3.34.29 \(\int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x (16-10 x+2 x^2)+e^{4+8 x-2 x^2+(-4+x) \log (x)} (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} (6-16 x+8 x^2)+e^x (-48+150 x-98 x^2+16 x^3)+(-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x (16 x-4 x^2)) \log (x))}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx\)
Optimal. Leaf size=33 \[ \frac {\left (-4+e^x+x\right )^2}{\left (x+e^{4+(4-x) (2 x-\log (x))} x\right )^2} \]
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Rubi [F] time = 93.03, 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 {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-32 + 8*x + E^(2*x)*(-2 + 2*x) + E^x*(16 - 10*x + 2*x^2) + E^(4 + 8*x - 2*x^2 + (-4 + x)*Log[x])*(96 - 34
4*x + 280*x^2 - 82*x^3 + 8*x^4 + E^(2*x)*(6 - 16*x + 8*x^2) + E^x*(-48 + 150*x - 98*x^2 + 16*x^3) + (-32*x - 2
*E^(2*x)*x + 16*x^2 - 2*x^3 + E^x*(16*x - 4*x^2))*Log[x]))/(x^3 + 3*E^(4 + 8*x - 2*x^2 + (-4 + x)*Log[x])*x^3
+ 3*E^(8 + 16*x - 4*x^2 + 2*(-4 + x)*Log[x])*x^3 + E^(12 + 24*x - 6*x^2 + 3*(-4 + x)*Log[x])*x^3),x]
[Out]
-128*Defer[Int][(E^(6*x^2)*x^9)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 64*Defer[Int][(E^(x + 6*x^2)*x^9)/(E
^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] - 8*Defer[Int][(E^(2*x + 6*x^2)*x^9)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3
, x] + 352*Defer[Int][(E^(6*x^2)*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 32*Log[x]*Defer[Int][(E^(6*x^
2)*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] - 160*Defer[Int][(E^(x + 6*x^2)*x^10)/(E^(2*x^2)*x^4 + E^(4 +
8*x)*x^x)^3, x] - 16*Log[x]*Defer[Int][(E^(x + 6*x^2)*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 18*Defe
r[Int][(E^(2*x + 6*x^2)*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 2*Log[x]*Defer[Int][(E^(2*x + 6*x^2)*x
^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] - 280*Defer[Int][(E^(6*x^2)*x^11)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^
x)^3, x] - 16*Log[x]*Defer[Int][(E^(6*x^2)*x^11)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 100*Defer[Int][(E^(
x + 6*x^2)*x^11)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 4*Log[x]*Defer[Int][(E^(x + 6*x^2)*x^11)/(E^(2*x^2)
*x^4 + E^(4 + 8*x)*x^x)^3, x] - 8*Defer[Int][(E^(2*x + 6*x^2)*x^11)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] +
82*Defer[Int][(E^(6*x^2)*x^12)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 2*Log[x]*Defer[Int][(E^(6*x^2)*x^12)/
(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] - 16*Defer[Int][(E^(x + 6*x^2)*x^12)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)
^3, x] - 8*Defer[Int][(E^(6*x^2)*x^13)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x] + 96*Defer[Int][(E^(4*x^2)*x^5)
/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 48*Defer[Int][(E^(x + 4*x^2)*x^5)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)
^2, x] + 6*Defer[Int][(E^(2*x + 4*x^2)*x^5)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 344*Defer[Int][(E^(4*x^2
)*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 32*Log[x]*Defer[Int][(E^(4*x^2)*x^6)/(E^(2*x^2)*x^4 + E^(4 +
8*x)*x^x)^2, x] + 150*Defer[Int][(E^(x + 4*x^2)*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] + 16*Log[x]*Defer
[Int][(E^(x + 4*x^2)*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 16*Defer[Int][(E^(2*x + 4*x^2)*x^6)/(E^(2*
x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 2*Log[x]*Defer[Int][(E^(2*x + 4*x^2)*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x
)^2, x] + 280*Defer[Int][(E^(4*x^2)*x^7)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] + 16*Log[x]*Defer[Int][(E^(4*
x^2)*x^7)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 98*Defer[Int][(E^(x + 4*x^2)*x^7)/(E^(2*x^2)*x^4 + E^(4 +
8*x)*x^x)^2, x] - 4*Log[x]*Defer[Int][(E^(x + 4*x^2)*x^7)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] + 8*Defer[In
t][(E^(2*x + 4*x^2)*x^7)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 82*Defer[Int][(E^(4*x^2)*x^8)/(E^(2*x^2)*x^
4 + E^(4 + 8*x)*x^x)^2, x] - 2*Log[x]*Defer[Int][(E^(4*x^2)*x^8)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] + 16*
Defer[Int][(E^(x + 4*x^2)*x^8)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x] + 8*Defer[Int][(E^(4*x^2)*x^9)/(E^(2*x^
2)*x^4 + E^(4 + 8*x)*x^x)^2, x] - 32*Defer[Int][Defer[Int][(E^(6*x^2)*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^
3, x]/x, x] - 2*Defer[Int][Defer[Int][(E^(2*x*(1 + 3*x))*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x]/x, x] +
16*Defer[Int][Defer[Int][(E^(x + 6*x^2)*x^10)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x]/x, x] + 16*Defer[Int][D
efer[Int][(E^(6*x^2)*x^11)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x]/x, x] - 4*Defer[Int][Defer[Int][(E^(x + 6*x
^2)*x^11)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^3, x]/x, x] - 2*Defer[Int][Defer[Int][(E^(6*x^2)*x^12)/(E^(2*x^2)*
x^4 + E^(4 + 8*x)*x^x)^3, x]/x, x] + 32*Defer[Int][Defer[Int][(E^(4*x^2)*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x
)^2, x]/x, x] + 2*Defer[Int][Defer[Int][(E^(2*x*(1 + 2*x))*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x]/x, x]
- 16*Defer[Int][Defer[Int][(E^(x + 4*x^2)*x^6)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x]/x, x] - 16*Defer[Int][D
efer[Int][(E^(4*x^2)*x^7)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x]/x, x] + 4*Defer[Int][Defer[Int][(E^(x + 4*x^
2)*x^7)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2, x]/x, x] + 2*Defer[Int][Defer[Int][(E^(4*x^2)*x^8)/(E^(2*x^2)*x^4
+ E^(4 + 8*x)*x^x)^2, x]/x, x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{4 x^2} \left (4-e^x-x\right ) x^5 \left (-4 e^{2 x^2} x^4-e^{x+2 x^2} (-1+x) x^4-e^{4+9 x} x^x \left (3-8 x+4 x^2\right )-e^{4+8 x} x^x \left (-12+40 x-25 x^2+4 x^3\right )+e^{4+8 x} x^{1+x} \left (-4+e^x+x\right ) \log (x)\right )}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^3} \, dx\\ &=2 \int \frac {e^{4 x^2} \left (4-e^x-x\right ) x^5 \left (-4 e^{2 x^2} x^4-e^{x+2 x^2} (-1+x) x^4-e^{4+9 x} x^x \left (3-8 x+4 x^2\right )-e^{4+8 x} x^x \left (-12+40 x-25 x^2+4 x^3\right )+e^{4+8 x} x^{1+x} \left (-4+e^x+x\right ) \log (x)\right )}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^3} \, dx\\ &=2 \int \left (-\frac {e^{6 x^2} x^9 \left (-4+e^x+x\right )^2 \left (4-9 x+4 x^2-x \log (x)\right )}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^3}+\frac {e^{4 x^2} x^5 \left (-4+e^x+x\right ) \left (-12+3 e^x+40 x-8 e^x x-25 x^2+4 e^x x^2+4 x^3+4 x \log (x)-e^x x \log (x)-x^2 \log (x)\right )}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{6 x^2} x^9 \left (-4+e^x+x\right )^2 \left (4-9 x+4 x^2-x \log (x)\right )}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^3} \, dx\right )+2 \int \frac {e^{4 x^2} x^5 \left (-4+e^x+x\right ) \left (-12+3 e^x+40 x-8 e^x x-25 x^2+4 e^x x^2+4 x^3+4 x \log (x)-e^x x \log (x)-x^2 \log (x)\right )}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 44, normalized size = 1.33 \begin {gather*} \frac {e^{4 x^2} x^6 \left (-4+e^x+x\right )^2}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-32 + 8*x + E^(2*x)*(-2 + 2*x) + E^x*(16 - 10*x + 2*x^2) + E^(4 + 8*x - 2*x^2 + (-4 + x)*Log[x])*(9
6 - 344*x + 280*x^2 - 82*x^3 + 8*x^4 + E^(2*x)*(6 - 16*x + 8*x^2) + E^x*(-48 + 150*x - 98*x^2 + 16*x^3) + (-32
*x - 2*E^(2*x)*x + 16*x^2 - 2*x^3 + E^x*(16*x - 4*x^2))*Log[x]))/(x^3 + 3*E^(4 + 8*x - 2*x^2 + (-4 + x)*Log[x]
)*x^3 + 3*E^(8 + 16*x - 4*x^2 + 2*(-4 + x)*Log[x])*x^3 + E^(12 + 24*x - 6*x^2 + 3*(-4 + x)*Log[x])*x^3),x]
[Out]
(E^(4*x^2)*x^6*(-4 + E^x + x)^2)/(E^(2*x^2)*x^4 + E^(4 + 8*x)*x^x)^2
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fricas [B] time = 0.56, size = 70, normalized size = 2.12 \begin {gather*} \frac {x^{2} + 2 \, {\left (x - 4\right )} e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 16}{2 \, x^{2} e^{\left (-2 \, x^{2} + {\left (x - 4\right )} \log \relax (x) + 8 \, x + 4\right )} + x^{2} e^{\left (-4 \, x^{2} + 2 \, {\left (x - 4\right )} \log \relax (x) + 16 \, x + 8\right )} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*log(x)+(8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x
^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x^2-344*x+96)*exp((x-4)*log(x)-2*x^2+8*x+4)+(2*x-2)*exp(x)^2+(2*x^2-10*x+
16)*exp(x)+8*x-32)/(x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*
log(x)-2*x^2+8*x+4)+x^3),x, algorithm="fricas")
[Out]
(x^2 + 2*(x - 4)*e^x - 8*x + e^(2*x) + 16)/(2*x^2*e^(-2*x^2 + (x - 4)*log(x) + 8*x + 4) + x^2*e^(-4*x^2 + 2*(x
- 4)*log(x) + 16*x + 8) + x^2)
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giac [B] time = 1.05, size = 4807, normalized size = 145.67 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*log(x)+(8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x
^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x^2-344*x+96)*exp((x-4)*log(x)-2*x^2+8*x+4)+(2*x-2)*exp(x)^2+(2*x^2-10*x+
16)*exp(x)+8*x-32)/(x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*
log(x)-2*x^2+8*x+4)+x^3),x, algorithm="giac")
[Out]
(64*x^24*e^(4*x^2) - 48*x^23*e^(4*x^2)*log(x) + 12*x^22*e^(4*x^2)*log(x)^2 - x^21*e^(4*x^2)*log(x)^3 - 944*x^2
3*e^(4*x^2) + 128*x^23*e^(4*x^2 + x) + 600*x^22*e^(4*x^2)*log(x) - 96*x^22*e^(4*x^2 + x)*log(x) - 123*x^21*e^(
4*x^2)*log(x)^2 + 24*x^21*e^(4*x^2 + x)*log(x)^2 + 8*x^20*e^(4*x^2)*log(x)^3 - 2*x^20*e^(4*x^2 + x)*log(x)^3 +
5644*x^22*e^(4*x^2) + 64*x^22*e^(4*x^2 + 2*x) - 1376*x^22*e^(4*x^2 + x) - 2835*x^21*e^(4*x^2)*log(x) - 48*x^2
1*e^(4*x^2 + 2*x)*log(x) + 816*x^21*e^(4*x^2 + x)*log(x) + 420*x^20*e^(4*x^2)*log(x)^2 + 12*x^20*e^(4*x^2 + 2*
x)*log(x)^2 - 150*x^20*e^(4*x^2 + x)*log(x)^2 - 16*x^19*e^(4*x^2)*log(x)^3 - x^19*e^(4*x^2 + 2*x)*log(x)^3 + 8
*x^19*e^(4*x^2 + x)*log(x)^3 - 17817*x^21*e^(4*x^2) - 432*x^21*e^(4*x^2 + 2*x) + 5784*x^21*e^(4*x^2 + x) + 638
4*x^20*e^(4*x^2)*log(x) + 216*x^20*e^(4*x^2 + 2*x)*log(x) - 2406*x^20*e^(4*x^2 + x)*log(x) - 528*x^19*e^(4*x^2
)*log(x)^2 - 27*x^19*e^(4*x^2 + 2*x)*log(x)^2 + 240*x^19*e^(4*x^2 + x)*log(x)^2 + 32532*x^20*e^(4*x^2) + 1164*
x^20*e^(4*x^2 + 2*x) - 12498*x^20*e^(4*x^2 + x) + 128*x^20*e^(2*x^2 + x*log(x) + 8*x + 4) - 7200*x^19*e^(4*x^2
)*log(x) - 339*x^19*e^(4*x^2 + 2*x)*log(x) + 3144*x^19*e^(4*x^2 + x)*log(x) - 96*x^19*e^(2*x^2 + x*log(x) + 8*
x + 4)*log(x) + 192*x^18*e^(4*x^2)*log(x)^2 + 12*x^18*e^(4*x^2 + 2*x)*log(x)^2 - 96*x^18*e^(4*x^2 + x)*log(x)^
2 + 24*x^18*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 - 2*x^17*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^3 - 35232*x
^19*e^(4*x^2) - 1593*x^19*e^(4*x^2 + 2*x) + 15072*x^19*e^(4*x^2 + x) + 256*x^19*e^(2*x^2 + x*log(x) + 9*x + 4)
- 1888*x^19*e^(2*x^2 + x*log(x) + 8*x + 4) + 3840*x^18*e^(4*x^2)*log(x) + 216*x^18*e^(4*x^2 + 2*x)*log(x) - 1
824*x^18*e^(4*x^2 + x)*log(x) - 192*x^18*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x) + 1200*x^18*e^(2*x^2 + x*log(x)
+ 8*x + 4)*log(x) + 48*x^17*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x)^2 - 246*x^17*e^(2*x^2 + x*log(x) + 8*x + 4)
*log(x)^2 - 4*x^16*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x)^3 + 16*x^16*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^3 +
22144*x^18*e^(4*x^2) + 1164*x^18*e^(4*x^2 + 2*x) - 10176*x^18*e^(4*x^2 + x) + 128*x^18*e^(2*x^2 + x*log(x) +
10*x + 4) - 2752*x^18*e^(2*x^2 + x*log(x) + 9*x + 4) + 11288*x^18*e^(2*x^2 + x*log(x) + 8*x + 4) - 768*x^17*e^
(4*x^2)*log(x) - 48*x^17*e^(4*x^2 + 2*x)*log(x) + 384*x^17*e^(4*x^2 + x)*log(x) - 96*x^17*e^(2*x^2 + x*log(x)
+ 10*x + 4)*log(x) + 1632*x^17*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x) - 5670*x^17*e^(2*x^2 + x*log(x) + 8*x + 4
)*log(x) + 24*x^16*e^(2*x^2 + x*log(x) + 10*x + 4)*log(x)^2 - 300*x^16*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x)^2
+ 840*x^16*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 - 2*x^15*e^(2*x^2 + x*log(x) + 10*x + 4)*log(x)^3 + 16*x^1
5*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x)^3 - 32*x^15*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^3 - 7424*x^17*e^(4*x
^2) - 432*x^17*e^(4*x^2 + 2*x) + 3584*x^17*e^(4*x^2 + x) - 864*x^17*e^(2*x^2 + x*log(x) + 10*x + 4) + 11568*x^
17*e^(2*x^2 + x*log(x) + 9*x + 4) - 35634*x^17*e^(2*x^2 + x*log(x) + 8*x + 4) + 432*x^16*e^(2*x^2 + x*log(x) +
10*x + 4)*log(x) - 4812*x^16*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x) + 12768*x^16*e^(2*x^2 + x*log(x) + 8*x + 4
)*log(x) - 54*x^15*e^(2*x^2 + x*log(x) + 10*x + 4)*log(x)^2 + 480*x^15*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x)^2
- 1056*x^15*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 + 1024*x^16*e^(4*x^2) + 64*x^16*e^(4*x^2 + 2*x) - 512*x^1
6*e^(4*x^2 + x) + 2328*x^16*e^(2*x^2 + x*log(x) + 10*x + 4) - 24996*x^16*e^(2*x^2 + x*log(x) + 9*x + 4) + 6506
4*x^16*e^(2*x^2 + x*log(x) + 8*x + 4) + 64*x^16*e^(2*x*log(x) + 16*x + 8) - 678*x^15*e^(2*x^2 + x*log(x) + 10*
x + 4)*log(x) + 6288*x^15*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x) - 14400*x^15*e^(2*x^2 + x*log(x) + 8*x + 4)*lo
g(x) - 48*x^15*e^(2*x*log(x) + 16*x + 8)*log(x) + 24*x^14*e^(2*x^2 + x*log(x) + 10*x + 4)*log(x)^2 - 192*x^14*
e^(2*x^2 + x*log(x) + 9*x + 4)*log(x)^2 + 384*x^14*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 + 12*x^14*e^(2*x*lo
g(x) + 16*x + 8)*log(x)^2 - x^13*e^(2*x*log(x) + 16*x + 8)*log(x)^3 - 3186*x^15*e^(2*x^2 + x*log(x) + 10*x + 4
) + 30144*x^15*e^(2*x^2 + x*log(x) + 9*x + 4) - 70464*x^15*e^(2*x^2 + x*log(x) + 8*x + 4) + 128*x^15*e^(2*x*lo
g(x) + 17*x + 8) - 944*x^15*e^(2*x*log(x) + 16*x + 8) + 432*x^14*e^(2*x^2 + x*log(x) + 10*x + 4)*log(x) - 3648
*x^14*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x) + 7680*x^14*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x) - 96*x^14*e^(2*x
*log(x) + 17*x + 8)*log(x) + 600*x^14*e^(2*x*log(x) + 16*x + 8)*log(x) + 24*x^13*e^(2*x*log(x) + 17*x + 8)*log
(x)^2 - 123*x^13*e^(2*x*log(x) + 16*x + 8)*log(x)^2 - 2*x^12*e^(2*x*log(x) + 17*x + 8)*log(x)^3 + 8*x^12*e^(2*
x*log(x) + 16*x + 8)*log(x)^3 + 2328*x^14*e^(2*x^2 + x*log(x) + 10*x + 4) - 20352*x^14*e^(2*x^2 + x*log(x) + 9
*x + 4) + 44288*x^14*e^(2*x^2 + x*log(x) + 8*x + 4) + 64*x^14*e^(2*x*log(x) + 18*x + 8) - 1376*x^14*e^(2*x*log
(x) + 17*x + 8) + 5644*x^14*e^(2*x*log(x) + 16*x + 8) - 96*x^13*e^(2*x^2 + x*log(x) + 10*x + 4)*log(x) + 768*x
^13*e^(2*x^2 + x*log(x) + 9*x + 4)*log(x) - 1536*x^13*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x) - 48*x^13*e^(2*x*l
og(x) + 18*x + 8)*log(x) + 816*x^13*e^(2*x*log(x) + 17*x + 8)*log(x) - 2835*x^13*e^(2*x*log(x) + 16*x + 8)*log
(x) + 12*x^12*e^(2*x*log(x) + 18*x + 8)*log(x)^2 - 150*x^12*e^(2*x*log(x) + 17*x + 8)*log(x)^2 + 420*x^12*e^(2
*x*log(x) + 16*x + 8)*log(x)^2 - x^11*e^(2*x*log(x) + 18*x + 8)*log(x)^3 + 8*x^11*e^(2*x*log(x) + 17*x + 8)*lo
g(x)^3 - 16*x^11*e^(2*x*log(x) + 16*x + 8)*log(x)^3 - 864*x^13*e^(2*x^2 + x*log(x) + 10*x + 4) + 7168*x^13*e^(
2*x^2 + x*log(x) + 9*x + 4) - 14848*x^13*e^(2*x^2 + x*log(x) + 8*x + 4) - 432*x^13*e^(2*x*log(x) + 18*x + 8) +
5784*x^13*e^(2*x*log(x) + 17*x + 8) - 17817*x^13*e^(2*x*log(x) + 16*x + 8) + 216*x^12*e^(2*x*log(x) + 18*x +
8)*log(x) - 2406*x^12*e^(2*x*log(x) + 17*x + 8)*log(x) + 6384*x^12*e^(2*x*log(x) + 16*x + 8)*log(x) - 27*x^11*
e^(2*x*log(x) + 18*x + 8)*log(x)^2 + 240*x^11*e^(2*x*log(x) + 17*x + 8)*log(x)^2 - 528*x^11*e^(2*x*log(x) + 16
*x + 8)*log(x)^2 + 128*x^12*e^(2*x^2 + x*log(x) + 10*x + 4) - 1024*x^12*e^(2*x^2 + x*log(x) + 9*x + 4) + 2048*
x^12*e^(2*x^2 + x*log(x) + 8*x + 4) + 1164*x^12*e^(2*x*log(x) + 18*x + 8) - 12498*x^12*e^(2*x*log(x) + 17*x +
8) + 32532*x^12*e^(2*x*log(x) + 16*x + 8) - 339*x^11*e^(2*x*log(x) + 18*x + 8)*log(x) + 3144*x^11*e^(2*x*log(x
) + 17*x + 8)*log(x) - 7200*x^11*e^(2*x*log(x) + 16*x + 8)*log(x) + 12*x^10*e^(2*x*log(x) + 18*x + 8)*log(x)^2
- 96*x^10*e^(2*x*log(x) + 17*x + 8)*log(x)^2 + 192*x^10*e^(2*x*log(x) + 16*x + 8)*log(x)^2 - 1593*x^11*e^(2*x
*log(x) + 18*x + 8) + 15072*x^11*e^(2*x*log(x) + 17*x + 8) - 35232*x^11*e^(2*x*log(x) + 16*x + 8) + 216*x^10*e
^(2*x*log(x) + 18*x + 8)*log(x) - 1824*x^10*e^(2*x*log(x) + 17*x + 8)*log(x) + 3840*x^10*e^(2*x*log(x) + 16*x
+ 8)*log(x) + 1164*x^10*e^(2*x*log(x) + 18*x + 8) - 10176*x^10*e^(2*x*log(x) + 17*x + 8) + 22144*x^10*e^(2*x*l
og(x) + 16*x + 8) - 48*x^9*e^(2*x*log(x) + 18*x + 8)*log(x) + 384*x^9*e^(2*x*log(x) + 17*x + 8)*log(x) - 768*x
^9*e^(2*x*log(x) + 16*x + 8)*log(x) - 432*x^9*e^(2*x*log(x) + 18*x + 8) + 3584*x^9*e^(2*x*log(x) + 17*x + 8) -
7424*x^9*e^(2*x*log(x) + 16*x + 8) + 64*x^8*e^(2*x*log(x) + 18*x + 8) - 512*x^8*e^(2*x*log(x) + 17*x + 8) + 1
024*x^8*e^(2*x*log(x) + 16*x + 8))/(64*x^24*e^(4*x^2) - 48*x^23*e^(4*x^2)*log(x) + 12*x^22*e^(4*x^2)*log(x)^2
- x^21*e^(4*x^2)*log(x)^3 - 432*x^23*e^(4*x^2) + 216*x^22*e^(4*x^2)*log(x) - 27*x^21*e^(4*x^2)*log(x)^2 + 1164
*x^22*e^(4*x^2) - 339*x^21*e^(4*x^2)*log(x) + 12*x^20*e^(4*x^2)*log(x)^2 - 1593*x^21*e^(4*x^2) + 216*x^20*e^(4
*x^2)*log(x) + 1164*x^20*e^(4*x^2) + 256*x^20*e^(2*x^2 + x*log(x) + 8*x + 4) - 48*x^19*e^(4*x^2)*log(x) - 192*
x^19*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x) + 48*x^18*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 - 4*x^17*e^(2*x^2
+ x*log(x) + 8*x + 4)*log(x)^3 - 432*x^19*e^(4*x^2) - 1728*x^19*e^(2*x^2 + x*log(x) + 8*x + 4) + 864*x^18*e^(
2*x^2 + x*log(x) + 8*x + 4)*log(x) - 108*x^17*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 + 64*x^18*e^(4*x^2) + 46
56*x^18*e^(2*x^2 + x*log(x) + 8*x + 4) - 1356*x^17*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x) + 48*x^16*e^(2*x^2 +
x*log(x) + 8*x + 4)*log(x)^2 - 6372*x^17*e^(2*x^2 + x*log(x) + 8*x + 4) + 864*x^16*e^(2*x^2 + x*log(x) + 8*x +
4)*log(x) + 4656*x^16*e^(2*x^2 + x*log(x) + 8*x + 4) + 384*x^16*e^(2*x*log(x) + 16*x + 8) - 192*x^15*e^(2*x^2
+ x*log(x) + 8*x + 4)*log(x) - 288*x^15*e^(2*x*log(x) + 16*x + 8)*log(x) + 72*x^14*e^(2*x*log(x) + 16*x + 8)*
log(x)^2 - 6*x^13*e^(2*x*log(x) + 16*x + 8)*log(x)^3 - 1728*x^15*e^(2*x^2 + x*log(x) + 8*x + 4) - 2592*x^15*e^
(2*x*log(x) + 16*x + 8) + 1296*x^14*e^(2*x*log(x) + 16*x + 8)*log(x) - 162*x^13*e^(2*x*log(x) + 16*x + 8)*log(
x)^2 + 256*x^14*e^(2*x^2 + x*log(x) + 8*x + 4) + 6984*x^14*e^(2*x*log(x) + 16*x + 8) - 2034*x^13*e^(2*x*log(x)
+ 16*x + 8)*log(x) + 72*x^12*e^(2*x*log(x) + 16*x + 8)*log(x)^2 - 9558*x^13*e^(2*x*log(x) + 16*x + 8) + 1296*
x^12*e^(2*x*log(x) + 16*x + 8)*log(x) + 256*x^12*e^(-2*x^2 + 3*x*log(x) + 24*x + 12) + 6984*x^12*e^(2*x*log(x)
+ 16*x + 8) - 192*x^11*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x) - 288*x^11*e^(2*x*log(x) + 16*x + 8)*log(x)
+ 48*x^10*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x)^2 - 4*x^9*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x)^3 -
1728*x^11*e^(-2*x^2 + 3*x*log(x) + 24*x + 12) - 2592*x^11*e^(2*x*log(x) + 16*x + 8) + 864*x^10*e^(-2*x^2 + 3*
x*log(x) + 24*x + 12)*log(x) - 108*x^9*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x)^2 + 4656*x^10*e^(-2*x^2 + 3*
x*log(x) + 24*x + 12) + 384*x^10*e^(2*x*log(x) + 16*x + 8) - 1356*x^9*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(
x) + 48*x^8*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x)^2 - 6372*x^9*e^(-2*x^2 + 3*x*log(x) + 24*x + 12) + 864*
x^8*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x) + 4656*x^8*e^(-2*x^2 + 3*x*log(x) + 24*x + 12) + 64*x^8*e^(-4*x
^2 + 4*x*log(x) + 32*x + 16) - 192*x^7*e^(-2*x^2 + 3*x*log(x) + 24*x + 12)*log(x) - 48*x^7*e^(-4*x^2 + 4*x*log
(x) + 32*x + 16)*log(x) + 12*x^6*e^(-4*x^2 + 4*x*log(x) + 32*x + 16)*log(x)^2 - x^5*e^(-4*x^2 + 4*x*log(x) + 3
2*x + 16)*log(x)^3 - 1728*x^7*e^(-2*x^2 + 3*x*log(x) + 24*x + 12) - 432*x^7*e^(-4*x^2 + 4*x*log(x) + 32*x + 16
) + 216*x^6*e^(-4*x^2 + 4*x*log(x) + 32*x + 16)*log(x) - 27*x^5*e^(-4*x^2 + 4*x*log(x) + 32*x + 16)*log(x)^2 +
256*x^6*e^(-2*x^2 + 3*x*log(x) + 24*x + 12) + 1164*x^6*e^(-4*x^2 + 4*x*log(x) + 32*x + 16) - 339*x^5*e^(-4*x^
2 + 4*x*log(x) + 32*x + 16)*log(x) + 12*x^4*e^(-4*x^2 + 4*x*log(x) + 32*x + 16)*log(x)^2 - 1593*x^5*e^(-4*x^2
+ 4*x*log(x) + 32*x + 16) + 216*x^4*e^(-4*x^2 + 4*x*log(x) + 32*x + 16)*log(x) + 1164*x^4*e^(-4*x^2 + 4*x*log(
x) + 32*x + 16) - 48*x^3*e^(-4*x^2 + 4*x*log(x) + 32*x + 16)*log(x) - 432*x^3*e^(-4*x^2 + 4*x*log(x) + 32*x +
16) + 64*x^2*e^(-4*x^2 + 4*x*log(x) + 32*x + 16))
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maple [A] time = 0.10, size = 47, normalized size = 1.42
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\(\frac {x^{2}+2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}-8 x -8 \,{\mathrm e}^{x}+16}{x^{2} \left (x^{x -4} {\mathrm e}^{-2 x^{2}+8 x +4}+1\right )^{2}}\) |
\(47\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*ln(x)+(8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x^2+150*
x-48)*exp(x)+8*x^4-82*x^3+280*x^2-344*x+96)*exp((x-4)*ln(x)-2*x^2+8*x+4)+(2*x-2)*exp(x)^2+(2*x^2-10*x+16)*exp(
x)+8*x-32)/(x^3*exp((x-4)*ln(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*ln(x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*ln(x)-2*x^
2+8*x+4)+x^3),x,method=_RETURNVERBOSE)
[Out]
(x^2+2*exp(x)*x+exp(2*x)-8*x-8*exp(x)+16)/x^2/(x^(x-4)*exp(-2*x^2+8*x+4)+1)^2
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maxima [B] time = 0.75, size = 86, normalized size = 2.61 \begin {gather*} \frac {{\left (x^{8} - 8 \, x^{7} + x^{6} e^{\left (2 \, x\right )} + 16 \, x^{6} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} e^{x}\right )} e^{\left (4 \, x^{2}\right )}}{x^{8} e^{\left (4 \, x^{2}\right )} + 2 \, x^{4} e^{\left (2 \, x^{2} + x \log \relax (x) + 8 \, x + 4\right )} + e^{\left (2 \, x \log \relax (x) + 16 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*log(x)+(8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x
^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x^2-344*x+96)*exp((x-4)*log(x)-2*x^2+8*x+4)+(2*x-2)*exp(x)^2+(2*x^2-10*x+
16)*exp(x)+8*x-32)/(x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*
log(x)-2*x^2+8*x+4)+x^3),x, algorithm="maxima")
[Out]
(x^8 - 8*x^7 + x^6*e^(2*x) + 16*x^6 + 2*(x^7 - 4*x^6)*e^x)*e^(4*x^2)/(x^8*e^(4*x^2) + 2*x^4*e^(2*x^2 + x*log(x
) + 8*x + 4) + e^(2*x*log(x) + 16*x + 8))
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mupad [B] time = 2.51, size = 168, normalized size = 5.09 \begin {gather*} \frac {176\,x-4\,{\mathrm {e}}^{2\,x}+32\,{\mathrm {e}}^x+9\,x\,{\mathrm {e}}^{2\,x}+50\,x^2\,{\mathrm {e}}^x-8\,x^3\,{\mathrm {e}}^x-8\,x^2\,\ln \relax (x)+x^3\,\ln \relax (x)-4\,x^2\,{\mathrm {e}}^{2\,x}-80\,x\,{\mathrm {e}}^x+16\,x\,\ln \relax (x)-140\,x^2+41\,x^3-4\,x^4-8\,x\,{\mathrm {e}}^x\,\ln \relax (x)+x\,{\mathrm {e}}^{2\,x}\,\ln \relax (x)+2\,x^2\,{\mathrm {e}}^x\,\ln \relax (x)-64}{x^2\,\left (x^{2\,x-8}\,{\mathrm {e}}^{-4\,x^2+16\,x+8}+2\,x^{x-4}\,{\mathrm {e}}^{-2\,x^2+8\,x+4}+1\right )\,\left (9\,x+x\,\ln \relax (x)-4\,x^2-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((8*x + exp(x)*(2*x^2 - 10*x + 16) + exp(8*x + log(x)*(x - 4) - 2*x^2 + 4)*(exp(2*x)*(8*x^2 - 16*x + 6) - 3
44*x - log(x)*(32*x + 2*x*exp(2*x) - exp(x)*(16*x - 4*x^2) - 16*x^2 + 2*x^3) + 280*x^2 - 82*x^3 + 8*x^4 + exp(
x)*(150*x - 98*x^2 + 16*x^3 - 48) + 96) + exp(2*x)*(2*x - 2) - 32)/(3*x^3*exp(8*x + log(x)*(x - 4) - 2*x^2 + 4
) + 3*x^3*exp(16*x + 2*log(x)*(x - 4) - 4*x^2 + 8) + x^3*exp(24*x + 3*log(x)*(x - 4) - 6*x^2 + 12) + x^3),x)
[Out]
(176*x - 4*exp(2*x) + 32*exp(x) + 9*x*exp(2*x) + 50*x^2*exp(x) - 8*x^3*exp(x) - 8*x^2*log(x) + x^3*log(x) - 4*
x^2*exp(2*x) - 80*x*exp(x) + 16*x*log(x) - 140*x^2 + 41*x^3 - 4*x^4 - 8*x*exp(x)*log(x) + x*exp(2*x)*log(x) +
2*x^2*exp(x)*log(x) - 64)/(x^2*(x^(2*x - 8)*exp(16*x - 4*x^2 + 8) + 2*x^(x - 4)*exp(8*x - 2*x^2 + 4) + 1)*(9*x
+ x*log(x) - 4*x^2 - 4))
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sympy [B] time = 0.70, size = 75, normalized size = 2.27 \begin {gather*} \frac {x^{2} + 2 x e^{x} - 8 x + e^{2 x} - 8 e^{x} + 16}{x^{2} e^{- 4 x^{2} + 16 x + 2 \left (x - 4\right ) \log {\relax (x )} + 8} + 2 x^{2} e^{- 2 x^{2} + 8 x + \left (x - 4\right ) \log {\relax (x )} + 4} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x*exp(x)**2+(-4*x**2+16*x)*exp(x)-2*x**3+16*x**2-32*x)*ln(x)+(8*x**2-16*x+6)*exp(x)**2+(16*x**
3-98*x**2+150*x-48)*exp(x)+8*x**4-82*x**3+280*x**2-344*x+96)*exp((x-4)*ln(x)-2*x**2+8*x+4)+(2*x-2)*exp(x)**2+(
2*x**2-10*x+16)*exp(x)+8*x-32)/(x**3*exp((x-4)*ln(x)-2*x**2+8*x+4)**3+3*x**3*exp((x-4)*ln(x)-2*x**2+8*x+4)**2+
3*x**3*exp((x-4)*ln(x)-2*x**2+8*x+4)+x**3),x)
[Out]
(x**2 + 2*x*exp(x) - 8*x + exp(2*x) - 8*exp(x) + 16)/(x**2*exp(-4*x**2 + 16*x + 2*(x - 4)*log(x) + 8) + 2*x**2
*exp(-2*x**2 + 8*x + (x - 4)*log(x) + 4) + x**2)
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