3.80.29 \(\int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x (-4 x^2-512 x^4)+2 e^{2 x} (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10})}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} (x+4 x^2)+e^x (4 x^2+512 x^5+2048 x^6)+2 e^{2 x} (e^{2 x} x^2+512 e^x x^6+65536 x^{10})} \, dx\)

Optimal. Leaf size=26 \[ \log \left (4+2 e^{2 x}+\frac {1}{x}+\frac {4}{e^x+256 x^4}\right ) \]

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Rubi [F]  time = 4.09, 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^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^(2*x) - 4096*x^5 - 65536*x^8 + E^x*(-4*x^2 - 512*x^4) + 2*E^(2*x)*(2*E^(2*x)*x^2 + 1024*E^x*x^6 + 1310
72*x^10))/(1024*x^6 + 65536*x^9 + 262144*x^10 + E^(2*x)*(x + 4*x^2) + E^x*(4*x^2 + 512*x^5 + 2048*x^6) + 2*E^(
2*x)*(E^(2*x)*x^2 + 512*E^x*x^6 + 65536*x^10)),x]

[Out]

2*x - 1024*Defer[Int][x^3/(E^x + 256*x^4), x] + 256*Defer[Int][x^4/(E^x + 256*x^4), x] - 2*Defer[Int][E^x/(E^x
+ 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x^5 + 512*E^(2*x)*x^5), x] - Defer[Int][E^x/(x*(E^x + 4*x + 4*
E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x^5 + 512*E^(2*x)*x^5)), x] - 12*Defer[Int][x/(E^x + 4*x + 4*E^x*x + 2*E^
(3*x)*x + 256*x^4 + 1024*x^5 + 512*E^(2*x)*x^5), x] - 8*Defer[Int][(E^x*x)/(E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x
+ 256*x^4 + 1024*x^5 + 512*E^(2*x)*x^5), x] + 768*Defer[Int][x^3/(E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4
+ 1024*x^5 + 512*E^(2*x)*x^5), x] + 3328*Defer[Int][x^4/(E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x^
5 + 512*E^(2*x)*x^5), x] + 2048*Defer[Int][(E^(2*x)*x^4)/(E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x
^5 + 512*E^(2*x)*x^5), x] - 3072*Defer[Int][x^5/(E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x^5 + 512*
E^(2*x)*x^5), x] - 512*Defer[Int][(E^(2*x)*x^5)/(E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x^5 + 512*
E^(2*x)*x^5), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {256 (-4+x) x^3}{e^x+256 x^4}-\frac {e^x+2 e^x x+12 x^2+8 e^x x^2-768 x^4-3328 x^5-2048 e^{2 x} x^5+3072 x^6+512 e^{2 x} x^6}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )}\right ) \, dx\\ &=2 x+256 \int \frac {(-4+x) x^3}{e^x+256 x^4} \, dx-\int \frac {e^x+2 e^x x+12 x^2+8 e^x x^2-768 x^4-3328 x^5-2048 e^{2 x} x^5+3072 x^6+512 e^{2 x} x^6}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )} \, dx\\ &=2 x+256 \int \left (-\frac {4 x^3}{e^x+256 x^4}+\frac {x^4}{e^x+256 x^4}\right ) \, dx-\int \left (\frac {2 e^x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {e^x}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )}+\frac {12 x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {8 e^x x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}-\frac {768 x^3}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}-\frac {3328 x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}-\frac {2048 e^{2 x} x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {3072 x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {512 e^{2 x} x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}\right ) \, dx\\ &=2 x-2 \int \frac {e^x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-8 \int \frac {e^x x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-12 \int \frac {x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx+256 \int \frac {x^4}{e^x+256 x^4} \, dx-512 \int \frac {e^{2 x} x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx+768 \int \frac {x^3}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-1024 \int \frac {x^3}{e^x+256 x^4} \, dx+2048 \int \frac {e^{2 x} x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-3072 \int \frac {x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx+3328 \int \frac {x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-\int \frac {e^x}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.15, size = 59, normalized size = 2.27 \begin {gather*} -\log (x)-\log \left (e^x+256 x^4\right )+\log \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(2*x) - 4096*x^5 - 65536*x^8 + E^x*(-4*x^2 - 512*x^4) + 2*E^(2*x)*(2*E^(2*x)*x^2 + 1024*E^x*x^6
+ 131072*x^10))/(1024*x^6 + 65536*x^9 + 262144*x^10 + E^(2*x)*(x + 4*x^2) + E^x*(4*x^2 + 512*x^5 + 2048*x^6) +
2*E^(2*x)*(E^(2*x)*x^2 + 512*E^x*x^6 + 65536*x^10)),x]

[Out]

-Log[x] - Log[E^x + 256*x^4] + Log[E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256*x^4 + 1024*x^5 + 512*E^(2*x)*x^5]

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fricas [B]  time = 0.70, size = 55, normalized size = 2.12 \begin {gather*} -\log \left (256 \, x^{4} + e^{x}\right ) + \log \left (\frac {512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + {\left (4 \, x + 1\right )} e^{x} + 4 \, x}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-exp(x)^2+(-512*x^4-4*x^2)*exp(x)-65536
*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6*exp(x)+65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+
4*x^2)*exp(x)+262144*x^10+65536*x^9+1024*x^6),x, algorithm="fricas")

[Out]

-log(256*x^4 + e^x) + log((512*x^5*e^(2*x) + 1024*x^5 + 256*x^4 + 2*x*e^(3*x) + (4*x + 1)*e^x + 4*x)/x)

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giac [B]  time = 2.85, size = 54, normalized size = 2.08 \begin {gather*} \log \left (512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + 4 \, x e^{x} + 4 \, x + e^{x}\right ) - \log \left (256 \, x^{4} + e^{x}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-exp(x)^2+(-512*x^4-4*x^2)*exp(x)-65536
*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6*exp(x)+65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+
4*x^2)*exp(x)+262144*x^10+65536*x^9+1024*x^6),x, algorithm="giac")

[Out]

log(512*x^5*e^(2*x) + 1024*x^5 + 256*x^4 + 2*x*e^(3*x) + 4*x*e^x + 4*x + e^x) - log(256*x^4 + e^x) - log(x)

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maple [A]  time = 0.07, size = 51, normalized size = 1.96

method	result	size
risch	\(-\ln \left ({\mathrm e}^{x}+256 x^{4}\right )+\ln \left ({\mathrm e}^{3 x}+256 \,{\mathrm e}^{2 x} x^{4}+\frac {\left (4 x +1\right ) {\mathrm e}^{x}}{2 x}+512 x^{4}+128 x^{3}+2\right )\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(ln(2)+2*x)-exp(x)^2+(-512*x^4-4*x^2)*exp(x)-65536*x^8-40
96*x^5)/((exp(x)^2*x^2+512*x^6*exp(x)+65536*x^10)*exp(ln(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+4*x^2)*e
xp(x)+262144*x^10+65536*x^9+1024*x^6),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(x)+256*x^4)+ln(exp(3*x)+256*exp(2*x)*x^4+1/2*(4*x+1)/x*exp(x)+512*x^4+128*x^3+2)

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maxima [B]  time = 0.44, size = 56, normalized size = 2.15 \begin {gather*} -\log \left (256 \, x^{4} + e^{x}\right ) + \log \left (\frac {512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + {\left (4 \, x + 1\right )} e^{x} + 4 \, x}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-exp(x)^2+(-512*x^4-4*x^2)*exp(x)-65536
*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6*exp(x)+65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+
4*x^2)*exp(x)+262144*x^10+65536*x^9+1024*x^6),x, algorithm="maxima")

[Out]

-log(256*x^4 + e^x) + log(1/2*(512*x^5*e^(2*x) + 1024*x^5 + 256*x^4 + 2*x*e^(3*x) + (4*x + 1)*e^x + 4*x)/x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (512\,x^4+4\,x^2\right )-{\mathrm {e}}^{2\,x+\ln \relax (2)}\,\left (1024\,x^6\,{\mathrm {e}}^x+2\,x^2\,{\mathrm {e}}^{2\,x}+131072\,x^{10}\right )+4096\,x^5+65536\,x^8}{{\mathrm {e}}^x\,\left (2048\,x^6+512\,x^5+4\,x^2\right )+{\mathrm {e}}^{2\,x+\ln \relax (2)}\,\left (512\,x^6\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+65536\,x^{10}\right )+{\mathrm {e}}^{2\,x}\,\left (4\,x^2+x\right )+1024\,x^6+65536\,x^9+262144\,x^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x) + exp(x)*(4*x^2 + 512*x^4) - exp(2*x + log(2))*(1024*x^6*exp(x) + 2*x^2*exp(2*x) + 131072*x^10)
+ 4096*x^5 + 65536*x^8)/(exp(x)*(4*x^2 + 512*x^5 + 2048*x^6) + exp(2*x + log(2))*(512*x^6*exp(x) + x^2*exp(2*
x) + 65536*x^10) + exp(2*x)*(x + 4*x^2) + 1024*x^6 + 65536*x^9 + 262144*x^10),x)

[Out]

int(-(exp(2*x) + exp(x)*(4*x^2 + 512*x^4) - exp(2*x + log(2))*(1024*x^6*exp(x) + 2*x^2*exp(2*x) + 131072*x^10)
+ 4096*x^5 + 65536*x^8)/(exp(x)*(4*x^2 + 512*x^5 + 2048*x^6) + exp(2*x + log(2))*(512*x^6*exp(x) + x^2*exp(2*
x) + 65536*x^10) + exp(2*x)*(x + 4*x^2) + 1024*x^6 + 65536*x^9 + 262144*x^10), x)

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sympy [B]  time = 1.70, size = 49, normalized size = 1.88 \begin {gather*} - \log {\left (256 x^{4} + e^{x} \right )} + \log {\left (256 x^{4} e^{2 x} + 512 x^{4} + 128 x^{3} + e^{3 x} + 2 + \frac {\left (4 x + 1\right ) e^{x}}{2 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)**2*x**2+1024*x**6*exp(x)+131072*x**10)*exp(ln(2)+2*x)-exp(x)**2+(-512*x**4-4*x**2)*exp(x)
-65536*x**8-4096*x**5)/((exp(x)**2*x**2+512*x**6*exp(x)+65536*x**10)*exp(ln(2)+2*x)+(4*x**2+x)*exp(x)**2+(2048
*x**6+512*x**5+4*x**2)*exp(x)+262144*x**10+65536*x**9+1024*x**6),x)

[Out]

-log(256*x**4 + exp(x)) + log(256*x**4*exp(2*x) + 512*x**4 + 128*x**3 + exp(3*x) + 2 + (4*x + 1)*exp(x)/(2*x))

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