3.67.90 \(\int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} (-16 x^4+7 x^5-x^6)+e^2 (16 x^4-8 x^5+x^6)}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6)+e^4 (32 x^4-16 x^5+2 x^6)+e^{\frac {2}{-4+x}} (32 x^4-16 x^5+2 x^6)+e^{\frac {1}{-4+x}} (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 (-64 x^4+32 x^5-4 x^6))} \, dx\)

Optimal. Leaf size=31 \[ 5+\frac {x}{2 \left (e^2-e^{\frac {1}{-4+x}}-4 \left (25-\frac {2}{x^2}\right )\right )} \]

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Rubi [F]  time = 4.98, 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 {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(384*x^2 - 192*x^3 - 1576*x^4 + 800*x^5 - 100*x^6 + E^(-4 + x)^(-1)*(-16*x^4 + 7*x^5 - x^6) + E^2*(16*x^4
- 8*x^5 + x^6))/(2048 - 1024*x - 51072*x^2 + 25600*x^3 + 316800*x^4 - 160000*x^5 + 20000*x^6 + E^2*(512*x^2 -
256*x^3 - 6368*x^4 + 3200*x^5 - 400*x^6) + E^4*(32*x^4 - 16*x^5 + 2*x^6) + E^(2/(-4 + x))*(32*x^4 - 16*x^5 + 2
*x^6) + E^(-4 + x)^(-1)*(-512*x^2 + 256*x^3 + 6368*x^4 - 3200*x^5 + 400*x^6 + E^2*(-64*x^4 + 32*x^5 - 4*x^6)))
,x]

[Out]

32*(399 - 4*E^2)*Defer[Int][(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)^(-2), x] + 256*(199 - 2*E^2)*Def
er[Int][1/((4 - x)^2*(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)^2), x] - 64*(997 - 10*E^2)*Defer[Int][1
/((4 - x)*(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)^2), x] + 4*(599 - 6*E^2)*Defer[Int][x/(8 - E^(-4 +
x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)^2, x] + 4*(102 - E^2)*Defer[Int][x^2/(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 -
E^2/100)*x^2)^2, x] + ((100 - E^2)*Defer[Int][x^3/(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)^2, x])/2
+ 4*Defer[Int][(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)^(-1), x] + 32*Defer[Int][1/((4 - x)^2*(8 - E^
(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2)), x] + 24*Defer[Int][1/((-4 + x)*(8 - E^(-4 + x)^(-1)*x^2 - 100*(1
- E^2/100)*x^2)), x] + Defer[Int][x/(8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2), x]/2 + Defer[Int][x^2/(
8 - E^(-4 + x)^(-1)*x^2 - 100*(1 - E^2/100)*x^2), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (384-192 x-8 \left (197-2 e^2+2 e^{\frac {1}{-4+x}}\right ) x^2-\left (-800+8 e^2-7 e^{\frac {1}{-4+x}}\right ) x^3-\left (100-e^2+e^{\frac {1}{-4+x}}\right ) x^4\right )}{2 (4-x)^2 \left (8-\left (100-e^2+e^{\frac {1}{-4+x}}\right ) x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {x^2 \left (384-192 x-8 \left (197-2 e^2+2 e^{\frac {1}{-4+x}}\right ) x^2-\left (-800+8 e^2-7 e^{\frac {1}{-4+x}}\right ) x^3-\left (100-e^2+e^{\frac {1}{-4+x}}\right ) x^4\right )}{(4-x)^2 \left (8-\left (100-e^2+e^{\frac {1}{-4+x}}\right ) x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {x^2 \left (16-7 x+x^2\right )}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )}+\frac {x^2 \left (256-136 x+16 x^2+\left (100-e^2\right ) x^3\right )}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x^2 \left (16-7 x+x^2\right )}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )} \, dx+\frac {1}{2} \int \frac {x^2 \left (256-136 x+16 x^2+\left (100-e^2\right ) x^3\right )}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {64 \left (399-4 e^2\right )}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}+\frac {512 \left (199-2 e^2\right )}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}+\frac {128 \left (-997+10 e^2\right )}{(4-x) \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}+\frac {8 \left (599-6 e^2\right ) x}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}+\frac {8 \left (102-e^2\right ) x^2}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}+\frac {\left (100-e^2\right ) x^3}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2}\right ) \, dx+\frac {1}{2} \int \left (\frac {8}{8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2}+\frac {64}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )}+\frac {48}{(-4+x) \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )}+\frac {x}{8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2}+\frac {x^2}{8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x}{8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2} \, dx+\frac {1}{2} \int \frac {x^2}{8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2} \, dx+4 \int \frac {1}{8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2} \, dx+24 \int \frac {1}{(-4+x) \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )} \, dx+32 \int \frac {1}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )} \, dx-\left (64 \left (997-10 e^2\right )\right ) \int \frac {1}{(4-x) \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx+\left (4 \left (599-6 e^2\right )\right ) \int \frac {x}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx+\left (32 \left (399-4 e^2\right )\right ) \int \frac {1}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx+\left (256 \left (199-2 e^2\right )\right ) \int \frac {1}{(4-x)^2 \left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx+\frac {1}{2} \left (100-e^2\right ) \int \frac {x^3}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx+\left (4 \left (102-e^2\right )\right ) \int \frac {x^2}{\left (8-e^{\frac {1}{-4+x}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 29, normalized size = 0.94 \begin {gather*} -\frac {x^3}{2 \left (-8+\left (100-e^2+e^{\frac {1}{-4+x}}\right ) x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(384*x^2 - 192*x^3 - 1576*x^4 + 800*x^5 - 100*x^6 + E^(-4 + x)^(-1)*(-16*x^4 + 7*x^5 - x^6) + E^2*(1
6*x^4 - 8*x^5 + x^6))/(2048 - 1024*x - 51072*x^2 + 25600*x^3 + 316800*x^4 - 160000*x^5 + 20000*x^6 + E^2*(512*
x^2 - 256*x^3 - 6368*x^4 + 3200*x^5 - 400*x^6) + E^4*(32*x^4 - 16*x^5 + 2*x^6) + E^(2/(-4 + x))*(32*x^4 - 16*x
^5 + 2*x^6) + E^(-4 + x)^(-1)*(-512*x^2 + 256*x^3 + 6368*x^4 - 3200*x^5 + 400*x^6 + E^2*(-64*x^4 + 32*x^5 - 4*
x^6))),x]

[Out]

-1/2*x^3/(-8 + (100 - E^2 + E^(-4 + x)^(-1))*x^2)

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fricas [A]  time = 0.48, size = 31, normalized size = 1.00 \begin {gather*} \frac {x^{3}}{2 \, {\left (x^{2} e^{2} - x^{2} e^{\left (\frac {1}{x - 4}\right )} - 100 \, x^{2} + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100*x^6+800*x^5-1576*x^4-192*x^3+384*x^2
)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x
^2)*exp(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4-256*x^3+512*x^2)*exp(2)+20000*x^6-
160000*x^5+316800*x^4+25600*x^3-51072*x^2-1024*x+2048),x, algorithm="fricas")

[Out]

1/2*x^3/(x^2*e^2 - x^2*e^(1/(x - 4)) - 100*x^2 + 8)

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giac [A]  time = 0.29, size = 41, normalized size = 1.32 \begin {gather*} \frac {x^{3} e^{\frac {1}{4}}}{2 \, {\left (x^{2} e^{\frac {9}{4}} - 100 \, x^{2} e^{\frac {1}{4}} - x^{2} e^{\left (\frac {x}{4 \, {\left (x - 4\right )}}\right )} + 8 \, e^{\frac {1}{4}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100*x^6+800*x^5-1576*x^4-192*x^3+384*x^2
)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x
^2)*exp(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4-256*x^3+512*x^2)*exp(2)+20000*x^6-
160000*x^5+316800*x^4+25600*x^3-51072*x^2-1024*x+2048),x, algorithm="giac")

[Out]

1/2*x^3*e^(1/4)/(x^2*e^(9/4) - 100*x^2*e^(1/4) - x^2*e^(1/4*x/(x - 4)) + 8*e^(1/4))

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maple [A]  time = 0.59, size = 32, normalized size = 1.03

method	result	size
risch	\(\frac {x^{3}}{2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}^{\frac {1}{x -4}}-200 x^{2}+16}\)	\(32\)
norman	\(\frac {-2 x^{3}+\frac {1}{2} x^{4}}{\left (x -4\right ) \left (x^{2} {\mathrm e}^{2}-x^{2} {\mathrm e}^{\frac {1}{x -4}}-100 x^{2}+8\right )}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100*x^6+800*x^5-1576*x^4-192*x^3+384*x^2)/((2*
x^6-16*x^5+32*x^4)*exp(1/(x-4))^2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x^2)*ex
p(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4-256*x^3+512*x^2)*exp(2)+20000*x^6-160000
*x^5+316800*x^4+25600*x^3-51072*x^2-1024*x+2048),x,method=_RETURNVERBOSE)

[Out]

1/2*x^3/(x^2*exp(2)-x^2*exp(1/(x-4))-100*x^2+8)

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maxima [A]  time = 0.43, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{3}}{2 \, {\left (x^{2} {\left (e^{2} - 100\right )} - x^{2} e^{\left (\frac {1}{x - 4}\right )} + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100*x^6+800*x^5-1576*x^4-192*x^3+384*x^2
)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x
^2)*exp(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4-256*x^3+512*x^2)*exp(2)+20000*x^6-
160000*x^5+316800*x^4+25600*x^3-51072*x^2-1024*x+2048),x, algorithm="maxima")

[Out]

1/2*x^3/(x^2*(e^2 - 100) - x^2*e^(1/(x - 4)) + 8)

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mupad [B]  time = 4.49, size = 129, normalized size = 4.16 \begin {gather*} -\frac {{\left (x^5-8\,x^4+16\,x^3\right )}^2\,\left (16\,x^2-x^3\,{\mathrm {e}}^2-136\,x+100\,x^3+256\right )}{2\,x^2\,\left ({\mathrm {e}}^{\frac {1}{x-4}}-\frac {x^2\,{\mathrm {e}}^2-100\,x^2+8}{x^2}\right )\,{\left (x-4\right )}^2\,\left (8\,x^7\,{\mathrm {e}}^2-16\,x^6\,{\mathrm {e}}^2-x^8\,{\mathrm {e}}^2+4096\,x^3-4224\,x^4+1600\,x^5+1336\,x^6-784\,x^7+100\,x^8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1/(x - 4))*(16*x^4 - 7*x^5 + x^6) - exp(2)*(16*x^4 - 8*x^5 + x^6) - 384*x^2 + 192*x^3 + 1576*x^4 - 8
00*x^5 + 100*x^6)/(exp(2/(x - 4))*(32*x^4 - 16*x^5 + 2*x^6) - exp(1/(x - 4))*(exp(2)*(64*x^4 - 32*x^5 + 4*x^6)
+ 512*x^2 - 256*x^3 - 6368*x^4 + 3200*x^5 - 400*x^6) - exp(2)*(256*x^3 - 512*x^2 + 6368*x^4 - 3200*x^5 + 400*
x^6) - 1024*x + exp(4)*(32*x^4 - 16*x^5 + 2*x^6) - 51072*x^2 + 25600*x^3 + 316800*x^4 - 160000*x^5 + 20000*x^6
+ 2048),x)

[Out]

-((16*x^3 - 8*x^4 + x^5)^2*(16*x^2 - x^3*exp(2) - 136*x + 100*x^3 + 256))/(2*x^2*(exp(1/(x - 4)) - (x^2*exp(2)
- 100*x^2 + 8)/x^2)*(x - 4)^2*(8*x^7*exp(2) - 16*x^6*exp(2) - x^8*exp(2) + 4096*x^3 - 4224*x^4 + 1600*x^5 + 1
336*x^6 - 784*x^7 + 100*x^8))

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sympy [A]  time = 0.29, size = 31, normalized size = 1.00 \begin {gather*} - \frac {x^{3}}{2 x^{2} e^{\frac {1}{x - 4}} - 2 x^{2} e^{2} + 200 x^{2} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**6+7*x**5-16*x**4)*exp(1/(x-4))+(x**6-8*x**5+16*x**4)*exp(2)-100*x**6+800*x**5-1576*x**4-192*x*
*3+384*x**2)/((2*x**6-16*x**5+32*x**4)*exp(1/(x-4))**2+((-4*x**6+32*x**5-64*x**4)*exp(2)+400*x**6-3200*x**5+63
68*x**4+256*x**3-512*x**2)*exp(1/(x-4))+(2*x**6-16*x**5+32*x**4)*exp(2)**2+(-400*x**6+3200*x**5-6368*x**4-256*
x**3+512*x**2)*exp(2)+20000*x**6-160000*x**5+316800*x**4+25600*x**3-51072*x**2-1024*x+2048),x)

[Out]

-x**3/(2*x**2*exp(1/(x - 4)) - 2*x**2*exp(2) + 200*x**2 - 16)

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