∫ −32+8x+e2x(−2+2x)+ex(16−10x+2x2)+e4+8x−2x2+(−4+x) log(x) (96−344x+280x2−82x3+8x4+e2x(6−16x+8x2)+ex(−48+150x−98x2+16x3)+(−32x−2e2xx+16x2−2x3+ex(16x−4x2)) log(x))__________________________________________________________________________________________________________________________________________ x3+3e4+8x−2x2+(−4+x) log(x)x3+3e8+16x−4x2+2(−4+x) log(x)x3+e12+24x−6x2+3(−4+x) log(x)x3 dx

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


method	result	size

risch	\(\frac {x^{2}+2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}-8 x -8 \,{\mathrm e}^{x}+16}{x^{2} \left (x^{x -4} {\mathrm e}^{-2 x^{2}+8 x +4}+1\right )^{2}}\)	\(47\)

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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