3.85.22 \(\int \frac {e^{2 e^4} (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 (-4+x^2))}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 (-128 x^3+128 x^4-32 x^5)} \, dx\)
Optimal. Leaf size=27 \[ \frac {e^{2 e^4}}{\left (-16+\frac {e^8}{(-2+x)^2 x}\right ) x^2} \]
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Rubi [B] time = 48.60, antiderivative size = 5694, normalized size of antiderivative =
210.89, number of steps used = 21, number of rules used = 11, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115,
Rules used = {12, 2074, 2101, 2081, 2079, 822, 800, 634, 618, 204, 628}
result too large to display
Antiderivative was successfully verified.
[In]
Int[(E^(2*E^4)*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + E^8*(-4 + x^2)))/(E^16*x^2 + 4096*x^4 - 8192*x
^5 + 6144*x^6 - 2048*x^7 + 256*x^8 + E^8*(-128*x^3 + 128*x^4 - 32*x^5)),x]
[Out]
(48*E^(-8 + 2*E^4)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3))/(2^(1/3)*(32*2^(1/3) + (-256 + 27*E^8 +
3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 4*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)*(4 - 3*x)) - (18*2^
(2/3)*E^(-8 + 2*E^4)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(4/3)*(48*E^4*(512 - 27*E^8) - (2^(1/3)*(243
*E^20 + 24576*E^4*(16 + (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 144*E^12*(176 + 9*(-512 + 54*E^8
+ 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + Sqrt[-1536 + 81*E^8]*(27*E^16 + 4096*(8 + (-512 + 54*E^8 + 6*E^4*Sqrt[
-1536 + 81*E^8])^(1/3)) - 16*E^8*(160 + 11*(-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3))))*(4 - 3*x))/(-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))/((4608*E^4 - 243*E^12 + 256*Sqrt[-1536 + 81*E^8] - 27*E^8*S
qrt[-1536 + 81*E^8])*(32*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))*((32*2^(2/3))/(-256 + 2
7*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 4*(4 - 3*x))*
(64 - (2048*2^(1/3))/(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3) - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 +
81*E^8])^(2/3) + 4*((32*2^(2/3))/(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-512 + 54*E^8 + 6*E^4*S
qrt[-1536 + 81*E^8])^(1/3))*(4 - 3*x) - 16*(4 - 3*x)^2)) + (4*E^(-8 + 2*E^4))/x + (8*E^(-8 + 2*E^4)*(32 - E^8)
)/(3*(E^8 - 64*x + 64*x^2 - 16*x^3)) + (3*2^(5/6)*E^(-8 + 2*E^4)*Sqrt[3/(729*2^(2/3)*E^16 + 81*2^(2/3)*E^12*Sq
rt[-1536 + 81*E^8] - 192*E^4*Sqrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1
/3)) - 1728*E^8*(8*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*2^(2/3) + 16*(-256
+ 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))]*
(27*E^16*(112*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 16384*(32*2^(1/3) + (-256 + 27*E
^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 512*E^8*(16*2^(1/3) + 5*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])
^(2/3)) + E^4*Sqrt[-1536 + 81*E^8]*(E^8*(336*2^(1/3) - 3*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) +
256*(16*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))))*ArcTan[(2^(1/3)*(27*E^8 + 3*E^4*Sqrt[
-1536 + 81*E^8] - 32*(8 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3))) - 8*(-256 + 27*E^8 + 3*E^4*Sqrt
[-1536 + 81*E^8])^(2/3)*(4 - 3*x))/Sqrt[6*(729*2^(2/3)*E^16 + 81*2^(2/3)*E^12*Sqrt[-1536 + 81*E^8] - 192*E^4*S
qrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 1728*E^8*(8*2^(2/3) + (
-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*2^(2/3) + 16*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 +
81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))]])/(27*2^(2/3)*E^8*(-256 + 27*E
^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536
+ 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*
E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))) - (5184*2^(5/6)*E^(2*E^4)*(512 - 27*E^8)*Sqrt[3/(729*2^(2/3)*E^16 +
81*2^(2/3)*E^12*Sqrt[-1536 + 81*E^8] - 192*E^4*Sqrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-
1536 + 81*E^8])^(1/3)) - 1728*E^8*(8*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*
2^(2/3) + 16*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 +
81*E^8])^(2/3)))]*(2239488*E^24*(16*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 19683*E^32
*(112*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 268435456*(32*2^(1/3) + (-256 + 27*E^8 +
3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 1990656*E^16*(192*2^(1/3) + 25*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^
8])^(2/3)) - 4194304*E^8*(1472*2^(1/3) + 73*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + E^4*Sqrt[-15
36 + 81*E^8]*(20736*E^16*(80*2^(1/3) - 11*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 2187*E^24*(112
*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 2097152*(128*2^(1/3) + 5*(-256 + 27*E^8 + 3*E
^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 73728*E^8*(640*2^(1/3) + 47*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/
3))))*ArcTan[(2^(1/3)*(27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8] - 32*(8 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^
8])^(1/3))) - 8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)*(4 - 3*x))/Sqrt[6*(729*2^(2/3)*E^16 + 81*2^
(2/3)*E^12*Sqrt[-1536 + 81*E^8] - 192*E^4*Sqrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536
+ 81*E^8])^(1/3)) - 1728*E^8*(8*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*2^(2/
3) + 16*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^
8])^(2/3)))]])/((27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536
+ 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 128*(16*2^(1/3) - 2*2^(2/3)*(-256 + 27*E^8 + 3*
E^4*Sqrt[-1536 + 81*E^8])^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*(27*2^(2/3)*E^8*(-256 +
27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-
1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256
+ 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))^3) + (2*E^(-8 + 2*E^4)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E
^8])^(1/3)*(3*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 8*(32*2^(2/3) - 32*(-256 + 27*E^8 + 3*E
^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*Log[2^(1/3)*(32*
2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 4*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])
^(1/3)*(4 - 3*x)])/(27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-15
36 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 +
3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))) - (864*E^(2*E^4)*(512
- 27*E^8)*(256 - 27*E^8 - 3*E^4*Sqrt[-1536 + 81*E^8])*(81*E^16*(144*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-15
36 + 81*E^8])^(2/3)) + 65536*(8*2^(1/3) - 2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 +
27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 768*E^8*(288*2^(1/3) - 9*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1
536 + 81*E^8])^(1/3) + 10*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + E^4*Sqrt[-1536 + 81*E^8]*(9*E^
8*(144*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 768*(16*2^(1/3) - 2^(2/3)*(-256 + 27*E^
8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))))*Log[2^(1/3)*(32*
2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 4*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])
^(1/3)*(4 - 3*x)])/((27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1
536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 128*(16*2^(1/3) - 2*2^(2/3)*(-256 + 27*E^8
+ 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*(27*2^(2/3)*E^8*(-2
56 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sq
rt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))^3) - (E^(-8 + 2*E^4)*(24*2^(1/3)*E^4*Sqrt[-1536 + 81*E^8] +
3*E^8*(72*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 256*(8*2^(1/3) - 2^(2/3)*(-256 + 27
*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*Log[64*(32*2^(
1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E
^8])^(2/3)) - 9*2^(1/3)*E^8*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 2^(1/3)*E^4*Sqrt[-1536
+ 81*E^8]*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 1024*2^(1/3)*x + 108*2^(1/3)*E^8*x + 12
*2^(1/3)*E^4*Sqrt[3*(-512 + 27*E^8)]*x + 128*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)*x - 12
8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)*x + 48*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)
*x^2])/(27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]
*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-
1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))) - (864*E^(2*E^4)*(512 - 27*E^8)*(
2187*E^24*(144*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 8388608*(8*2^(1/3) - 2^(2/3)*(-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 20736
*E^16*(432*2^(1/3) - 9*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 11*(-256 + 27*E^8 + 3*E^4*
Sqrt[-1536 + 81*E^8])^(2/3)) + 98304*E^8*(648*2^(1/3) - 36*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8]
)^(1/3) + 37*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + E^4*Sqrt[-1536 + 81*E^8]*(243*E^16*(144*2^(
1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 196608*(12*2^(1/3) - 2^(2/3)*(-256 + 27*E^8 + 3*E
^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 2304*E^8*(288*2^(1/3) -
9*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 10*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8]
)^(2/3))))*Log[64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8
+ 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 9*2^(1/3)*E^8*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3))
- 2^(1/3)*E^4*Sqrt[-1536 + 81*E^8]*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 1024*2^(1/3)*x
+ 108*2^(1/3)*E^8*x + 12*2^(1/3)*E^4*Sqrt[3*(-512 + 27*E^8)]*x + 128*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-153
6 + 81*E^8])^(1/3)*x - 128*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)*x + 48*(-256 + 27*E^8 + 3*E^4*Sq
rt[-1536 + 81*E^8])^(2/3)*x^2])/((27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3
)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 128*(16*2^(1/3) - 2*2^(2/3)*(-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*(27*2^
(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E
^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8
])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))^3)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 800
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]
Rule 822
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 2074
Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /; !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]
Rule 2079
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/
3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]
Rule 2081
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]
Rule 2101
Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)
)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P
m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,
-1]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^{2 e^4} \int \frac {512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx\\ &=e^{2 e^4} \int \left (-\frac {4}{e^8 x^2}+\frac {16384-768 e^8+3 e^{16}-64 \left (256-9 e^8\right ) x+128 \left (32-e^8\right ) x^2}{e^8 \left (e^8-64 x+64 x^2-16 x^3\right )^2}-\frac {2 \left (-128+e^8+32 x\right )}{e^8 \left (e^8-64 x+64 x^2-16 x^3\right )}\right ) \, dx\\ &=\frac {4 e^{-8+2 e^4}}{x}+e^{-8+2 e^4} \int \frac {16384-768 e^8+3 e^{16}-64 \left (256-9 e^8\right ) x+128 \left (32-e^8\right ) x^2}{\left (e^8-64 x+64 x^2-16 x^3\right )^2} \, dx-\left (2 e^{-8+2 e^4}\right ) \int \frac {-128+e^8+32 x}{e^8-64 x+64 x^2-16 x^3} \, dx\\ &=\frac {4 e^{-8+2 e^4}}{x}+\frac {8 e^{-8+2 e^4} \left (32-e^8\right )}{3 \left (e^8-64 x+64 x^2-16 x^3\right )}-\frac {1}{48} e^{-8+2 e^4} \int \frac {-16 \left (32768-1792 e^8+9 e^{16}\right )+1024 \left (256-11 e^8\right ) x}{\left (e^8-64 x+64 x^2-16 x^3\right )^2} \, dx-\left (2 e^{-8+2 e^4}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{48} \left (2048+48 \left (-128+e^8\right )\right )+32 x}{\frac {1}{27} \left (-256+27 e^8\right )+\frac {64 x}{3}-16 x^3} \, dx,x,-\frac {4}{3}+x\right )\\ &=\frac {4 e^{-8+2 e^4}}{x}+\frac {8 e^{-8+2 e^4} \left (32-e^8\right )}{3 \left (e^8-64 x+64 x^2-16 x^3\right )}-\frac {1}{48} e^{-8+2 e^4} \operatorname {Subst}\left (\int \frac {\frac {1}{48} \left (65536 \left (256-11 e^8\right )-768 \left (32768-1792 e^8+9 e^{16}\right )\right )+1024 \left (256-11 e^8\right ) x}{\left (\frac {1}{27} \left (-256+27 e^8\right )+\frac {64 x}{3}-16 x^3\right )^2} \, dx,x,-\frac {4}{3}+x\right )-\left (512 e^{-8+2 e^4}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{48} \left (2048+48 \left (-128+e^8\right )\right )+32 x}{\left (\frac {4}{3} \left (\frac {32\ 2^{2/3}}{\sqrt [3]{-256+27 e^8+3 e^4 \sqrt {-1536+81 e^8}}}+\sqrt [3]{-512+54 e^8+6 e^4 \sqrt {-1536+81 e^8}}\right )-16 x\right ) \left (-\frac {16}{9} \left (64-\frac {2048 \sqrt [3]{2}}{\left (-256+27 e^8+3 e^4 \sqrt {-1536+81 e^8}\right )^{2/3}}-\left (-512+54 e^8+6 e^4 \sqrt {-1536+81 e^8}\right )^{2/3}\right )+\frac {64}{3} \left (\frac {32\ 2^{2/3}}{\sqrt [3]{-256+27 e^8+3 e^4 \sqrt {-1536+81 e^8}}}+\sqrt [3]{-512+54 e^8+6 e^4 \sqrt {-1536+81 e^8}}\right ) x+256 x^2\right )} \, dx,x,-\frac {4}{3}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 33, normalized size = 1.22 \begin {gather*} -\frac {e^{2 e^4} (-2+x)^2}{x \left (-e^8+16 (-2+x)^2 x\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^(2*E^4)*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + E^8*(-4 + x^2)))/(E^16*x^2 + 4096*x^4 -
8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x^8 + E^8*(-128*x^3 + 128*x^4 - 32*x^5)),x]
[Out]
-((E^(2*E^4)*(-2 + x)^2)/(x*(-E^8 + 16*(-2 + x)^2*x)))
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fricas [A] time = 0.59, size = 38, normalized size = 1.41 \begin {gather*} -\frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e^{4}\right )}}{16 \, x^{4} - 64 \, x^{3} + 64 \, x^{2} - x e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^
4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x, algorithm="fricas")
[Out]
-(x^2 - 4*x + 4)*e^(2*e^4)/(16*x^4 - 64*x^3 + 64*x^2 - x*e^8)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^
4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to convert to real -32*exp(8)-8192 Error: Bad Argument Value
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maple [A] time = 0.38, size = 35, normalized size = 1.30
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| method |
result |
size |
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| gosper |
\(\frac {\left (x -2\right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}}{x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\) |
\(35\) |
| risch |
\(\frac {{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (x^{2}-4 x +4\right )}{x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\) |
\(36\) |
| norman |
\(\frac {x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}}{x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\) |
\(50\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^4-128*
x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x,method=_RETURNVERBOSE)
[Out]
1/x*(x-2)^2*exp(2*exp(4))/(-16*x^3+exp(4)^2+64*x^2-64*x)
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maxima [A] time = 0.37, size = 38, normalized size = 1.41 \begin {gather*} -\frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e^{4}\right )}}{16 \, x^{4} - 64 \, x^{3} + 64 \, x^{2} - x e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^
4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x, algorithm="maxima")
[Out]
-(x^2 - 4*x + 4)*e^(2*e^4)/(16*x^4 - 64*x^3 + 64*x^2 - x*e^8)
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(2*exp(4))*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + exp(8)*(x^2 - 4)))/(x^2*exp(16) - exp(8)*(
128*x^3 - 128*x^4 + 32*x^5) + 4096*x^4 - 8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x^8),x)
[Out]
\text{Hanged}
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sympy [B] time = 1.42, size = 48, normalized size = 1.78 \begin {gather*} \frac {- x^{2} e^{2 e^{4}} + 4 x e^{2 e^{4}} - 4 e^{2 e^{4}}}{16 x^{4} - 64 x^{3} + 64 x^{2} - x e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x**2-4)*exp(4)**2+32*x**5-256*x**4+768*x**3-1024*x**2+512*x)*exp(2*exp(4))/(x**2*exp(4)**4+(-32*x*
*5+128*x**4-128*x**3)*exp(4)**2+256*x**8-2048*x**7+6144*x**6-8192*x**5+4096*x**4),x)
[Out]
(-x**2*exp(2*exp(4)) + 4*x*exp(2*exp(4)) - 4*exp(2*exp(4)))/(16*x**4 - 64*x**3 + 64*x**2 - x*exp(8))
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