3.5.22 \(\int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} (262144+196608 x^2+e^{4 e^x} (4+3 x^2)+e^{2 e^x} (2048+1536 x^2+e^x (64 x+16 x^3)))}{e^{\frac {16}{256+e^{2 e^x}}} (327680+2560 e^{2 e^x}+5 e^{4 e^x})+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {x \left (4+x^2\right )}{5 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )} \]

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Rubi [F]  time = 28.76, 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 {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-131072*x^3 - 1024*E^(2*E^x)*x^3 - 2*E^(4*E^x)*x^3 + E^(8/(256 + E^(2*E^x)))*(262144 + 196608*x^2 + E^(4*
E^x)*(4 + 3*x^2) + E^(2*E^x)*(2048 + 1536*x^2 + E^x*(64*x + 16*x^3))))/(E^(16/(256 + E^(2*E^x)))*(327680 + 256
0*E^(2*E^x) + 5*E^(4*E^x)) + 327680*x^2 + 2560*E^(2*E^x)*x^2 + 5*E^(4*E^x)*x^2 + E^(8/(256 + E^(2*E^x)))*(-655
360*x - 5120*E^(2*E^x)*x - 10*E^(4*E^x)*x)),x]

[Out]

(262144*Defer[Int][E^(8/(256 + E^(2*E^x)))/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5 + (204
8*Defer[Int][E^(2*E^x + 8/(256 + E^(2*E^x)))/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5 + (4
*Defer[Int][E^(4*E^x + 8/(256 + E^(2*E^x)))/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5 + (64
*Defer[Int][(E^(2*E^x + 8/(256 + E^(2*E^x)) + x)*x)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])
/5 + (196608*Defer[Int][(E^(8/(256 + E^(2*E^x)))*x^2)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x
])/5 + (1536*Defer[Int][(E^(2*E^x + 8/(256 + E^(2*E^x)))*x^2)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) -
x)^2), x])/5 + (3*Defer[Int][(E^(4*E^x + 8/(256 + E^(2*E^x)))*x^2)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x)
)) - x)^2), x])/5 - (131072*Defer[Int][x^3/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5 - (102
4*Defer[Int][(E^(2*E^x)*x^3)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5 - (2*Defer[Int][(E^(
4*E^x)*x^3)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5 + (16*Defer[Int][(E^(2*E^x + 8/(256 +
E^(2*E^x)) + x)*x^3)/((256 + E^(2*E^x))^2*(E^(8/(256 + E^(2*E^x))) - x)^2), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{5 \left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {262144 e^{\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2048 e^{2 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {196608 e^{\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {1536 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {16 e^{2 e^x+\frac {8}{256+e^{2 e^x}}+x} x \left (4+x^2\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (4+3 x^2\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (4+3 x^2\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx-\frac {2}{5} \int \frac {e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {16}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}+x} x \left (4+x^2\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx-\frac {1024}{5} \int \frac {e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {1536}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {2048}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx-\frac {131072}{5} \int \frac {x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {196608}{5} \int \frac {e^{\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {262144}{5} \int \frac {e^{\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {4 e^{4 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {3 e^{4 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right ) \, dx-\frac {2}{5} \int \frac {e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {16}{5} \int \left (\frac {4 e^{2 e^x+\frac {8}{256+e^{2 e^x}}+x} x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}+x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right ) \, dx-\frac {1024}{5} \int \frac {e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {1536}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {2048}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx-\frac {131072}{5} \int \frac {x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {196608}{5} \int \frac {e^{\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {262144}{5} \int \frac {e^{\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx\\ &=-\left (\frac {2}{5} \int \frac {e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx\right )+\frac {3}{5} \int \frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {4}{5} \int \frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {16}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}+x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {64}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}+x} x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx-\frac {1024}{5} \int \frac {e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {1536}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {2048}{5} \int \frac {e^{2 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx-\frac {131072}{5} \int \frac {x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {196608}{5} \int \frac {e^{\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx+\frac {262144}{5} \int \frac {e^{\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 31, normalized size = 1.00 \begin {gather*} \frac {x \left (4+x^2\right )}{5 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-131072*x^3 - 1024*E^(2*E^x)*x^3 - 2*E^(4*E^x)*x^3 + E^(8/(256 + E^(2*E^x)))*(262144 + 196608*x^2 +
E^(4*E^x)*(4 + 3*x^2) + E^(2*E^x)*(2048 + 1536*x^2 + E^x*(64*x + 16*x^3))))/(E^(16/(256 + E^(2*E^x)))*(327680
+ 2560*E^(2*E^x) + 5*E^(4*E^x)) + 327680*x^2 + 2560*E^(2*E^x)*x^2 + 5*E^(4*E^x)*x^2 + E^(8/(256 + E^(2*E^x)))
*(-655360*x - 5120*E^(2*E^x)*x - 10*E^(4*E^x)*x)),x]

[Out]

(x*(4 + x^2))/(5*(E^(8/(256 + E^(2*E^x))) - x))

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fricas [A]  time = 0.94, size = 27, normalized size = 0.87 \begin {gather*} -\frac {x^{3} + 4 \, x}{5 \, {\left (x - e^{\left (\frac {8}{e^{\left (2 \, e^{x}\right )} + 256}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*exp(exp(x))^2+196608*x^2+262144)*exp(
4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp
(x))^2+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp(exp(x))^2-655360*x)*exp(4/(exp(exp
(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560*x^2*exp(exp(x))^2+327680*x^2),x, algorithm="fricas")

[Out]

-1/5*(x^3 + 4*x)/(x - e^(8/(e^(2*e^x) + 256)))

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giac [B]  time = 8.01, size = 566, normalized size = 18.26 \begin {gather*} -\frac {16 \, x^{4} e^{\left (x + 7 \, e^{x}\right )} + 8192 \, x^{4} e^{\left (x + 5 \, e^{x}\right )} + 1048576 \, x^{4} e^{\left (x + 3 \, e^{x}\right )} + x^{3} e^{\left (9 \, e^{x}\right )} + 1024 \, x^{3} e^{\left (7 \, e^{x}\right )} + 393216 \, x^{3} e^{\left (5 \, e^{x}\right )} + 67108864 \, x^{3} e^{\left (3 \, e^{x}\right )} + 4294967296 \, x^{3} e^{\left (e^{x}\right )} + 64 \, x^{2} e^{\left (x + 7 \, e^{x}\right )} + 32768 \, x^{2} e^{\left (x + 5 \, e^{x}\right )} + 4194304 \, x^{2} e^{\left (x + 3 \, e^{x}\right )} + 4 \, x e^{\left (9 \, e^{x}\right )} + 4096 \, x e^{\left (7 \, e^{x}\right )} + 1572864 \, x e^{\left (5 \, e^{x}\right )} + 268435456 \, x e^{\left (3 \, e^{x}\right )} + 17179869184 \, x e^{\left (e^{x}\right )}}{5 \, {\left (16 \, x^{2} e^{\left (x + 7 \, e^{x}\right )} + 8192 \, x^{2} e^{\left (x + 5 \, e^{x}\right )} + 1048576 \, x^{2} e^{\left (x + 3 \, e^{x}\right )} - 16 \, x e^{\left (x + \frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 6 \, e^{x} + \frac {1}{32}\right )} - 8192 \, x e^{\left (x + \frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 4 \, e^{x} + \frac {1}{32}\right )} - 1048576 \, x e^{\left (x + \frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 2 \, e^{x} + \frac {1}{32}\right )} + x e^{\left (9 \, e^{x}\right )} + 1024 \, x e^{\left (7 \, e^{x}\right )} + 393216 \, x e^{\left (5 \, e^{x}\right )} + 67108864 \, x e^{\left (3 \, e^{x}\right )} + 4294967296 \, x e^{\left (e^{x}\right )} - e^{\left (\frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 8 \, e^{x} + \frac {1}{32}\right )} - 1024 \, e^{\left (\frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 6 \, e^{x} + \frac {1}{32}\right )} - 393216 \, e^{\left (\frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 4 \, e^{x} + \frac {1}{32}\right )} - 67108864 \, e^{\left (\frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + 2 \, e^{x} + \frac {1}{32}\right )} - 4294967296 \, e^{\left (\frac {32 \, e^{\left (x + 2 \, e^{x}\right )} + 8192 \, e^{x} - e^{\left (2 \, e^{x}\right )}}{32 \, {\left (e^{\left (2 \, e^{x}\right )} + 256\right )}} + \frac {1}{32}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*exp(exp(x))^2+196608*x^2+262144)*exp(
4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp
(x))^2+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp(exp(x))^2-655360*x)*exp(4/(exp(exp
(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560*x^2*exp(exp(x))^2+327680*x^2),x, algorithm="giac")

[Out]

-1/5*(16*x^4*e^(x + 7*e^x) + 8192*x^4*e^(x + 5*e^x) + 1048576*x^4*e^(x + 3*e^x) + x^3*e^(9*e^x) + 1024*x^3*e^(
7*e^x) + 393216*x^3*e^(5*e^x) + 67108864*x^3*e^(3*e^x) + 4294967296*x^3*e^(e^x) + 64*x^2*e^(x + 7*e^x) + 32768
*x^2*e^(x + 5*e^x) + 4194304*x^2*e^(x + 3*e^x) + 4*x*e^(9*e^x) + 4096*x*e^(7*e^x) + 1572864*x*e^(5*e^x) + 2684
35456*x*e^(3*e^x) + 17179869184*x*e^(e^x))/(16*x^2*e^(x + 7*e^x) + 8192*x^2*e^(x + 5*e^x) + 1048576*x^2*e^(x +
3*e^x) - 16*x*e^(x + 1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 6*e^x + 1/32) - 8192*
x*e^(x + 1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 4*e^x + 1/32) - 1048576*x*e^(x + 1
/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 2*e^x + 1/32) + x*e^(9*e^x) + 1024*x*e^(7*e^
x) + 393216*x*e^(5*e^x) + 67108864*x*e^(3*e^x) + 4294967296*x*e^(e^x) - e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x -
e^(2*e^x))/(e^(2*e^x) + 256) + 8*e^x + 1/32) - 1024*e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e
^x) + 256) + 6*e^x + 1/32) - 393216*e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 4*e^
x + 1/32) - 67108864*e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 2*e^x + 1/32) - 429
4967296*e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 1/32))

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maple [A]  time = 0.11, size = 27, normalized size = 0.87

method	result	size
risch	\(-\frac {x \left (x^{2}+4\right )}{5 \left (-{\mathrm e}^{\frac {8}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+256}}+x \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*exp(exp(x))^2+196608*x^2+262144)*exp(4/(exp
(exp(x))^2+256))^2-2*x^3*exp(exp(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp(x))^2
+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp(exp(x))^2-655360*x)*exp(4/(exp(exp(x))^2
+256))^2+5*x^2*exp(exp(x))^4+2560*x^2*exp(exp(x))^2+327680*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/5*x*(x^2+4)/(-exp(8/(exp(2*exp(x))+256))+x)

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maxima [A]  time = 0.74, size = 27, normalized size = 0.87 \begin {gather*} -\frac {x^{3} + 4 \, x}{5 \, {\left (x - e^{\left (\frac {8}{e^{\left (2 \, e^{x}\right )} + 256}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*exp(exp(x))^2+196608*x^2+262144)*exp(
4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp
(x))^2+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp(exp(x))^2-655360*x)*exp(4/(exp(exp
(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560*x^2*exp(exp(x))^2+327680*x^2),x, algorithm="maxima")

[Out]

-1/5*(x^3 + 4*x)/(x - e^(8/(e^(2*e^x) + 256)))

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mupad [B]  time = 0.69, size = 168, normalized size = 5.42 \begin {gather*} -\frac {x\,{\left (512\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+{\mathrm {e}}^{4\,{\mathrm {e}}^x}+65536\right )}^2\,\left (2048\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+4\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+64\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+16\,x^3\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+65536\,x^2+512\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+262144\right )}{5\,{\left ({\mathrm {e}}^{2\,{\mathrm {e}}^x}+256\right )}^2\,\left (x-{\mathrm {e}}^{\frac {8}{{\mathrm {e}}^{2\,{\mathrm {e}}^x}+256}}\right )\,\left (67108864\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+393216\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+1024\,{\mathrm {e}}^{6\,{\mathrm {e}}^x}+{\mathrm {e}}^{8\,{\mathrm {e}}^x}+1048576\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+8192\,x\,{\mathrm {e}}^{x+4\,{\mathrm {e}}^x}+16\,x\,{\mathrm {e}}^{x+6\,{\mathrm {e}}^x}[phone]\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(131072*x^3 + 1024*x^3*exp(2*exp(x)) + 2*x^3*exp(4*exp(x)) - exp(8/(exp(2*exp(x)) + 256))*(exp(2*exp(x))*
(exp(x)*(64*x + 16*x^3) + 1536*x^2 + 2048) + exp(4*exp(x))*(3*x^2 + 4) + 196608*x^2 + 262144))/(327680*x^2 - e
xp(8/(exp(2*exp(x)) + 256))*(655360*x + 5120*x*exp(2*exp(x)) + 10*x*exp(4*exp(x))) + 2560*x^2*exp(2*exp(x)) +
5*x^2*exp(4*exp(x)) + exp(16/(exp(2*exp(x)) + 256))*(2560*exp(2*exp(x)) + 5*exp(4*exp(x)) + 327680)),x)

[Out]

-(x*(512*exp(2*exp(x)) + exp(4*exp(x)) + 65536)^2*(2048*exp(2*exp(x)) + 4*exp(4*exp(x)) + 64*x*exp(x + 2*exp(x
)) + 16*x^3*exp(x + 2*exp(x)) + 65536*x^2 + 512*x^2*exp(2*exp(x)) + x^2*exp(4*exp(x)) + 262144))/(5*(exp(2*exp
(x)) + 256)^2*(x - exp(8/(exp(2*exp(x)) + 256)))*(67108864*exp(2*exp(x)) + 393216*exp(4*exp(x)) + 1024*exp(6*e
xp(x)) + exp(8*exp(x)) + 1048576*x*exp(x + 2*exp(x)) + 8192*x*exp(x + 4*exp(x)) + 16*x*exp(x + 6*exp(x)) + 429
4967296))

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sympy [A]  time = 0.40, size = 22, normalized size = 0.71 \begin {gather*} \frac {x^{3} + 4 x}{- 5 x + 5 e^{\frac {8}{e^{2 e^{x}} + 256}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**2+4)*exp(exp(x))**4+((16*x**3+64*x)*exp(x)+1536*x**2+2048)*exp(exp(x))**2+196608*x**2+262144
)*exp(4/(exp(exp(x))**2+256))**2-2*x**3*exp(exp(x))**4-1024*x**3*exp(exp(x))**2-131072*x**3)/((5*exp(exp(x))**
4+2560*exp(exp(x))**2+327680)*exp(4/(exp(exp(x))**2+256))**4+(-10*x*exp(exp(x))**4-5120*x*exp(exp(x))**2-65536
0*x)*exp(4/(exp(exp(x))**2+256))**2+5*x**2*exp(exp(x))**4+2560*x**2*exp(exp(x))**2+327680*x**2),x)

[Out]

(x**3 + 4*x)/(-5*x + 5*exp(8/(exp(2*exp(x)) + 256)))

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