e^{-2 x}}{x^2 (x+\log (x))^3}\right ) \, dx+128 \int \left (\frac {e^{-2 x}}{x^3 (x+\log (x))^2}+\frac {e^{-2 x}}{x^2 (x+\log (x))^2}\right ) \, dx-256 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^3} \, dx-256 \int \frac {e^{-4 x}}{x^3 (x+\log (x))^3} \, dx-256 \int \left (-\frac {e^{-4 x}}{x^4 (x+\log (x))^3}-\frac {e^{-4 x}}{x^3 (x+\log (x))^3}\right ) \, dx-256 \int \left (\frac {e^{-4 x}}{x^5 (x+\log (x))^2}+\frac {e^{-4 x}}{x^4 (x+\log (x))^2}\right ) \, dx+512 \int \left (\frac {e^{-2 x}}{x^3 (x+\log (x))}+\frac {e^{-2 x}}{x^2 (x+\log (x))}\right ) \, dx\\ &=\frac {16}{(x+\log (x))^2}-\frac {128}{x+\log (x)}-128 \int \frac {e^{-4 x}}{x^5 (x+\log (x))^3} \, dx-128 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^3} \, dx-128 \int \frac {e^{-2 x}}{x^3 (x+\log (x))^3} \, dx-128 \int \frac {e^{-2 x}}{x^2 (x+\log (x))^3} \, dx+128 \int \frac {e^{-2 x}}{x^3 (x+\log (x))^2} \, dx+128 \int \frac {e^{-2 x}}{x^2 (x+\log (x))^2} \, dx-256 \int \frac {e^{-4 x}}{x^5 (x+\log (x))^2} \, dx-256 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^2} \, dx+512 \int \frac {e^{-2 x}}{x^3 (x+\log (x))} \, dx+512 \int \frac {e^{-2 x}}{x^2 (x+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 42, normalized size = 1.83 \begin {gather*} \frac {16 \left (2 e^{-2 x}+x^2\right ) \left (2 e^{-2 x}+x^2-8 x^2 (x+\log (x))\right )}{x^4 (x+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-128 - 384*x - 256*x^2 + E^(2*x)*(-128*x^2 + 640*x^4 + 512*x^5) + E^(4*x)*(-32*x^4 + 96*x^5 + 128*x
^6) + (-256 - 256*x + E^(2*x)*(128*x^2 + 1152*x^3 + 1024*x^4) + E^(4*x)*(128*x^4 + 128*x^5))*Log[x] + E^(2*x)*
(512*x^2 + 512*x^3)*Log[x]^2)/(E^(4*x)*x^8 + 3*E^(4*x)*x^7*Log[x] + 3*E^(4*x)*x^6*Log[x]^2 + E^(4*x)*x^5*Log[x
]^3),x]

[Out]

(16*(2/E^(2*x) + x^2)*(2/E^(2*x) + x^2 - 8*x^2*(x + Log[x])))/(x^4*(x + Log[x])^2)

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fricas [B]  time = 0.78, size = 93, normalized size = 4.04 \begin {gather*} -\frac {16 \, {\left ({\left (8 \, x^{5} - x^{4}\right )} e^{\left (4 \, x\right )} + 4 \, {\left (4 \, x^{3} - x^{2}\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{4} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )}\right )} \log \relax (x) - 4\right )}}{x^{6} e^{\left (4 \, x\right )} + 2 \, x^{5} e^{\left (4 \, x\right )} \log \relax (x) + x^{4} e^{\left (4 \, x\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4+(1024*x^4+1152*x^3+128*x^2)*exp(x)^
2-256*x-256)*log(x)+(128*x^6+96*x^5-32*x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^
5*exp(x)^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x)^4),x, algorithm="fricas")

[Out]

-16*((8*x^5 - x^4)*e^(4*x) + 4*(4*x^3 - x^2)*e^(2*x) + 8*(x^4*e^(4*x) + 2*x^2*e^(2*x))*log(x) - 4)/(x^6*e^(4*x
) + 2*x^5*e^(4*x)*log(x) + x^4*e^(4*x)*log(x)^2)

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giac [B]  time = 0.68, size = 76, normalized size = 3.30 \begin {gather*} -\frac {16 \, {\left (8 \, x^{5} + 8 \, x^{4} \log \relax (x) - x^{4} + 16 \, x^{3} e^{\left (-2 \, x\right )} + 16 \, x^{2} e^{\left (-2 \, x\right )} \log \relax (x) - 4 \, x^{2} e^{\left (-2 \, x\right )} - 4 \, e^{\left (-4 \, x\right )}\right )}}{x^{6} + 2 \, x^{5} \log \relax (x) + x^{4} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4+(1024*x^4+1152*x^3+128*x^2)*exp(x)^
2-256*x-256)*log(x)+(128*x^6+96*x^5-32*x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^
5*exp(x)^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x)^4),x, algorithm="giac")

[Out]

-16*(8*x^5 + 8*x^4*log(x) - x^4 + 16*x^3*e^(-2*x) + 16*x^2*e^(-2*x)*log(x) - 4*x^2*e^(-2*x) - 4*e^(-4*x))/(x^6
+ 2*x^5*log(x) + x^4*log(x)^2)

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maple [B]  time = 0.05, size = 76, normalized size = 3.30

method	result	size
risch	\(-\frac {16 \left (8 \,{\mathrm e}^{4 x} x^{5}+8 \ln \relax (x ) {\mathrm e}^{4 x} x^{4}-{\mathrm e}^{4 x} x^{4}+16 \,{\mathrm e}^{2 x} x^{3}+16 \ln \relax (x ) {\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{2 x} x^{2}-4\right ) {\mathrm e}^{-4 x}}{x^{4} \left (x +\ln \relax (x )\right )^{2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((512*x^3+512*x^2)*exp(x)^2*ln(x)^2+((128*x^5+128*x^4)*exp(x)^4+(1024*x^4+1152*x^3+128*x^2)*exp(x)^2-256*x
-256)*ln(x)+(128*x^6+96*x^5-32*x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^5*exp(x)
^4*ln(x)^3+3*x^6*exp(x)^4*ln(x)^2+3*x^7*exp(x)^4*ln(x)+x^8*exp(x)^4),x,method=_RETURNVERBOSE)

[Out]

-16*(8*exp(4*x)*x^5+8*ln(x)*exp(4*x)*x^4-exp(4*x)*x^4+16*exp(2*x)*x^3+16*ln(x)*exp(2*x)*x^2-4*exp(2*x)*x^2-4)/
x^4*exp(-4*x)/(x+ln(x))^2

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maxima [B]  time = 0.41, size = 71, normalized size = 3.09 \begin {gather*} -\frac {16 \, {\left (8 \, x^{5} + 8 \, x^{4} \log \relax (x) - x^{4} + 4 \, {\left (4 \, x^{3} + 4 \, x^{2} \log \relax (x) - x^{2}\right )} e^{\left (-2 \, x\right )} - 4 \, e^{\left (-4 \, x\right )}\right )}}{x^{6} + 2 \, x^{5} \log \relax (x) + x^{4} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4+(1024*x^4+1152*x^3+128*x^2)*exp(x)^
2-256*x-256)*log(x)+(128*x^6+96*x^5-32*x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^
5*exp(x)^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x)^4),x, algorithm="maxima")

[Out]

-16*(8*x^5 + 8*x^4*log(x) - x^4 + 4*(4*x^3 + 4*x^2*log(x) - x^2)*e^(-2*x) - 4*e^(-4*x))/(x^6 + 2*x^5*log(x) +
x^4*log(x)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {384\,x+\ln \relax (x)\,\left (256\,x-{\mathrm {e}}^{4\,x}\,\left (128\,x^5+128\,x^4\right )-{\mathrm {e}}^{2\,x}\,\left (1024\,x^4+1152\,x^3+128\,x^2\right )+256\right )-{\mathrm {e}}^{4\,x}\,\left (128\,x^6+96\,x^5-32\,x^4\right )-{\mathrm {e}}^{2\,x}\,\left (512\,x^5+640\,x^4-128\,x^2\right )+256\,x^2-{\mathrm {e}}^{2\,x}\,{\ln \relax (x)}^2\,\left (512\,x^3+512\,x^2\right )+128}{x^8\,{\mathrm {e}}^{4\,x}+3\,x^7\,{\mathrm {e}}^{4\,x}\,\ln \relax (x)+x^5\,{\mathrm {e}}^{4\,x}\,{\ln \relax (x)}^3+3\,x^6\,{\mathrm {e}}^{4\,x}\,{\ln \relax (x)}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(384*x + log(x)*(256*x - exp(4*x)*(128*x^4 + 128*x^5) - exp(2*x)*(128*x^2 + 1152*x^3 + 1024*x^4) + 256) -
exp(4*x)*(96*x^5 - 32*x^4 + 128*x^6) - exp(2*x)*(640*x^4 - 128*x^2 + 512*x^5) + 256*x^2 - exp(2*x)*log(x)^2*(
512*x^2 + 512*x^3) + 128)/(x^8*exp(4*x) + 3*x^7*exp(4*x)*log(x) + x^5*exp(4*x)*log(x)^3 + 3*x^6*exp(4*x)*log(x
)^2),x)

[Out]

-int((384*x + log(x)*(256*x - exp(4*x)*(128*x^4 + 128*x^5) - exp(2*x)*(128*x^2 + 1152*x^3 + 1024*x^4) + 256) -
exp(4*x)*(96*x^5 - 32*x^4 + 128*x^6) - exp(2*x)*(640*x^4 - 128*x^2 + 512*x^5) + 256*x^2 - exp(2*x)*log(x)^2*(
512*x^2 + 512*x^3) + 128)/(x^8*exp(4*x) + 3*x^7*exp(4*x)*log(x) + x^5*exp(4*x)*log(x)^3 + 3*x^6*exp(4*x)*log(x
)^2), x)

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sympy [B]  time = 0.54, size = 156, normalized size = 6.78 \begin {gather*} \frac {\left (64 x^{4} + 128 x^{3} \log {\relax (x )} + 64 x^{2} \log {\relax (x )}^{2}\right ) e^{- 4 x} + \left (- 256 x^{7} - 768 x^{6} \log {\relax (x )} + 64 x^{6} - 768 x^{5} \log {\relax (x )}^{2} + 128 x^{5} \log {\relax (x )} - 256 x^{4} \log {\relax (x )}^{3} + 64 x^{4} \log {\relax (x )}^{2}\right ) e^{- 2 x}}{x^{10} + 4 x^{9} \log {\relax (x )} + 6 x^{8} \log {\relax (x )}^{2} + 4 x^{7} \log {\relax (x )}^{3} + x^{6} \log {\relax (x )}^{4}} + \frac {- 128 x - 128 \log {\relax (x )} + 16}{x^{2} + 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x**3+512*x**2)*exp(x)**2*ln(x)**2+((128*x**5+128*x**4)*exp(x)**4+(1024*x**4+1152*x**3+128*x**2
)*exp(x)**2-256*x-256)*ln(x)+(128*x**6+96*x**5-32*x**4)*exp(x)**4+(512*x**5+640*x**4-128*x**2)*exp(x)**2-256*x
**2-384*x-128)/(x**5*exp(x)**4*ln(x)**3+3*x**6*exp(x)**4*ln(x)**2+3*x**7*exp(x)**4*ln(x)+x**8*exp(x)**4),x)

[Out]

((64*x**4 + 128*x**3*log(x) + 64*x**2*log(x)**2)*exp(-4*x) + (-256*x**7 - 768*x**6*log(x) + 64*x**6 - 768*x**5
*log(x)**2 + 128*x**5*log(x) - 256*x**4*log(x)**3 + 64*x**4*log(x)**2)*exp(-2*x))/(x**10 + 4*x**9*log(x) + 6*x
**8*log(x)**2 + 4*x**7*log(x)**3 + x**6*log(x)**4) + (-128*x - 128*log(x) + 16)/(x**2 + 2*x*log(x) + log(x)**2
)

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