3.92.67 \(\int \frac {1+75 x^2+5 x^3-5 x^4+e^x (15 x^2+6 x^3-2 x^4)+(-75 x^2+20 x^3+e^x (-15 x^2-x^3+x^4)) \log (\frac {e^x}{2})}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} (25 x^6-10 x^7+x^8)+e^x (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8)} \, dx\)

Optimal. Leaf size=36 \[ \frac {-x+x \log \left (\frac {e^x}{2}\right )}{x \left (1+\left (5+e^x\right ) (5-x) x^3\right )} \]

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Rubi [F]  time = 12.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 75*x^2 + 5*x^3 - 5*x^4 + E^x*(15*x^2 + 6*x^3 - 2*x^4) + (-75*x^2 + 20*x^3 + E^x*(-15*x^2 - x^3 + x^4)
)*Log[E^x/2])/(1 + 50*x^3 - 10*x^4 + 625*x^6 - 250*x^7 + 25*x^8 + E^(2*x)*(25*x^6 - 10*x^7 + x^8) + E^x*(10*x^
3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8)),x]

[Out]

-((1 + Log[2])*Defer[Int][(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^(-2), x]) + Log[E^x]*Defer[Int][(-1 - 25
*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^(-2), x] - (1 + Log[2])*Defer[Int][1/((-5 + x)*(-1 - 25*x^3 - 5*E^x*x^3 +
5*x^4 + E^x*x^4)^2), x] + Log[E^x]*Defer[Int][1/((-5 + x)*(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^2), x] -
(3 + Log[8])*Defer[Int][1/(x*(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^2), x] + 3*Log[E^x]*Defer[Int][1/(x*
(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^2), x] - 25*Defer[Int][x^3/(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*
x^4)^2, x] + 25*Log[E^x/2]*Defer[Int][x^3/(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^2, x] + 5*Defer[Int][x^4
/(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^2, x] - 5*Log[E^x/2]*Defer[Int][x^4/(-1 - 25*x^3 - 5*E^x*x^3 + 5*
x^4 + E^x*x^4)^2, x] - 2*Defer[Int][(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^(-1), x] + Log[E^x/2]*Defer[In
t][(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)^(-1), x] - Defer[Int][1/((-5 + x)*(-1 - 25*x^3 - 5*E^x*x^3 + 5*
x^4 + E^x*x^4)), x] + Log[E^x/2]*Defer[Int][1/((-5 + x)*(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)), x] - 3*D
efer[Int][1/(x*(-1 - 25*x^3 - 5*E^x*x^3 + 5*x^4 + E^x*x^4)), x] + 3*Log[E^x/2]*Defer[Int][1/(x*(-1 - 25*x^3 -
5*E^x*x^3 + 5*x^4 + E^x*x^4)), x] - Defer[Int][Defer[Int][(1 + 5*(5 + E^x)*x^3 - (5 + E^x)*x^4)^(-2), x], x] -
Defer[Int][Defer[Int][1/((-5 + x)*(1 + 5*(5 + E^x)*x^3 - (5 + E^x)*x^4)^2), x], x] - 3*Defer[Int][Defer[Int][
1/(x*(1 + 5*(5 + E^x)*x^3 - (5 + E^x)*x^4)^2), x], x] - 25*Defer[Int][Defer[Int][x^3/(1 + 5*(5 + E^x)*x^3 - (5
+ E^x)*x^4)^2, x], x] + 5*Defer[Int][Defer[Int][x^4/(1 + 5*(5 + E^x)*x^3 - (5 + E^x)*x^4)^2, x], x] - Defer[I
nt][Defer[Int][(-1 - 5*(5 + E^x)*x^3 + (5 + E^x)*x^4)^(-1), x], x] - Defer[Int][Defer[Int][1/((-5 + x)*(-1 - 5
*(5 + E^x)*x^3 + (5 + E^x)*x^4)), x], x] - 3*Defer[Int][Defer[Int][1/(x*(-1 - 5*(5 + E^x)*x^3 + (5 + E^x)*x^4)
), x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+15 \left (5+e^x\right ) x^2+\left (5+6 e^x\right ) x^3-\left (5+2 e^x\right ) x^4+x^2 \left (-75+20 x+e^x \left (-15-x+x^2\right )\right ) \log \left (\frac {e^x}{2}\right )}{\left (1+5 \left (5+e^x\right ) x^3-\left (5+e^x\right ) x^4\right )^2} \, dx\\ &=\int \left (-\frac {\left (15+x-x^2+125 x^4-50 x^5+5 x^6\right ) \left (-1+\log \left (\frac {e^x}{2}\right )\right )}{(-5+x) x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}+\frac {15+6 x-2 x^2-15 \log \left (\frac {e^x}{2}\right )-x \log \left (\frac {e^x}{2}\right )+x^2 \log \left (\frac {e^x}{2}\right )}{(-5+x) x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}\right ) \, dx\\ &=-\int \frac {\left (15+x-x^2+125 x^4-50 x^5+5 x^6\right ) \left (-1+\log \left (\frac {e^x}{2}\right )\right )}{(-5+x) x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx+\int \frac {15+6 x-2 x^2-15 \log \left (\frac {e^x}{2}\right )-x \log \left (\frac {e^x}{2}\right )+x^2 \log \left (\frac {e^x}{2}\right )}{(-5+x) x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx\\ &=\int \left (\frac {15+6 x-2 x^2-15 \log \left (\frac {e^x}{2}\right )-x \log \left (\frac {e^x}{2}\right )+x^2 \log \left (\frac {e^x}{2}\right )}{5 (-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}-\frac {15+6 x-2 x^2-15 \log \left (\frac {e^x}{2}\right )-x \log \left (\frac {e^x}{2}\right )+x^2 \log \left (\frac {e^x}{2}\right )}{5 x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}\right ) \, dx-\int \left (-\frac {25 x^3 \left (-1+\log \left (\frac {e^x}{2}\right )\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}+\frac {5 x^4 \left (-1+\log \left (\frac {e^x}{2}\right )\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}+\frac {3 (1+\log (2))-3 \log \left (e^x\right )}{x \left (1+25 x^3+5 e^x x^3-5 x^4-e^x x^4\right )^2}+\frac {1+\log (2)-\log \left (e^x\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}+\frac {1+\log (2)-\log \left (e^x\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {15+6 x-2 x^2-15 \log \left (\frac {e^x}{2}\right )-x \log \left (\frac {e^x}{2}\right )+x^2 \log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx-\frac {1}{5} \int \frac {15+6 x-2 x^2-15 \log \left (\frac {e^x}{2}\right )-x \log \left (\frac {e^x}{2}\right )+x^2 \log \left (\frac {e^x}{2}\right )}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx-5 \int \frac {x^4 \left (-1+\log \left (\frac {e^x}{2}\right )\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx+25 \int \frac {x^3 \left (-1+\log \left (\frac {e^x}{2}\right )\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-\int \frac {3 (1+\log (2))-3 \log \left (e^x\right )}{x \left (1+25 x^3+5 e^x x^3-5 x^4-e^x x^4\right )^2} \, dx-\int \frac {1+\log (2)-\log \left (e^x\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-\int \frac {1+\log (2)-\log \left (e^x\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx\\ &=-\left (\frac {1}{5} \int \frac {-15-6 x+2 x^2-\left (-15-x+x^2\right ) \log \left (\frac {e^x}{2}\right )}{x \left (1+5 \left (5+e^x\right ) x^3-\left (5+e^x\right ) x^4\right )} \, dx\right )+\frac {1}{5} \int \left (\frac {15}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}+\frac {6 x}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}-\frac {2 x^2}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}-\frac {15 \log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}-\frac {x \log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}+\frac {x^2 \log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}\right ) \, dx-5 \int \left (-\frac {x^4}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}+\frac {x^4 \log \left (\frac {e^x}{2}\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}\right ) \, dx+25 \int \left (-\frac {x^3}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}+\frac {x^3 \log \left (\frac {e^x}{2}\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}\right ) \, dx-\int \left (\frac {1+\log (2)}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}-\frac {\log \left (e^x\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}\right ) \, dx-\int \left (\frac {1+\log (2)}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}-\frac {\log \left (e^x\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}\right ) \, dx-\int \left (\frac {3+\log (8)}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}-\frac {3 \log \left (e^x\right )}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {x \log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx\right )+\frac {1}{5} \int \frac {x^2 \log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx-\frac {1}{5} \int \left (\frac {6}{-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4}+\frac {15}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}-\frac {2 x}{-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4}-\frac {\log \left (\frac {e^x}{2}\right )}{-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4}-\frac {15 \log \left (\frac {e^x}{2}\right )}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )}+\frac {x \log \left (\frac {e^x}{2}\right )}{-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4}\right ) \, dx-\frac {2}{5} \int \frac {x^2}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx+\frac {6}{5} \int \frac {x}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx+3 \int \frac {1}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx-3 \int \frac {\log \left (\frac {e^x}{2}\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )} \, dx+3 \int \frac {\log \left (e^x\right )}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx+5 \int \frac {x^4}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-5 \int \frac {x^4 \log \left (\frac {e^x}{2}\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-25 \int \frac {x^3}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx+25 \int \frac {x^3 \log \left (\frac {e^x}{2}\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-(1+\log (2)) \int \frac {1}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-(1+\log (2)) \int \frac {1}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx-(3+\log (8)) \int \frac {1}{x \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx+\int \frac {\log \left (e^x\right )}{\left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx+\int \frac {\log \left (e^x\right )}{(-5+x) \left (-1-25 x^3-5 e^x x^3+5 x^4+e^x x^4\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 34, normalized size = 0.94 \begin {gather*} \frac {1+\log (2)-\log \left (e^x\right )}{-1-5 \left (5+e^x\right ) x^3+\left (5+e^x\right ) x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 75*x^2 + 5*x^3 - 5*x^4 + E^x*(15*x^2 + 6*x^3 - 2*x^4) + (-75*x^2 + 20*x^3 + E^x*(-15*x^2 - x^3
+ x^4))*Log[E^x/2])/(1 + 50*x^3 - 10*x^4 + 625*x^6 - 250*x^7 + 25*x^8 + E^(2*x)*(25*x^6 - 10*x^7 + x^8) + E^x*
(10*x^3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8)),x]

[Out]

(1 + Log[2] - Log[E^x])/(-1 - 5*(5 + E^x)*x^3 + (5 + E^x)*x^4)

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fricas [A]  time = 0.55, size = 35, normalized size = 0.97 \begin {gather*} -\frac {x - \log \relax (2) - 1}{5 \, x^{4} - 25 \, x^{3} + {\left (x^{4} - 5 \, x^{3}\right )} e^{x} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*log(1/2*exp(x))+(-2*x^4+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75
*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x
^4+50*x^3+1),x, algorithm="fricas")

[Out]

-(x - log(2) - 1)/(5*x^4 - 25*x^3 + (x^4 - 5*x^3)*e^x - 1)

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giac [A]  time = 0.16, size = 36, normalized size = 1.00 \begin {gather*} -\frac {x - \log \relax (2) - 1}{x^{4} e^{x} + 5 \, x^{4} - 5 \, x^{3} e^{x} - 25 \, x^{3} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*log(1/2*exp(x))+(-2*x^4+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75
*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x
^4+50*x^3+1),x, algorithm="giac")

[Out]

-(x - log(2) - 1)/(x^4*e^x + 5*x^4 - 5*x^3*e^x - 25*x^3 - 1)

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maple [A]  time = 0.28, size = 38, normalized size = 1.06

method	result	size
default	\(\frac {1-\ln \left (\frac {{\mathrm e}^{x}}{2}\right )}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}\)	\(38\)
risch	\(-\frac {\ln \left ({\mathrm e}^{x}\right )}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}+\frac {1}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}+\frac {\ln \relax (2)}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*ln(1/2*exp(x))+(-2*x^4+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75*x^2+1)
/((x^8-10*x^7+25*x^6)*exp(x)^2+(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x^4+50*x
^3+1),x,method=_RETURNVERBOSE)

[Out]

(1-ln(1/2*exp(x)))/(exp(x)*x^4-5*exp(x)*x^3+5*x^4-25*x^3-1)

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maxima [A]  time = 0.52, size = 35, normalized size = 0.97 \begin {gather*} -\frac {x - \log \relax (2) - 1}{5 \, x^{4} - 25 \, x^{3} + {\left (x^{4} - 5 \, x^{3}\right )} e^{x} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*log(1/2*exp(x))+(-2*x^4+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75
*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x
^4+50*x^3+1),x, algorithm="maxima")

[Out]

-(x - log(2) - 1)/(5*x^4 - 25*x^3 + (x^4 - 5*x^3)*e^x - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (-2\,x^4+6\,x^3+15\,x^2\right )-\ln \left (\frac {{\mathrm {e}}^x}{2}\right )\,\left (75\,x^2-20\,x^3+{\mathrm {e}}^x\,\left (-x^4+x^3+15\,x^2\right )\right )+75\,x^2+5\,x^3-5\,x^4+1}{{\mathrm {e}}^{2\,x}\,\left (x^8-10\,x^7+25\,x^6\right )+{\mathrm {e}}^x\,\left (10\,x^8-100\,x^7+250\,x^6-2\,x^4+10\,x^3\right )+50\,x^3-10\,x^4+625\,x^6-250\,x^7+25\,x^8+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(15*x^2 + 6*x^3 - 2*x^4) - log(exp(x)/2)*(75*x^2 - 20*x^3 + exp(x)*(15*x^2 + x^3 - x^4)) + 75*x^2
+ 5*x^3 - 5*x^4 + 1)/(exp(2*x)*(25*x^6 - 10*x^7 + x^8) + exp(x)*(10*x^3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8)
+ 50*x^3 - 10*x^4 + 625*x^6 - 250*x^7 + 25*x^8 + 1),x)

[Out]

int((exp(x)*(15*x^2 + 6*x^3 - 2*x^4) - log(exp(x)/2)*(75*x^2 - 20*x^3 + exp(x)*(15*x^2 + x^3 - x^4)) + 75*x^2
+ 5*x^3 - 5*x^4 + 1)/(exp(2*x)*(25*x^6 - 10*x^7 + x^8) + exp(x)*(10*x^3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8)
+ 50*x^3 - 10*x^4 + 625*x^6 - 250*x^7 + 25*x^8 + 1), x)

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sympy [A]  time = 0.34, size = 29, normalized size = 0.81 \begin {gather*} \frac {- x + \log {\relax (2 )} + 1}{5 x^{4} - 25 x^{3} + \left (x^{4} - 5 x^{3}\right ) e^{x} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4-x**3-15*x**2)*exp(x)+20*x**3-75*x**2)*ln(1/2*exp(x))+(-2*x**4+6*x**3+15*x**2)*exp(x)-5*x**4+
5*x**3+75*x**2+1)/((x**8-10*x**7+25*x**6)*exp(x)**2+(10*x**8-100*x**7+250*x**6-2*x**4+10*x**3)*exp(x)+25*x**8-
250*x**7+625*x**6-10*x**4+50*x**3+1),x)

[Out]

(-x + log(2) + 1)/(5*x**4 - 25*x**3 + (x**4 - 5*x**3)*exp(x) - 1)

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