3.86.12 \(\int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} (16-4 e^x)+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} (512-400 x+4 e^x x+96 x^2-8 x^3)}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{4} \left (-e^x+\frac {1}{2 e^{(-4+x)^4}-x}\right )+x \]

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Rubi [F]  time = 16.86, 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 {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4)*(16 - 4*E^x) + 4*x^2 - E^x*x^2 + E^(256 - 256*x + 96*x^2 -
16*x^3 + x^4)*(512 - 400*x + 4*E^x*x + 96*x^2 - 8*x^3))/(16*E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4) - 16*E
^(256 - 256*x + 96*x^2 - 16*x^3 + x^4)*x + 4*x^2),x]

[Out]

Defer[Int][E^(32*x*(16 + x^2))/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x]/4 + 128*Defer[Int][E^(
256 + 256*x + 96*x^2 + 16*x^3 + x^4)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] + 4*Defer[Int][E
^(512 + 192*x^2 + 2*x^4)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] - Defer[Int][E^(512 + x + 19
2*x^2 + 2*x^4)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] - 100*Defer[Int][(E^(256 + 256*x + 96*
x^2 + 16*x^3 + x^4)*x)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] + Defer[Int][(E^(256 + 257*x +
96*x^2 + 16*x^3 + x^4)*x)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] - Defer[Int][(E^(513*x + 3
2*x^3)*x^2)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x]/4 + 24*Defer[Int][(E^(256 + 256*x + 96*x^
2 + 16*x^3 + x^4)*x^2)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] - 2*Defer[Int][(E^(256 + 256*x
+ 96*x^2 + 16*x^3 + x^4)*x^3)/(2*E^(256 + 96*x^2 + x^4) - E^(16*x*(16 + x^2))*x)^2, x] + Defer[Int][(E^(32*x*
(16 + x^2))*x^2)/(-2*E^(256 + 96*x^2 + x^4) + E^(16*x*(16 + x^2))*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{32 x \left (16+x^2\right )} \left (1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )\right )}{4 \left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{32 x \left (16+x^2\right )} \left (1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )\right )}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{32 x \left (16+x^2\right )} \left (1-4 e^{2 (-4+x)^4} \left (-4+e^x\right )+4 x^2-e^x x^2+4 e^{(-4+x)^4} \left (128+\left (-100+e^x\right ) x+24 x^2-2 x^3\right )\right )}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^{32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}+\frac {16 e^{2 (-4+x)^4+32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}+\frac {512 e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}-\frac {4 \exp \left (512-511 x+192 x^2-32 x^3+2 x^4+32 x \left (16+x^2\right )\right )}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}-\frac {400 e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}+\frac {4 e^{256-255 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}+\frac {96 e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x^2}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}-\frac {8 e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x^3}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2}+\frac {4 e^{32 x \left (16+x^2\right )} x^2}{\left (-2 e^{256+96 x^2+x^4}+e^{16 x \left (16+x^2\right )} x\right )^2}-\frac {e^{x+32 x \left (16+x^2\right )} x^2}{\left (-2 e^{256+96 x^2+x^4}+e^{16 x \left (16+x^2\right )} x\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-\frac {1}{4} \int \frac {e^{x+32 x \left (16+x^2\right )} x^2}{\left (-2 e^{256+96 x^2+x^4}+e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-2 \int \frac {e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x^3}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+4 \int \frac {e^{2 (-4+x)^4+32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+24 \int \frac {e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x^2}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-100 \int \frac {e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+128 \int \frac {e^{256-256 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-\int \frac {\exp \left (512-511 x+192 x^2-32 x^3+2 x^4+32 x \left (16+x^2\right )\right )}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+\int \frac {e^{256-255 x+96 x^2-16 x^3+x^4+32 x \left (16+x^2\right )} x}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+\int \frac {e^{32 x \left (16+x^2\right )} x^2}{\left (-2 e^{256+96 x^2+x^4}+e^{16 x \left (16+x^2\right )} x\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{32 x \left (16+x^2\right )}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-\frac {1}{4} \int \frac {e^{513 x+32 x^3} x^2}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-2 \int \frac {e^{256+256 x+96 x^2+16 x^3+x^4} x^3}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+4 \int \frac {e^{512+192 x^2+2 x^4}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+24 \int \frac {e^{256+256 x+96 x^2+16 x^3+x^4} x^2}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-100 \int \frac {e^{256+256 x+96 x^2+16 x^3+x^4} x}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+128 \int \frac {e^{256+256 x+96 x^2+16 x^3+x^4}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx-\int \frac {e^{512+x+192 x^2+2 x^4}}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+\int \frac {e^{256+257 x+96 x^2+16 x^3+x^4} x}{\left (2 e^{256+96 x^2+x^4}-e^{16 x \left (16+x^2\right )} x\right )^2} \, dx+\int \frac {e^{32 x \left (16+x^2\right )} x^2}{\left (-2 e^{256+96 x^2+x^4}+e^{16 x \left (16+x^2\right )} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4)*(16 - 4*E^x) + 4*x^2 - E^x*x^2 + E^(256 - 256*x + 96
*x^2 - 16*x^3 + x^4)*(512 - 400*x + 4*E^x*x + 96*x^2 - 8*x^3))/(16*E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4)
- 16*E^(256 - 256*x + 96*x^2 - 16*x^3 + x^4)*x + 4*x^2),x]

[Out]

$Aborted

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fricas [B]  time = 0.67, size = 68, normalized size = 2.52 \begin {gather*} \frac {4 \, x^{2} - 2 \, {\left (4 \, x - e^{x}\right )} e^{\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )} - x e^{x} - 1}{4 \, {\left (x - 2 \, e^{\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8*x^3+96*x^2-400*x+512)*exp(x^4-16*x^
3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*
x+256)+4*x^2),x, algorithm="fricas")

[Out]

1/4*(4*x^2 - 2*(4*x - e^x)*e^(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - x*e^x - 1)/(x - 2*e^(x^4 - 16*x^3 + 96*x^
2 - 256*x + 256))

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giac [B]  time = 0.31, size = 908, normalized size = 33.63 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8*x^3+96*x^2-400*x+512)*exp(x^4-16*x^
3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*
x+256)+4*x^2),x, algorithm="giac")

[Out]

1/4*(16*x^7*e^x - 64*x^6*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - 4*x^6*e^(2*x) - 192*x^6*e^x + 64*x^5*e^(2*x
^4 - 32*x^3 + 192*x^2 - 511*x + 512) + 16*x^5*e^(x^4 - 16*x^3 + 96*x^2 - 254*x + 256) + 768*x^5*e^(x^4 - 16*x^
3 + 96*x^2 - 255*x + 256) + 48*x^5*e^(2*x) + 764*x^5*e^x - 16*x^4*e^(2*x^4 - 32*x^3 + 192*x^2 - 510*x + 512) -
768*x^4*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) - 192*x^4*e^(x^4 - 16*x^3 + 96*x^2 - 254*x + 256) - 3064*x
^4*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - 192*x^4*e^(2*x) - 976*x^4*e^x + 192*x^3*e^(2*x^4 - 32*x^3 + 192*x
^2 - 510*x + 512) + 3072*x^3*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) + 768*x^3*e^(x^4 - 16*x^3 + 96*x^2 - 2
54*x + 256) + 4000*x^3*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) + 256*x^3*e^(2*x) - 196*x^3*e^x - 768*x^2*e^(2*
x^4 - 32*x^3 + 192*x^2 - 510*x + 512) - 4096*x^2*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) - 1024*x^2*e^(x^4
- 16*x^3 + 96*x^2 - 254*x + 256) + 400*x^2*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) + x^2*e^(2*x) + 256*x^2*e^x
+ 1024*x*e^(2*x^4 - 32*x^3 + 192*x^2 - 510*x + 512) - 16*x*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) - 4*x*e
^(x^4 - 16*x^3 + 96*x^2 - 254*x + 256) - 512*x*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) + x*e^x + 4*e^(2*x^4 -
32*x^3 + 192*x^2 - 510*x + 512) - 2*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256))/(4*x^6*e^x - 16*x^5*e^(x^4 - 16*x
^3 + 96*x^2 - 255*x + 256) - 48*x^5*e^x + 16*x^4*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) + 192*x^4*e^(x^4 -
16*x^3 + 96*x^2 - 255*x + 256) + 192*x^4*e^x - 192*x^3*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) - 768*x^3*e
^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - 256*x^3*e^x + 768*x^2*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) + 10
24*x^2*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - x^2*e^x - 1024*x*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) +
4*x*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - 4*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512))

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maple [A]  time = 0.10, size = 21, normalized size = 0.78

method	result	size
risch	\(x -\frac {{\mathrm e}^{x}}{4}-\frac {1}{4 \left (x -2 \,{\mathrm e}^{\left (x -4\right )^{4}}\right )}\)	\(21\)
norman	\(\frac {-\frac {1}{4}+x^{2}-2 x \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}-\frac {{\mathrm e}^{x} x}{4}+\frac {{\mathrm e}^{x} {\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}{2}}{x -2 \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8*x^3+96*x^2-400*x+512)*exp(x^4-16*x^3+96*x
^2-256*x+256)-exp(x)*x^2+4*x^2+1)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*x+256)
+4*x^2),x,method=_RETURNVERBOSE)

[Out]

x-1/4*exp(x)-1/4/(x-2*exp((x-4)^4))

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maxima [B]  time = 0.46, size = 82, normalized size = 3.04 \begin {gather*} -\frac {2 \, {\left (4 \, x e^{256} - e^{\left (x + 256\right )}\right )} e^{\left (x^{4} + 96 \, x^{2}\right )} + {\left (x e^{\left (257 \, x\right )} - {\left (4 \, x^{2} - 1\right )} e^{\left (256 \, x\right )}\right )} e^{\left (16 \, x^{3}\right )}}{4 \, {\left (x e^{\left (16 \, x^{3} + 256 \, x\right )} - 2 \, e^{\left (x^{4} + 96 \, x^{2} + 256\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8*x^3+96*x^2-400*x+512)*exp(x^4-16*x^
3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*
x+256)+4*x^2),x, algorithm="maxima")

[Out]

-1/4*(2*(4*x*e^256 - e^(x + 256))*e^(x^4 + 96*x^2) + (x*e^(257*x) - (4*x^2 - 1)*e^(256*x))*e^(16*x^3))/(x*e^(1
6*x^3 + 256*x) - 2*e^(x^4 + 96*x^2 + 256))

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mupad [B]  time = 6.10, size = 38, normalized size = 1.41 \begin {gather*} x-\frac {{\mathrm {e}}^x}{4}-\frac {1}{4\,\left (x-2\,{\mathrm {e}}^{-256\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{256}\,{\mathrm {e}}^{-16\,x^3}\,{\mathrm {e}}^{96\,x^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(96*x^2 - 256*x - 16*x^3 + x^4 + 256)*(4*x*exp(x) - 400*x + 96*x^2 - 8*x^3 + 512) - exp(192*x^2 - 512*
x - 32*x^3 + 2*x^4 + 512)*(4*exp(x) - 16) - x^2*exp(x) + 4*x^2 + 1)/(16*exp(192*x^2 - 512*x - 32*x^3 + 2*x^4 +
512) - 16*x*exp(96*x^2 - 256*x - 16*x^3 + x^4 + 256) + 4*x^2),x)

[Out]

x - exp(x)/4 - 1/(4*(x - 2*exp(-256*x)*exp(x^4)*exp(256)*exp(-16*x^3)*exp(96*x^2)))

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sympy [A]  time = 0.29, size = 32, normalized size = 1.19 \begin {gather*} x - \frac {e^{x}}{4} + \frac {1}{- 4 x + 8 e^{x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)+16)*exp(x**4-16*x**3+96*x**2-256*x+256)**2+(4*exp(x)*x-8*x**3+96*x**2-400*x+512)*exp(x**
4-16*x**3+96*x**2-256*x+256)-exp(x)*x**2+4*x**2+1)/(16*exp(x**4-16*x**3+96*x**2-256*x+256)**2-16*x*exp(x**4-16
*x**3+96*x**2-256*x+256)+4*x**2),x)

[Out]

x - exp(x)/4 + 1/(-4*x + 8*exp(x**4 - 16*x**3 + 96*x**2 - 256*x + 256))

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