\(\int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} (-128 x^2-128 x^3)+e^{\frac {20-x}{4}} (-64 x^2+240 x^3+256 x^4)+e^x (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} (4-15 x-16 x^2))+(64 x^2-256 x^3-256 x^4+e^x (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2)+e^{\frac {20-x}{4}} (256 x^2+256 x^3)) \log (e^{2 x}-32 e^x x^2+256 x^4)+(-128 x^2-128 x^3+e^x (8+8 x)) \log ^2(e^{2 x}-32 e^x x^2+256 x^4)}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2)+(128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x (-8 e^{\frac {20-x}{4}}+8 x)) \log (e^{2 x}-32 e^x x^2+256 x^4)+(4 e^x-64 x^2) \log ^2(e^{2 x}-32 e^x x^2+256 x^4)} \, dx\) [5381]

Optimal result

Integrand size = 434, antiderivative size = 34 \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=x \left (2+x-\frac {1}{-e^{5-\frac {x}{4}}+x+\log \left (\left (e^x-16 x^2\right )^2\right )}\right ) \]

[Out]

x*(2+x-1/(x+ln((exp(x)-16*x^2)^2)-exp(-1/4*x+5)))

Rubi [F(-1)]

Timed out. \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=\text {\$Aborted} \]

[In]

Int[(-256*x^2 - 128*x^4 - 128*x^5 + E^((20 - x)/2)*(-128*x^2 - 128*x^3) + E^((20 - x)/4)*(-64*x^2 + 240*x^3 +
256*x^4) + E^x*(8*x + 8*x^2 + 8*x^3 + E^((20 - x)/2)*(8 + 8*x) + E^((20 - x)/4)*(4 - 15*x - 16*x^2)) + (64*x^2
- 256*x^3 - 256*x^4 + E^x*(-4 + E^((20 - x)/4)*(-16 - 16*x) + 16*x + 16*x^2) + E^((20 - x)/4)*(256*x^2 + 256*
x^3))*Log[E^(2*x) - 32*E^x*x^2 + 256*x^4] + (-128*x^2 - 128*x^3 + E^x*(8 + 8*x))*Log[E^(2*x) - 32*E^x*x^2 + 25
6*x^4]^2)/(-64*E^((20 - x)/2)*x^2 + 128*E^((20 - x)/4)*x^3 - 64*x^4 + E^x*(4*E^((20 - x)/2) - 8*E^((20 - x)/4)
*x + 4*x^2) + (128*E^((20 - x)/4)*x^2 - 128*x^3 + E^x*(-8*E^((20 - x)/4) + 8*x))*Log[E^(2*x) - 32*E^x*x^2 + 25
6*x^4] + (4*E^x - 64*x^2)*Log[E^(2*x) - 32*E^x*x^2 + 256*x^4]^2),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=x \left (2+x-\frac {e^{x/4}}{-e^5+e^{x/4} x+e^{x/4} \log \left (\left (e^x-16 x^2\right )^2\right )}\right ) \]

[In]

Integrate[(-256*x^2 - 128*x^4 - 128*x^5 + E^((20 - x)/2)*(-128*x^2 - 128*x^3) + E^((20 - x)/4)*(-64*x^2 + 240*
x^3 + 256*x^4) + E^x*(8*x + 8*x^2 + 8*x^3 + E^((20 - x)/2)*(8 + 8*x) + E^((20 - x)/4)*(4 - 15*x - 16*x^2)) + (
64*x^2 - 256*x^3 - 256*x^4 + E^x*(-4 + E^((20 - x)/4)*(-16 - 16*x) + 16*x + 16*x^2) + E^((20 - x)/4)*(256*x^2
+ 256*x^3))*Log[E^(2*x) - 32*E^x*x^2 + 256*x^4] + (-128*x^2 - 128*x^3 + E^x*(8 + 8*x))*Log[E^(2*x) - 32*E^x*x^
2 + 256*x^4]^2)/(-64*E^((20 - x)/2)*x^2 + 128*E^((20 - x)/4)*x^3 - 64*x^4 + E^x*(4*E^((20 - x)/2) - 8*E^((20 -
x)/4)*x + 4*x^2) + (128*E^((20 - x)/4)*x^2 - 128*x^3 + E^x*(-8*E^((20 - x)/4) + 8*x))*Log[E^(2*x) - 32*E^x*x^
2 + 256*x^4] + (4*E^x - 64*x^2)*Log[E^(2*x) - 32*E^x*x^2 + 256*x^4]^2),x]

[Out]

x*(2 + x - E^(x/4)/(-E^5 + E^(x/4)*x + E^(x/4)*Log[(E^x - 16*x^2)^2]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs. \(2(30)=60\).

Time = 4.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.26

method	result	size
parallelrisch	\(-\frac {-128 x^{3}+128 x^{2} {\mathrm e}^{-\frac {x}{4}+5}-128 \ln \left ({\mathrm e}^{2 x}-32 \,{\mathrm e}^{x} x^{2}+256 x^{4}\right ) x^{2}-256 x^{2}+256 x \,{\mathrm e}^{-\frac {x}{4}+5}-256 x \ln \left ({\mathrm e}^{2 x}-32 \,{\mathrm e}^{x} x^{2}+256 x^{4}\right )+128 x}{128 \left (\ln \left ({\mathrm e}^{2 x}-32 \,{\mathrm e}^{x} x^{2}+256 x^{4}\right )-{\mathrm e}^{-\frac {x}{4}+5}+x \right )}\)	\(111\)
risch	\(x^{2}+2 x -\frac {2 x \,{\mathrm e}^{\frac {x}{4}}}{-i \pi {\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{x}}{16}-x^{2}\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{x}}{16}-x^{2}\right )^{2}\right ) {\mathrm e}^{\frac {x}{4}}-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{x}}{16}-x^{2}\right )\right ) {\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{x}}{16}-x^{2}\right )^{2}\right )}^{2} {\mathrm e}^{\frac {x}{4}}-i \pi {\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{x}}{16}-x^{2}\right )^{2}\right )}^{3} {\mathrm e}^{\frac {x}{4}}+2 x \,{\mathrm e}^{\frac {x}{4}}+4 \,{\mathrm e}^{\frac {x}{4}} \ln \left (-\frac {{\mathrm e}^{x}}{16}+x^{2}\right )-2 \,{\mathrm e}^{5}}\)	\(150\)

[In]

int((((8*x+8)*exp(x)-128*x^3-128*x^2)*ln(exp(x)^2-32*exp(x)*x^2+256*x^4)^2+(((-16*x-16)*exp(-1/4*x+5)+16*x^2+1
6*x-4)*exp(x)+(256*x^3+256*x^2)*exp(-1/4*x+5)-256*x^4-256*x^3+64*x^2)*ln(exp(x)^2-32*exp(x)*x^2+256*x^4)+((8*x
+8)*exp(-1/4*x+5)^2+(-16*x^2-15*x+4)*exp(-1/4*x+5)+8*x^3+8*x^2+8*x)*exp(x)+(-128*x^3-128*x^2)*exp(-1/4*x+5)^2+
(256*x^4+240*x^3-64*x^2)*exp(-1/4*x+5)-128*x^5-128*x^4-256*x^2)/((4*exp(x)-64*x^2)*ln(exp(x)^2-32*exp(x)*x^2+2
56*x^4)^2+((-8*exp(-1/4*x+5)+8*x)*exp(x)+128*x^2*exp(-1/4*x+5)-128*x^3)*ln(exp(x)^2-32*exp(x)*x^2+256*x^4)+(4*
exp(-1/4*x+5)^2-8*x*exp(-1/4*x+5)+4*x^2)*exp(x)-64*x^2*exp(-1/4*x+5)^2+128*x^3*exp(-1/4*x+5)-64*x^4),x,method=
_RETURNVERBOSE)

[Out]

-1/128*(-128*x^3+128*x^2*exp(-1/4*x+5)-128*x^2*ln(exp(x)^2-32*exp(x)*x^2+256*x^4)-256*x^2+256*x*exp(-1/4*x+5)-
256*x*ln(exp(x)^2-32*exp(x)*x^2+256*x^4)+128*x)/(ln(exp(x)^2-32*exp(x)*x^2+256*x^4)-exp(-1/4*x+5)+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (32) = 64\).

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.35 \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=\frac {x^{3} + 2 \, x^{2} - {\left (x^{2} + 2 \, x\right )} e^{\left (-\frac {1}{4} \, x + 5\right )} + {\left (x^{2} + 2 \, x\right )} \log \left ({\left (256 \, x^{4} e^{\left (-2 \, x + 40\right )} - 32 \, x^{2} e^{\left (-x + 40\right )} + e^{40}\right )} e^{\left (2 \, x - 40\right )}\right ) - x}{x - e^{\left (-\frac {1}{4} \, x + 5\right )} + \log \left ({\left (256 \, x^{4} e^{\left (-2 \, x + 40\right )} - 32 \, x^{2} e^{\left (-x + 40\right )} + e^{40}\right )} e^{\left (2 \, x - 40\right )}\right )} \]

[In]

integrate((((8*x+8)*exp(x)-128*x^3-128*x^2)*log(exp(x)^2-32*exp(x)*x^2+256*x^4)^2+(((-16*x-16)*exp(-1/4*x+5)+1
6*x^2+16*x-4)*exp(x)+(256*x^3+256*x^2)*exp(-1/4*x+5)-256*x^4-256*x^3+64*x^2)*log(exp(x)^2-32*exp(x)*x^2+256*x^
4)+((8*x+8)*exp(-1/4*x+5)^2+(-16*x^2-15*x+4)*exp(-1/4*x+5)+8*x^3+8*x^2+8*x)*exp(x)+(-128*x^3-128*x^2)*exp(-1/4
*x+5)^2+(256*x^4+240*x^3-64*x^2)*exp(-1/4*x+5)-128*x^5-128*x^4-256*x^2)/((4*exp(x)-64*x^2)*log(exp(x)^2-32*exp
(x)*x^2+256*x^4)^2+((-8*exp(-1/4*x+5)+8*x)*exp(x)+128*x^2*exp(-1/4*x+5)-128*x^3)*log(exp(x)^2-32*exp(x)*x^2+25
6*x^4)+(4*exp(-1/4*x+5)^2-8*x*exp(-1/4*x+5)+4*x^2)*exp(x)-64*x^2*exp(-1/4*x+5)^2+128*x^3*exp(-1/4*x+5)-64*x^4)
,x, algorithm="fricas")

[Out]

(x^3 + 2*x^2 - (x^2 + 2*x)*e^(-1/4*x + 5) + (x^2 + 2*x)*log((256*x^4*e^(-2*x + 40) - 32*x^2*e^(-x + 40) + e^40
)*e^(2*x - 40)) - x)/(x - e^(-1/4*x + 5) + log((256*x^4*e^(-2*x + 40) - 32*x^2*e^(-x + 40) + e^40)*e^(2*x - 40
)))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).

Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=x^{2} + 2 x - \frac {x \sqrt [4]{e^{x}}}{x \sqrt [4]{e^{x}} + \sqrt [4]{e^{x}} \log {\left (256 x^{4} - 32 x^{2} e^{x} + e^{2 x} \right )} - e^{5}} \]

[In]

integrate((((8*x+8)*exp(x)-128*x**3-128*x**2)*ln(exp(x)**2-32*exp(x)*x**2+256*x**4)**2+(((-16*x-16)*exp(-1/4*x
+5)+16*x**2+16*x-4)*exp(x)+(256*x**3+256*x**2)*exp(-1/4*x+5)-256*x**4-256*x**3+64*x**2)*ln(exp(x)**2-32*exp(x)
*x**2+256*x**4)+((8*x+8)*exp(-1/4*x+5)**2+(-16*x**2-15*x+4)*exp(-1/4*x+5)+8*x**3+8*x**2+8*x)*exp(x)+(-128*x**3
-128*x**2)*exp(-1/4*x+5)**2+(256*x**4+240*x**3-64*x**2)*exp(-1/4*x+5)-128*x**5-128*x**4-256*x**2)/((4*exp(x)-6
4*x**2)*ln(exp(x)**2-32*exp(x)*x**2+256*x**4)**2+((-8*exp(-1/4*x+5)+8*x)*exp(x)+128*x**2*exp(-1/4*x+5)-128*x**
3)*ln(exp(x)**2-32*exp(x)*x**2+256*x**4)+(4*exp(-1/4*x+5)**2-8*x*exp(-1/4*x+5)+4*x**2)*exp(x)-64*x**2*exp(-1/4
*x+5)**2+128*x**3*exp(-1/4*x+5)-64*x**4),x)

[Out]

x**2 + 2*x - x*exp(x)**(1/4)/(x*exp(x)**(1/4) + exp(x)**(1/4)*log(256*x**4 - 32*x**2*exp(x) + exp(2*x)) - exp(
5))

Maxima [F]

\[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=\int { \frac {128 \, x^{5} + 128 \, x^{4} + 8 \, {\left (16 \, x^{3} + 16 \, x^{2} - {\left (x + 1\right )} e^{x}\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right )^{2} + 256 \, x^{2} - {\left (8 \, x^{3} + 8 \, x^{2} - {\left (16 \, x^{2} + 15 \, x - 4\right )} e^{\left (-\frac {1}{4} \, x + 5\right )} + 8 \, {\left (x + 1\right )} e^{\left (-\frac {1}{2} \, x + 10\right )} + 8 \, x\right )} e^{x} - 16 \, {\left (16 \, x^{4} + 15 \, x^{3} - 4 \, x^{2}\right )} e^{\left (-\frac {1}{4} \, x + 5\right )} + 128 \, {\left (x^{3} + x^{2}\right )} e^{\left (-\frac {1}{2} \, x + 10\right )} + 4 \, {\left (64 \, x^{4} + 64 \, x^{3} - 16 \, x^{2} - {\left (4 \, x^{2} - 4 \, {\left (x + 1\right )} e^{\left (-\frac {1}{4} \, x + 5\right )} + 4 \, x - 1\right )} e^{x} - 64 \, {\left (x^{3} + x^{2}\right )} e^{\left (-\frac {1}{4} \, x + 5\right )}\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right )}{4 \, {\left (16 \, x^{4} - 32 \, x^{3} e^{\left (-\frac {1}{4} \, x + 5\right )} + 16 \, x^{2} e^{\left (-\frac {1}{2} \, x + 10\right )} + {\left (16 \, x^{2} - e^{x}\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right )^{2} - {\left (x^{2} - 2 \, x e^{\left (-\frac {1}{4} \, x + 5\right )} + e^{\left (-\frac {1}{2} \, x + 10\right )}\right )} e^{x} + 2 \, {\left (16 \, x^{3} - 16 \, x^{2} e^{\left (-\frac {1}{4} \, x + 5\right )} - {\left (x - e^{\left (-\frac {1}{4} \, x + 5\right )}\right )} e^{x}\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right )\right )}} \,d x } \]

[In]

integrate((((8*x+8)*exp(x)-128*x^3-128*x^2)*log(exp(x)^2-32*exp(x)*x^2+256*x^4)^2+(((-16*x-16)*exp(-1/4*x+5)+1
6*x^2+16*x-4)*exp(x)+(256*x^3+256*x^2)*exp(-1/4*x+5)-256*x^4-256*x^3+64*x^2)*log(exp(x)^2-32*exp(x)*x^2+256*x^
4)+((8*x+8)*exp(-1/4*x+5)^2+(-16*x^2-15*x+4)*exp(-1/4*x+5)+8*x^3+8*x^2+8*x)*exp(x)+(-128*x^3-128*x^2)*exp(-1/4
*x+5)^2+(256*x^4+240*x^3-64*x^2)*exp(-1/4*x+5)-128*x^5-128*x^4-256*x^2)/((4*exp(x)-64*x^2)*log(exp(x)^2-32*exp
(x)*x^2+256*x^4)^2+((-8*exp(-1/4*x+5)+8*x)*exp(x)+128*x^2*exp(-1/4*x+5)-128*x^3)*log(exp(x)^2-32*exp(x)*x^2+25
6*x^4)+(4*exp(-1/4*x+5)^2-8*x*exp(-1/4*x+5)+4*x^2)*exp(x)-64*x^2*exp(-1/4*x+5)^2+128*x^3*exp(-1/4*x+5)-64*x^4)
,x, algorithm="maxima")

[Out]

(256*x^6*e^10 + 512*x^5*e^10 + 4*(16*x^5 + 96*x^4 + 80*x^3 - 3*(x^3 + 2*x^2 - x)*e^x)*e^(3/2*x) + (16*x^5*e^5
- 32*x^4*e^5 - 400*x^3*e^5 - 512*x^2*e^5 - (x^3*e^5 - 10*x^2*e^5 - 25*x*e^5)*e^x)*e^(5/4*x) - 64*(16*x^7 + 96*
x^6 + 112*x^5 - 64*x^4 - (3*x^5 + 6*x^4 - x^3 - 4*x^2)*e^x)*e^(1/2*x) - 16*(16*x^7*e^5 - 32*x^6*e^5 - 400*x^5*
e^5 - 512*x^4*e^5 - (x^5*e^5 - 10*x^4*e^5 - 25*x^3*e^5)*e^x)*e^(1/4*x) - (16*x^4*e^10 + 32*x^3*e^10 - (x^2*e^1
0 + 2*x*e^10)*e^x)*e^x - 16*(x^4*e^10 + 2*x^3*e^10)*e^x + 2*(4*(16*x^4 + 96*x^3 + 128*x^2 - 3*(x^2 + 2*x)*e^x)
*e^(3/2*x) + (16*x^4*e^5 + 32*x^3*e^5 - (x^2*e^5 + 2*x*e^5)*e^x)*e^(5/4*x) - 64*(16*x^6 + 96*x^5 + 128*x^4 - 3
*(x^4 + 2*x^3)*e^x)*e^(1/2*x) - 16*(16*x^6*e^5 + 32*x^5*e^5 - (x^4*e^5 + 2*x^3*e^5)*e^x)*e^(1/4*x))*log(-16*x^
2 + e^x))/(256*x^4*e^10 - 16*x^2*e^(x + 10) + 4*(16*x^3 + 64*x^2 - 3*x*e^x)*e^(3/2*x) + (16*x^3*e^5 - 64*x^2*e
^5 - 256*x*e^5 - (x*e^5 - 12*e^5)*e^x)*e^(5/4*x) - 64*(16*x^5 + 64*x^4 - 3*x^3*e^x)*e^(1/2*x) - 16*(16*x^5*e^5
- 64*x^4*e^5 - 256*x^3*e^5 - (x^3*e^5 - 12*x^2*e^5)*e^x)*e^(1/4*x) - (16*x^2*e^10 - e^(x + 10))*e^x + 2*(4*(1
6*x^2 + 64*x - 3*e^x)*e^(3/2*x) + (16*x^2*e^5 - e^(x + 5))*e^(5/4*x) - 64*(16*x^4 + 64*x^3 - 3*x^2*e^x)*e^(1/2
*x) - 16*(16*x^4*e^5 - x^2*e^(x + 5))*e^(1/4*x))*log(-16*x^2 + e^x)) + 1/4*integrate(-128*(16*(16*x^4 + 128*x^
3 - 128*x^2 + 3*(x^3 - 5*x^2 + 4*x)*e^x)*e^(11/4*x) + (16*x^5*e^5 + 32*x^4*e^5 + (3*x^3*e^5 - 18*x^2*e^5 + 16*
x*e^5)*e^x)*e^(5/2*x) - 16*(256*x^7 + 256*x^6 - 1024*x^5 + 2048*x^4 + 3*(x^3 - 5*x^2 + 4*x)*e^(2*x) - 32*(x^4
- 4*x^3 + 4*x^2)*e^x)*e^(7/4*x) - (1280*x^7*e^5 - 3584*x^6*e^5 + 4096*x^5*e^5 + (3*x^3*e^5 - 18*x^2*e^5 + 16*x
*e^5)*e^(2*x))*e^(3/2*x) - 256*(3*x^4*e^(2*x) - 16*(x^7 + x^6 - 4*x^5 + 8*x^4)*e^x)*e^(3/4*x) - 16*((x^5*e^5 +
2*x^4*e^5)*e^(2*x) - 16*(5*x^7*e^5 - 14*x^6*e^5 + 16*x^5*e^5)*e^x)*e^(1/2*x))/(65536*x^8*e^15 - 8192*x^6*e^(x
+ 15) + 256*x^4*e^(2*x + 15) - 16*(256*x^5 + 2048*x^4 + 4096*x^3 + 9*x*e^(2*x) - 96*(x^3 + 4*x^2)*e^x)*e^(11/
4*x) - 8*(256*x^5*e^5 + 512*x^4*e^5 - 4096*x^3*e^5 - 8192*x^2*e^5 + 3*(x*e^5 - 6*e^5)*e^(2*x) - 64*(x^3*e^5 -
2*x^2*e^5 - 12*x*e^5)*e^x)*e^(5/2*x) - (256*x^5*e^10 - 2048*x^4*e^10 - 8192*x^3*e^10 + (x*e^10 - 24*e^10)*e^(2
*x) - 32*(x^3*e^10 - 16*x^2*e^10 - 16*x*e^10)*e^x)*e^(9/4*x) + (256*x^4*e^15 - 32*x^2*e^(x + 15) + e^(2*x + 15
))*e^(2*x) + 512*(256*x^7 + 2048*x^6 + 4096*x^5 + 9*x^3*e^(2*x) - 96*(x^5 + 4*x^4)*e^x)*e^(7/4*x) + 256*(256*x
^7*e^5 + 512*x^6*e^5 - 4096*x^5*e^5 - 8192*x^4*e^5 + 3*(x^3*e^5 - 6*x^2*e^5)*e^(2*x) - 64*(x^5*e^5 - 2*x^4*e^5
- 12*x^3*e^5)*e^x)*e^(3/2*x) + 32*(256*x^7*e^10 - 2048*x^6*e^10 - 8192*x^5*e^10 + (x^3*e^10 - 24*x^2*e^10)*e^
(2*x) - 32*(x^5*e^10 - 16*x^4*e^10 - 16*x^3*e^10)*e^x)*e^(5/4*x) - 4096*(256*x^9 + 2048*x^8 + 4096*x^7 + 9*x^5
*e^(2*x) - 96*(x^7 + 4*x^6)*e^x)*e^(3/4*x) - 2048*(256*x^9*e^5 + 512*x^8*e^5 - 4096*x^7*e^5 - 8192*x^6*e^5 + 3
*(x^5*e^5 - 6*x^4*e^5)*e^(2*x) - 64*(x^7*e^5 - 2*x^6*e^5 - 12*x^5*e^5)*e^x)*e^(1/2*x) - 256*(256*x^9*e^10 - 20
48*x^8*e^10 - 8192*x^7*e^10 + (x^5*e^10 - 24*x^4*e^10)*e^(2*x) - 32*(x^7*e^10 - 16*x^6*e^10 - 16*x^5*e^10)*e^x
)*e^(1/4*x) - 32*(256*x^6*e^15 - 32*x^4*e^(x + 15) + x^2*e^(2*x + 15))*e^x - 2*(16*(256*x^4 + 2048*x^3 + 4096*
x^2 - 96*(x^2 + 4*x)*e^x + 9*e^(2*x))*e^(11/4*x) + 8*(256*x^4*e^5 + 1024*x^3*e^5 - 64*(x^2*e^5 + x*e^5)*e^x +
3*e^(2*x + 5))*e^(5/2*x) + (256*x^4*e^10 - 32*x^2*e^(x + 10) + e^(2*x + 10))*e^(9/4*x) - 512*(256*x^6 + 2048*x
^5 + 4096*x^4 + 9*x^2*e^(2*x) - 96*(x^4 + 4*x^3)*e^x)*e^(7/4*x) - 256*(256*x^6*e^5 + 1024*x^5*e^5 + 3*x^2*e^(2
*x + 5) - 64*(x^4*e^5 + x^3*e^5)*e^x)*e^(3/2*x) - 32*(256*x^6*e^10 - 32*x^4*e^(x + 10) + x^2*e^(2*x + 10))*e^(
5/4*x) + 4096*(256*x^8 + 2048*x^7 + 4096*x^6 + 9*x^4*e^(2*x) - 96*(x^6 + 4*x^5)*e^x)*e^(3/4*x) + 2048*(256*x^8
*e^5 + 1024*x^7*e^5 + 3*x^4*e^(2*x + 5) - 64*(x^6*e^5 + x^5*e^5)*e^x)*e^(1/2*x) + 256*(256*x^8*e^10 - 32*x^6*e
^(x + 10) + x^4*e^(2*x + 10))*e^(1/4*x))*log(-16*x^2 + e^x)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (32) = 64\).

Time = 2.44 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.68 \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=\frac {x^{3} e^{\left (\frac {1}{4} \, x\right )} + x^{2} e^{\left (\frac {1}{4} \, x\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right ) - x^{2} e^{5} + 2 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} + 2 \, x e^{\left (\frac {1}{4} \, x\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right ) - 2 \, x e^{5} - x e^{\left (\frac {1}{4} \, x\right )}}{x e^{\left (\frac {1}{4} \, x\right )} + e^{\left (\frac {1}{4} \, x\right )} \log \left (256 \, x^{4} - 32 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right ) - e^{5}} \]

[In]

integrate((((8*x+8)*exp(x)-128*x^3-128*x^2)*log(exp(x)^2-32*exp(x)*x^2+256*x^4)^2+(((-16*x-16)*exp(-1/4*x+5)+1
6*x^2+16*x-4)*exp(x)+(256*x^3+256*x^2)*exp(-1/4*x+5)-256*x^4-256*x^3+64*x^2)*log(exp(x)^2-32*exp(x)*x^2+256*x^
4)+((8*x+8)*exp(-1/4*x+5)^2+(-16*x^2-15*x+4)*exp(-1/4*x+5)+8*x^3+8*x^2+8*x)*exp(x)+(-128*x^3-128*x^2)*exp(-1/4
*x+5)^2+(256*x^4+240*x^3-64*x^2)*exp(-1/4*x+5)-128*x^5-128*x^4-256*x^2)/((4*exp(x)-64*x^2)*log(exp(x)^2-32*exp
(x)*x^2+256*x^4)^2+((-8*exp(-1/4*x+5)+8*x)*exp(x)+128*x^2*exp(-1/4*x+5)-128*x^3)*log(exp(x)^2-32*exp(x)*x^2+25
6*x^4)+(4*exp(-1/4*x+5)^2-8*x*exp(-1/4*x+5)+4*x^2)*exp(x)-64*x^2*exp(-1/4*x+5)^2+128*x^3*exp(-1/4*x+5)-64*x^4)
,x, algorithm="giac")

[Out]

(x^3*e^(1/4*x) + x^2*e^(1/4*x)*log(256*x^4 - 32*x^2*e^x + e^(2*x)) - x^2*e^5 + 2*x^2*e^(1/4*x) + 2*x*e^(1/4*x)
*log(256*x^4 - 32*x^2*e^x + e^(2*x)) - 2*x*e^5 - x*e^(1/4*x))/(x*e^(1/4*x) + e^(1/4*x)*log(256*x^4 - 32*x^2*e^
x + e^(2*x)) - e^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {-256 x^2-128 x^4-128 x^5+e^{\frac {20-x}{2}} \left (-128 x^2-128 x^3\right )+e^{\frac {20-x}{4}} \left (-64 x^2+240 x^3+256 x^4\right )+e^x \left (8 x+8 x^2+8 x^3+e^{\frac {20-x}{2}} (8+8 x)+e^{\frac {20-x}{4}} \left (4-15 x-16 x^2\right )\right )+\left (64 x^2-256 x^3-256 x^4+e^x \left (-4+e^{\frac {20-x}{4}} (-16-16 x)+16 x+16 x^2\right )+e^{\frac {20-x}{4}} \left (256 x^2+256 x^3\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (-128 x^2-128 x^3+e^x (8+8 x)\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )}{-64 e^{\frac {20-x}{2}} x^2+128 e^{\frac {20-x}{4}} x^3-64 x^4+e^x \left (4 e^{\frac {20-x}{2}}-8 e^{\frac {20-x}{4}} x+4 x^2\right )+\left (128 e^{\frac {20-x}{4}} x^2-128 x^3+e^x \left (-8 e^{\frac {20-x}{4}}+8 x\right )\right ) \log \left (e^{2 x}-32 e^x x^2+256 x^4\right )+\left (4 e^x-64 x^2\right ) \log ^2\left (e^{2 x}-32 e^x x^2+256 x^4\right )} \, dx=-\int \frac {{\ln \left ({\mathrm {e}}^{2\,x}-32\,x^2\,{\mathrm {e}}^x+256\,x^4\right )}^2\,\left (128\,x^2-{\mathrm {e}}^x\,\left (8\,x+8\right )+128\,x^3\right )-\ln \left ({\mathrm {e}}^{2\,x}-32\,x^2\,{\mathrm {e}}^x+256\,x^4\right )\,\left ({\mathrm {e}}^{5-\frac {x}{4}}\,\left (256\,x^3+256\,x^2\right )+64\,x^2-256\,x^3-256\,x^4+{\mathrm {e}}^x\,\left (16\,x-{\mathrm {e}}^{5-\frac {x}{4}}\,\left (16\,x+16\right )+16\,x^2-4\right )\right )-{\mathrm {e}}^x\,\left (8\,x+{\mathrm {e}}^{10-\frac {x}{2}}\,\left (8\,x+8\right )-{\mathrm {e}}^{5-\frac {x}{4}}\,\left (16\,x^2+15\,x-4\right )+8\,x^2+8\,x^3\right )+{\mathrm {e}}^{10-\frac {x}{2}}\,\left (128\,x^3+128\,x^2\right )+256\,x^2+128\,x^4+128\,x^5-{\mathrm {e}}^{5-\frac {x}{4}}\,\left (256\,x^4+240\,x^3-64\,x^2\right )}{{\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^{10-\frac {x}{2}}-8\,x\,{\mathrm {e}}^{5-\frac {x}{4}}+4\,x^2\right )+\ln \left ({\mathrm {e}}^{2\,x}-32\,x^2\,{\mathrm {e}}^x+256\,x^4\right )\,\left ({\mathrm {e}}^x\,\left (8\,x-8\,{\mathrm {e}}^{5-\frac {x}{4}}\right )+128\,x^2\,{\mathrm {e}}^{5-\frac {x}{4}}-128\,x^3\right )+128\,x^3\,{\mathrm {e}}^{5-\frac {x}{4}}-64\,x^2\,{\mathrm {e}}^{10-\frac {x}{2}}-64\,x^4+{\ln \left ({\mathrm {e}}^{2\,x}-32\,x^2\,{\mathrm {e}}^x+256\,x^4\right )}^2\,\left (4\,{\mathrm {e}}^x-64\,x^2\right )} \,d x \]

[In]

int(-(log(exp(2*x) - 32*x^2*exp(x) + 256*x^4)^2*(128*x^2 - exp(x)*(8*x + 8) + 128*x^3) - log(exp(2*x) - 32*x^2
*exp(x) + 256*x^4)*(exp(5 - x/4)*(256*x^2 + 256*x^3) + 64*x^2 - 256*x^3 - 256*x^4 + exp(x)*(16*x - exp(5 - x/4
)*(16*x + 16) + 16*x^2 - 4)) - exp(x)*(8*x + exp(10 - x/2)*(8*x + 8) - exp(5 - x/4)*(15*x + 16*x^2 - 4) + 8*x^
2 + 8*x^3) + exp(10 - x/2)*(128*x^2 + 128*x^3) + 256*x^2 + 128*x^4 + 128*x^5 - exp(5 - x/4)*(240*x^3 - 64*x^2
+ 256*x^4))/(exp(x)*(4*exp(10 - x/2) - 8*x*exp(5 - x/4) + 4*x^2) + log(exp(2*x) - 32*x^2*exp(x) + 256*x^4)*(ex
p(x)*(8*x - 8*exp(5 - x/4)) + 128*x^2*exp(5 - x/4) - 128*x^3) + 128*x^3*exp(5 - x/4) - 64*x^2*exp(10 - x/2) -
64*x^4 + log(exp(2*x) - 32*x^2*exp(x) + 256*x^4)^2*(4*exp(x) - 64*x^2)),x)

[Out]

-int((log(exp(2*x) - 32*x^2*exp(x) + 256*x^4)^2*(128*x^2 - exp(x)*(8*x + 8) + 128*x^3) - log(exp(2*x) - 32*x^2
*exp(x) + 256*x^4)*(exp(5 - x/4)*(256*x^2 + 256*x^3) + 64*x^2 - 256*x^3 - 256*x^4 + exp(x)*(16*x - exp(5 - x/4
)*(16*x + 16) + 16*x^2 - 4)) - exp(x)*(8*x + exp(10 - x/2)*(8*x + 8) - exp(5 - x/4)*(15*x + 16*x^2 - 4) + 8*x^
2 + 8*x^3) + exp(10 - x/2)*(128*x^2 + 128*x^3) + 256*x^2 + 128*x^4 + 128*x^5 - exp(5 - x/4)*(240*x^3 - 64*x^2
+ 256*x^4))/(exp(x)*(4*exp(10 - x/2) - 8*x*exp(5 - x/4) + 4*x^2) + log(exp(2*x) - 32*x^2*exp(x) + 256*x^4)*(ex
p(x)*(8*x - 8*exp(5 - x/4)) + 128*x^2*exp(5 - x/4) - 128*x^3) + 128*x^3*exp(5 - x/4) - 64*x^2*exp(10 - x/2) -
64*x^4 + log(exp(2*x) - 32*x^2*exp(x) + 256*x^4)^2*(4*exp(x) - 64*x^2)), x)