3.17.31 \(\int \frac {512 x^5+768 x^6+256 x^7+e^x (-512 x^3+512 x^4+1152 x^5+384 x^6)+(-1024 x^4-1280 x^5-384 x^6+e^x (-1024 x^3-1280 x^4-384 x^5)) \log (\frac {e^x+x}{x})}{-e^x x^3-x^4+(3 e^x x^2+3 x^3) \log (\frac {e^x+x}{x})+(-3 e^x x-3 x^2) \log ^2(\frac {e^x+x}{x})+(e^x+x) \log ^3(\frac {e^x+x}{x})} \, dx\)

Optimal. Leaf size=29 \[ -3+e-\frac {64 x^4 (2+x)^2}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2} \]

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Rubi [F]  time = 5.42, 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 {512 x^5+768 x^6+256 x^7+e^x \left (-512 x^3+512 x^4+1152 x^5+384 x^6\right )+\left (-1024 x^4-1280 x^5-384 x^6+e^x \left (-1024 x^3-1280 x^4-384 x^5\right )\right ) \log \left (\frac {e^x+x}{x}\right )}{-e^x x^3-x^4+\left (3 e^x x^2+3 x^3\right ) \log \left (\frac {e^x+x}{x}\right )+\left (-3 e^x x-3 x^2\right ) \log ^2\left (\frac {e^x+x}{x}\right )+\left (e^x+x\right ) \log ^3\left (\frac {e^x+x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(512*x^5 + 768*x^6 + 256*x^7 + E^x*(-512*x^3 + 512*x^4 + 1152*x^5 + 384*x^6) + (-1024*x^4 - 1280*x^5 - 384
*x^6 + E^x*(-1024*x^3 - 1280*x^4 - 384*x^5))*Log[(E^x + x)/x])/(-(E^x*x^3) - x^4 + (3*E^x*x^2 + 3*x^3)*Log[(E^
x + x)/x] + (-3*E^x*x - 3*x^2)*Log[(E^x + x)/x]^2 + (E^x + x)*Log[(E^x + x)/x]^3),x]

[Out]

512*Defer[Int][x^3/(x - Log[(E^x + x)/x])^3, x] + 512*Defer[Int][x^4/(x - Log[(E^x + x)/x])^3, x] + 128*Defer[
Int][x^5/(x - Log[(E^x + x)/x])^3, x] - 512*Defer[Int][x^4/((E^x + x)*(x - Log[(E^x + x)/x])^3), x] + 384*Defe
r[Int][x^6/((E^x + x)*(x - Log[(E^x + x)/x])^3), x] + 128*Defer[Int][x^7/((E^x + x)*(x - Log[(E^x + x)/x])^3),
x] - 1024*Defer[Int][x^3/(x - Log[(E^x + x)/x])^2, x] - 1280*Defer[Int][x^4/(x - Log[(E^x + x)/x])^2, x] - 38
4*Defer[Int][x^5/(x - Log[(E^x + x)/x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {128 x^3 (2+x) \left (-2 x^2 (1+x)-e^x \left (-2+3 x+3 x^2\right )+\left (e^x+x\right ) (4+3 x) \log \left (\frac {e^x+x}{x}\right )\right )}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx\\ &=128 \int \frac {x^3 (2+x) \left (-2 x^2 (1+x)-e^x \left (-2+3 x+3 x^2\right )+\left (e^x+x\right ) (4+3 x) \log \left (\frac {e^x+x}{x}\right )\right )}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx\\ &=128 \int \left (\frac {(-1+x) x^4 (2+x)^2}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}-\frac {x^3 (2+x) \left (-2+3 x+3 x^2-4 \log \left (\frac {e^x+x}{x}\right )-3 x \log \left (\frac {e^x+x}{x}\right )\right )}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}\right ) \, dx\\ &=128 \int \frac {(-1+x) x^4 (2+x)^2}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-128 \int \frac {x^3 (2+x) \left (-2+3 x+3 x^2-4 \log \left (\frac {e^x+x}{x}\right )-3 x \log \left (\frac {e^x+x}{x}\right )\right )}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx\\ &=128 \int \left (-\frac {4 x^4}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}+\frac {3 x^6}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}+\frac {x^7}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}\right ) \, dx-128 \int \left (-\frac {x^3 (2+x)^2}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}+\frac {x^3 \left (8+10 x+3 x^2\right )}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2}\right ) \, dx\\ &=128 \int \frac {x^3 (2+x)^2}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx+128 \int \frac {x^7}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-128 \int \frac {x^3 \left (8+10 x+3 x^2\right )}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2} \, dx+384 \int \frac {x^6}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-512 \int \frac {x^4}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx\\ &=128 \int \left (\frac {4 x^3}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}+\frac {4 x^4}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}+\frac {x^5}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3}\right ) \, dx-128 \int \left (\frac {8 x^3}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2}+\frac {10 x^4}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2}+\frac {3 x^5}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2}\right ) \, dx+128 \int \frac {x^7}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx+384 \int \frac {x^6}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-512 \int \frac {x^4}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx\\ &=128 \int \frac {x^5}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx+128 \int \frac {x^7}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx+384 \int \frac {x^6}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-384 \int \frac {x^5}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2} \, dx+512 \int \frac {x^3}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx+512 \int \frac {x^4}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-512 \int \frac {x^4}{\left (e^x+x\right ) \left (x-\log \left (\frac {e^x+x}{x}\right )\right )^3} \, dx-1024 \int \frac {x^3}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2} \, dx-1280 \int \frac {x^4}{\left (x-\log \left (\frac {e^x+x}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.19, size = 26, normalized size = 0.90 \begin {gather*} -\frac {64 x^4 (2+x)^2}{\left (-x+\log \left (\frac {e^x+x}{x}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(512*x^5 + 768*x^6 + 256*x^7 + E^x*(-512*x^3 + 512*x^4 + 1152*x^5 + 384*x^6) + (-1024*x^4 - 1280*x^5
- 384*x^6 + E^x*(-1024*x^3 - 1280*x^4 - 384*x^5))*Log[(E^x + x)/x])/(-(E^x*x^3) - x^4 + (3*E^x*x^2 + 3*x^3)*L
og[(E^x + x)/x] + (-3*E^x*x - 3*x^2)*Log[(E^x + x)/x]^2 + (E^x + x)*Log[(E^x + x)/x]^3),x]

[Out]

(-64*x^4*(2 + x)^2)/(-x + Log[(E^x + x)/x])^2

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fricas [A]  time = 0.70, size = 45, normalized size = 1.55 \begin {gather*} -\frac {64 \, {\left (x^{6} + 4 \, x^{5} + 4 \, x^{4}\right )}}{x^{2} - 2 \, x \log \left (\frac {x + e^{x}}{x}\right ) + \log \left (\frac {x + e^{x}}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*log(1/x*(exp(x)+x))+(384*x^6+1152*x
^5+512*x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*log(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*log(
1/x*(exp(x)+x))^2+(3*exp(x)*x^2+3*x^3)*log(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x, algorithm="fricas")

[Out]

-64*(x^6 + 4*x^5 + 4*x^4)/(x^2 - 2*x*log((x + e^x)/x) + log((x + e^x)/x)^2)

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giac [A]  time = 0.80, size = 45, normalized size = 1.55 \begin {gather*} -\frac {64 \, {\left (x^{6} + 4 \, x^{5} + 4 \, x^{4}\right )}}{x^{2} - 2 \, x \log \left (\frac {x + e^{x}}{x}\right ) + \log \left (\frac {x + e^{x}}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*log(1/x*(exp(x)+x))+(384*x^6+1152*x
^5+512*x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*log(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*log(
1/x*(exp(x)+x))^2+(3*exp(x)*x^2+3*x^3)*log(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x, algorithm="giac")

[Out]

-64*(x^6 + 4*x^5 + 4*x^4)/(x^2 - 2*x*log((x + e^x)/x) + log((x + e^x)/x)^2)

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maple [C]  time = 0.48, size = 127, normalized size = 4.38

method	result	size
risch	\(-\frac {256 \left (x^{2}+4 x +4\right ) x^{4}}{\left (i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 x +2 \ln \relax (x )-2 \ln \left ({\mathrm e}^{x}+x \right )\right )^{2}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*ln(1/x*(exp(x)+x))+(384*x^6+1152*x^5+512*
x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*ln(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*ln(1/x*(exp(
x)+x))^2+(3*exp(x)*x^2+3*x^3)*ln(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x,method=_RETURNVERBOSE)

[Out]

-256*(x^2+4*x+4)*x^4/(I*Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))-I*Pi*csgn(I/x)*csgn(I/x*(exp(x)+x
))^2-I*Pi*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2+I*Pi*csgn(I/x*(exp(x)+x))^3+2*x+2*ln(x)-2*ln(exp(x)+x))^2

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maxima [A]  time = 0.68, size = 49, normalized size = 1.69 \begin {gather*} -\frac {64 \, {\left (x^{6} + 4 \, x^{5} + 4 \, x^{4}\right )}}{x^{2} - 2 \, {\left (x + \log \relax (x)\right )} \log \left (x + e^{x}\right ) + \log \left (x + e^{x}\right )^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*log(1/x*(exp(x)+x))+(384*x^6+1152*x
^5+512*x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*log(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*log(
1/x*(exp(x)+x))^2+(3*exp(x)*x^2+3*x^3)*log(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x, algorithm="maxima")

[Out]

-64*(x^6 + 4*x^5 + 4*x^4)/(x^2 - 2*(x + log(x))*log(x + e^x) + log(x + e^x)^2 + 2*x*log(x) + log(x)^2)

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mupad [B]  time = 1.64, size = 540, normalized size = 18.62 \begin {gather*} \frac {\frac {64\,x\,\left (x+{\mathrm {e}}^x\right )\,\left (24\,x^5\,{\mathrm {e}}^x+48\,x^6\,{\mathrm {e}}^x+43\,x^7\,{\mathrm {e}}^x+19\,x^8\,{\mathrm {e}}^x+3\,x^9\,{\mathrm {e}}^x-8\,x^3\,{\mathrm {e}}^{2\,x}+22\,x^4\,{\mathrm {e}}^{2\,x}+47\,x^5\,{\mathrm {e}}^{2\,x}+18\,x^6\,{\mathrm {e}}^{2\,x}+16\,x^7+30\,x^8+12\,x^9\right )}{{\left ({\mathrm {e}}^x+x^2\right )}^3}-\frac {64\,x\,\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )\,\left (x+{\mathrm {e}}^x\right )\,\left (40\,x^4\,{\mathrm {e}}^x+68\,x^5\,{\mathrm {e}}^x+49\,x^6\,{\mathrm {e}}^x+19\,x^7\,{\mathrm {e}}^x+3\,x^8\,{\mathrm {e}}^x+32\,x^3\,{\mathrm {e}}^{2\,x}+50\,x^4\,{\mathrm {e}}^{2\,x}+18\,x^5\,{\mathrm {e}}^{2\,x}+24\,x^6+40\,x^7+15\,x^8\right )}{{\left ({\mathrm {e}}^x+x^2\right )}^3}}{x-\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )}+\frac {\frac {64\,x^4\,\left (x+2\right )\,\left (3\,x^2\,{\mathrm {e}}^x-2\,{\mathrm {e}}^x+3\,x\,{\mathrm {e}}^x+2\,x^2+2\,x^3\right )}{{\mathrm {e}}^x+x^2}-\frac {64\,x^4\,\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )\,\left (x+{\mathrm {e}}^x\right )\,\left (3\,x^2+10\,x+8\right )}{{\mathrm {e}}^x+x^2}}{x^2-2\,x\,\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )+{\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )}^2}-2048\,x^4-3200\,x^5-1152\,x^6-\frac {64\,\left (-3\,x^{11}+41\,x^{10}+13\,x^9-188\,x^8-28\,x^7+144\,x^6\right )}{\left ({\mathrm {e}}^x+x^2\right )\,\left (2\,x-x^2\right )}-\frac {64\,\left (-3\,x^{15}+8\,x^{14}+13\,x^{13}-46\,x^{12}+4\,x^{11}+56\,x^{10}-32\,x^9\right )}{\left (2\,x-x^2\right )\,\left ({\mathrm {e}}^{3\,x}+3\,x^4\,{\mathrm {e}}^x+3\,x^2\,{\mathrm {e}}^{2\,x}+x^6\right )}+\frac {64\,\left (-6\,x^{13}+31\,x^{12}+12\,x^{11}-151\,x^{10}+50\,x^9+144\,x^8-80\,x^7\right )}{\left (2\,x-x^2\right )\,\left ({\mathrm {e}}^{2\,x}+2\,x^2\,{\mathrm {e}}^x+x^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(512*x^4 - 512*x^3 + 1152*x^5 + 384*x^6) + 512*x^5 + 768*x^6 + 256*x^7 - log((x + exp(x))/x)*(exp
(x)*(1024*x^3 + 1280*x^4 + 384*x^5) + 1024*x^4 + 1280*x^5 + 384*x^6))/(x^3*exp(x) + log((x + exp(x))/x)^2*(3*x
*exp(x) + 3*x^2) - log((x + exp(x))/x)*(3*x^2*exp(x) + 3*x^3) - log((x + exp(x))/x)^3*(x + exp(x)) + x^4),x)

[Out]

((64*x*(x + exp(x))*(24*x^5*exp(x) + 48*x^6*exp(x) + 43*x^7*exp(x) + 19*x^8*exp(x) + 3*x^9*exp(x) - 8*x^3*exp(
2*x) + 22*x^4*exp(2*x) + 47*x^5*exp(2*x) + 18*x^6*exp(2*x) + 16*x^7 + 30*x^8 + 12*x^9))/(exp(x) + x^2)^3 - (64
*x*log((x + exp(x))/x)*(x + exp(x))*(40*x^4*exp(x) + 68*x^5*exp(x) + 49*x^6*exp(x) + 19*x^7*exp(x) + 3*x^8*exp
(x) + 32*x^3*exp(2*x) + 50*x^4*exp(2*x) + 18*x^5*exp(2*x) + 24*x^6 + 40*x^7 + 15*x^8))/(exp(x) + x^2)^3)/(x -
log((x + exp(x))/x)) + ((64*x^4*(x + 2)*(3*x^2*exp(x) - 2*exp(x) + 3*x*exp(x) + 2*x^2 + 2*x^3))/(exp(x) + x^2)
- (64*x^4*log((x + exp(x))/x)*(x + exp(x))*(10*x + 3*x^2 + 8))/(exp(x) + x^2))/(log((x + exp(x))/x)^2 - 2*x*l
og((x + exp(x))/x) + x^2) - 2048*x^4 - 3200*x^5 - 1152*x^6 - (64*(144*x^6 - 28*x^7 - 188*x^8 + 13*x^9 + 41*x^1
0 - 3*x^11))/((exp(x) + x^2)*(2*x - x^2)) - (64*(56*x^10 - 32*x^9 + 4*x^11 - 46*x^12 + 13*x^13 + 8*x^14 - 3*x^
15))/((2*x - x^2)*(exp(3*x) + 3*x^4*exp(x) + 3*x^2*exp(2*x) + x^6)) + (64*(144*x^8 - 80*x^7 + 50*x^9 - 151*x^1
0 + 12*x^11 + 31*x^12 - 6*x^13))/((2*x - x^2)*(exp(2*x) + 2*x^2*exp(x) + x^4))

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sympy [A]  time = 0.27, size = 41, normalized size = 1.41 \begin {gather*} \frac {- 64 x^{6} - 256 x^{5} - 256 x^{4}}{x^{2} - 2 x \log {\left (\frac {x + e^{x}}{x} \right )} + \log {\left (\frac {x + e^{x}}{x} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x**5-1280*x**4-1024*x**3)*exp(x)-384*x**6-1280*x**5-1024*x**4)*ln(1/x*(exp(x)+x))+(384*x**6+
1152*x**5+512*x**4-512*x**3)*exp(x)+256*x**7+768*x**6+512*x**5)/((exp(x)+x)*ln(1/x*(exp(x)+x))**3+(-3*exp(x)*x
-3*x**2)*ln(1/x*(exp(x)+x))**2+(3*exp(x)*x**2+3*x**3)*ln(1/x*(exp(x)+x))-exp(x)*x**3-x**4),x)

[Out]

(-64*x**6 - 256*x**5 - 256*x**4)/(x**2 - 2*x*log((x + exp(x))/x) + log((x + exp(x))/x)**2)

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