3.98.30 \(\int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8)}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx\)

Optimal. Leaf size=24 \[ e^{x-x^3 \left (2+\frac {1}{x^2+(16+x)^2}\right )^2} \]

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Rubi [F]  time = 10.53, 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 {\exp \left (\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}\right ) \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((65536*x + 16384*x^2 - 261121*x^3 - 65536*x^4 - 8196*x^5 - 512*x^6 - 16*x^7)/(65536 + 16384*x + 2048*x
^2 + 128*x^3 + 4*x^4))*(8388608 + 3145728*x - 100467072*x^2 - 37765136*x^3 - 7081471*x^4 - 786624*x^5 - 55300*
x^6 - 2304*x^7 - 48*x^8))/(8388608 + 3145728*x + 589824*x^2 + 65536*x^3 + 4608*x^4 + 192*x^5 + 4*x^6),x]

[Out]

-Defer[Int][E^(-1/4*(x*(-65536 - 16384*x + 261121*x^2 + 65536*x^3 + 8196*x^4 + 512*x^5 + 16*x^6))/(128 + 16*x
+ x^2)^2), x] + (16 + 16*I)*Defer[Int][1/(E^((x*(-65536 - 16384*x + 261121*x^2 + 65536*x^3 + 8196*x^4 + 512*x^
5 + 16*x^6))/(4*(128 + 16*x + x^2)^2))*((-16 + 16*I) - 2*x)^3), x] + (1025/2 - 511*I)*Defer[Int][1/(E^((x*(-65
536 - 16384*x + 261121*x^2 + 65536*x^3 + 8196*x^4 + 512*x^5 + 16*x^6))/(4*(128 + 16*x + x^2)^2))*((-16 + 16*I)
- 2*x)^2), x] - 12*Defer[Int][x^2/E^((x*(-65536 - 16384*x + 261121*x^2 + 65536*x^3 + 8196*x^4 + 512*x^5 + 16*
x^6))/(4*(128 + 16*x + x^2)^2)), x] - (16 - 16*I)*Defer[Int][1/(E^((x*(-65536 - 16384*x + 261121*x^2 + 65536*x
^3 + 8196*x^4 + 512*x^5 + 16*x^6))/(4*(128 + 16*x + x^2)^2))*((16 + 16*I) + 2*x)^3), x] + (1025/2 + 511*I)*Def
er[Int][1/(E^((x*(-65536 - 16384*x + 261121*x^2 + 65536*x^3 + 8196*x^4 + 512*x^5 + 16*x^6))/(4*(128 + 16*x + x
^2)^2))*((16 + 16*I) + 2*x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{4 \left (128+16 x+x^2\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{\left (128+16 x+x^2\right )^3} \, dx\\ &=\frac {1}{4} \int \left (-4 \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right )-48 \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) x^2+\frac {4096 \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) x}{\left (128+16 x+x^2\right )^3}+\frac {16 \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) (-8+1021 x)}{\left (128+16 x+x^2\right )^2}+\frac {1025 \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right )}{128+16 x+x^2}\right ) \, dx\\ &=4 \int \frac {\exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) (-8+1021 x)}{\left (128+16 x+x^2\right )^2} \, dx-12 \int \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) x^2 \, dx+\frac {1025}{4} \int \frac {\exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right )}{128+16 x+x^2} \, dx+1024 \int \frac {\exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) x}{\left (128+16 x+x^2\right )^3} \, dx-\int \exp \left (-\frac {x \left (-65536-16384 x+261121 x^2+65536 x^3+8196 x^4+512 x^5+16 x^6\right )}{4 \left (128+16 x+x^2\right )^2}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 46, normalized size = 1.92 \begin {gather*} e^{32-x-4 x^3-\frac {32 (16+x)}{\left (128+16 x+x^2\right )^2}+\frac {-16368-1025 x}{4 \left (128+16 x+x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((65536*x + 16384*x^2 - 261121*x^3 - 65536*x^4 - 8196*x^5 - 512*x^6 - 16*x^7)/(65536 + 16384*x +
2048*x^2 + 128*x^3 + 4*x^4))*(8388608 + 3145728*x - 100467072*x^2 - 37765136*x^3 - 7081471*x^4 - 786624*x^5 -
55300*x^6 - 2304*x^7 - 48*x^8))/(8388608 + 3145728*x + 589824*x^2 + 65536*x^3 + 4608*x^4 + 192*x^5 + 4*x^6),x]

[Out]

E^(32 - x - 4*x^3 - (32*(16 + x))/(128 + 16*x + x^2)^2 + (-16368 - 1025*x)/(4*(128 + 16*x + x^2)))

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fricas [B]  time = 0.77, size = 57, normalized size = 2.38 \begin {gather*} e^{\left (-\frac {16 \, x^{7} + 512 \, x^{6} + 8196 \, x^{5} + 65536 \, x^{4} + 261121 \, x^{3} - 16384 \, x^{2} - 65536 \, x}{4 \, {\left (x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467072*x^2+3145728*x+8388608)*exp
((-16*x^7-512*x^6-8196*x^5-65536*x^4-261121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*
x^6+192*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x, algorithm="fricas")

[Out]

e^(-1/4*(16*x^7 + 512*x^6 + 8196*x^5 + 65536*x^4 + 261121*x^3 - 16384*x^2 - 65536*x)/(x^4 + 32*x^3 + 512*x^2 +
4096*x + 16384))

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giac [B]  time = 0.19, size = 175, normalized size = 7.29 \begin {gather*} e^{\left (-\frac {4 \, x^{7}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {128 \, x^{6}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {2049 \, x^{5}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {16384 \, x^{4}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {261121 \, x^{3}}{4 \, {\left (x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384\right )}} + \frac {4096 \, x^{2}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} + \frac {16384 \, x}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467072*x^2+3145728*x+8388608)*exp
((-16*x^7-512*x^6-8196*x^5-65536*x^4-261121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*
x^6+192*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x, algorithm="giac")

[Out]

e^(-4*x^7/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 128*x^6/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 2049
*x^5/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 16384*x^4/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 261121/
4*x^3/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) + 4096*x^2/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) + 16384*x
/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384))

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maple [A]  time = 8.26, size = 45, normalized size = 1.88

method	result	size
risch	\({\mathrm e}^{-\frac {x \left (4 x^{3}+62 x^{2}+481 x -256\right ) \left (4 x^{3}+66 x^{2}+545 x +256\right )}{4 \left (x^{2}+16 x +128\right )^{2}}}\)	\(45\)
gosper	\({\mathrm e}^{-\frac {x \left (16 x^{6}+512 x^{5}+8196 x^{4}+65536 x^{3}+261121 x^{2}-16384 x -65536\right )}{4 \left (x^{4}+32 x^{3}+512 x^{2}+4096 x +16384\right )}}\)	\(55\)
norman	\(\frac {x^{4} {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+4096 x \,{\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+512 x^{2} {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+32 x^{3} {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+16384 \,{\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}}{\left (x^{2}+16 x +128\right )^{2}}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467072*x^2+3145728*x+8388608)*exp((-16*
x^7-512*x^6-8196*x^5-65536*x^4-261121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^6+19
2*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x,method=_RETURNVERBOSE)

[Out]

exp(-1/4*x*(4*x^3+62*x^2+481*x-256)*(4*x^3+66*x^2+545*x+256)/(x^2+16*x+128)^2)

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maxima [B]  time = 0.75, size = 81, normalized size = 3.38 \begin {gather*} e^{\left (-4 \, x^{3} - x - \frac {32 \, x}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {1025 \, x}{4 \, {\left (x^{2} + 16 \, x + 128\right )}} - \frac {512}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {4092}{x^{2} + 16 \, x + 128} + 32\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467072*x^2+3145728*x+8388608)*exp
((-16*x^7-512*x^6-8196*x^5-65536*x^4-261121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*
x^6+192*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x, algorithm="maxima")

[Out]

e^(-4*x^3 - x - 32*x/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 1025/4*x/(x^2 + 16*x + 128) - 512/(x^4 + 32*x
^3 + 512*x^2 + 4096*x + 16384) - 4092/(x^2 + 16*x + 128) + 32)

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mupad [B]  time = 6.38, size = 183, normalized size = 7.62 \begin {gather*} {\mathrm {e}}^{\frac {16384\,x}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {4\,x^7}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {128\,x^6}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {2049\,x^5}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{\frac {4096\,x^2}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {16384\,x^4}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {261121\,x^3}{4\,x^4+128\,x^3+2048\,x^2+16384\,x+65536}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(261121*x^3 - 16384*x^2 - 65536*x + 65536*x^4 + 8196*x^5 + 512*x^6 + 16*x^7)/(16384*x + 2048*x^2 +
128*x^3 + 4*x^4 + 65536))*(100467072*x^2 - 3145728*x + 37765136*x^3 + 7081471*x^4 + 786624*x^5 + 55300*x^6 + 2
304*x^7 + 48*x^8 - 8388608))/(3145728*x + 589824*x^2 + 65536*x^3 + 4608*x^4 + 192*x^5 + 4*x^6 + 8388608),x)

[Out]

exp((16384*x)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(4*x^7)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384
))*exp(-(128*x^6)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(2049*x^5)/(4096*x + 512*x^2 + 32*x^3 + x^4
+ 16384))*exp((4096*x^2)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(16384*x^4)/(4096*x + 512*x^2 + 32*x^
3 + x^4 + 16384))*exp(-(261121*x^3)/(16384*x + 2048*x^2 + 128*x^3 + 4*x^4 + 65536))

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sympy [B]  time = 0.55, size = 54, normalized size = 2.25 \begin {gather*} e^{\frac {- 16 x^{7} - 512 x^{6} - 8196 x^{5} - 65536 x^{4} - 261121 x^{3} + 16384 x^{2} + 65536 x}{4 x^{4} + 128 x^{3} + 2048 x^{2} + 16384 x + 65536}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*x**8-2304*x**7-55300*x**6-786624*x**5-7081471*x**4-37765136*x**3-100467072*x**2+3145728*x+83886
08)*exp((-16*x**7-512*x**6-8196*x**5-65536*x**4-261121*x**3+16384*x**2+65536*x)/(4*x**4+128*x**3+2048*x**2+163
84*x+65536))/(4*x**6+192*x**5+4608*x**4+65536*x**3+589824*x**2+3145728*x+8388608),x)

[Out]

exp((-16*x**7 - 512*x**6 - 8196*x**5 - 65536*x**4 - 261121*x**3 + 16384*x**2 + 65536*x)/(4*x**4 + 128*x**3 + 2
048*x**2 + 16384*x + 65536))

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