3.94.23 \(\int \frac {-3 x+e^{2 x} (4 e^5+2 x)+e^x (-x+x^2+e^5 (4+x))+(24 x+e^{2 x} (-2 e^5+18 x)+e^x (e^5 (-2-4 x)+42 x-4 x^2)) \log (x)+(-54 x-53 e^{2 x} x+e^x (-107 x+e^5 x+x^2)) \log ^2(x)+(24 x+48 e^x x+24 e^{2 x} x) \log ^3(x)+(-3 x-6 e^x x-3 e^{2 x} x) \log ^4(x)}{x+2 e^x x+e^{2 x} x+(-8 x-16 e^x x-8 e^{2 x} x) \log (x)+(18 x+36 e^x x+18 e^{2 x} x) \log ^2(x)+(-8 x-16 e^x x-8 e^{2 x} x) \log ^3(x)+(x+2 e^x x+e^{2 x} x) \log ^4(x)} \, dx\)

Optimal. Leaf size=30 \[ -3 x+\frac {e^x \left (e^5+x\right )}{\left (1+e^x\right ) \left (-3+(-2+\log (x))^2\right )} \]

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Rubi [F]  time = 6.87, 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 {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x + E^(2*x)*(4*E^5 + 2*x) + E^x*(-x + x^2 + E^5*(4 + x)) + (24*x + E^(2*x)*(-2*E^5 + 18*x) + E^x*(E^5*
(-2 - 4*x) + 42*x - 4*x^2))*Log[x] + (-54*x - 53*E^(2*x)*x + E^x*(-107*x + E^5*x + x^2))*Log[x]^2 + (24*x + 48
*E^x*x + 24*E^(2*x)*x)*Log[x]^3 + (-3*x - 6*E^x*x - 3*E^(2*x)*x)*Log[x]^4)/(x + 2*E^x*x + E^(2*x)*x + (-8*x -
16*E^x*x - 8*E^(2*x)*x)*Log[x] + (18*x + 36*E^x*x + 18*E^(2*x)*x)*Log[x]^2 + (-8*x - 16*E^x*x - 8*E^(2*x)*x)*L
og[x]^3 + (x + 2*E^x*x + E^(2*x)*x)*Log[x]^4),x]

[Out]

-3*x + (E^(2 + Sqrt[3])*ExpIntegralEi[-2 - Sqrt[3] + Log[x]])/3 + (E^(2 + Sqrt[3])*ExpIntegralEi[-2 - Sqrt[3]
+ Log[x]])/(2*Sqrt[3]) - ((2 + Sqrt[3])*E^(2 + Sqrt[3])*ExpIntegralEi[-2 - Sqrt[3] + Log[x]])/6 + (E^(2 - Sqrt
[3])*ExpIntegralEi[-2 + Sqrt[3] + Log[x]])/3 - (E^(2 - Sqrt[3])*ExpIntegralEi[-2 + Sqrt[3] + Log[x]])/(2*Sqrt[
3]) - ((2 - Sqrt[3])*E^(2 - Sqrt[3])*ExpIntegralEi[-2 + Sqrt[3] + Log[x]])/6 + x/(3*(2 - Sqrt[3] - Log[x])) -
((2 - Sqrt[3])*x)/(6*(2 - Sqrt[3] - Log[x])) + x/(3*(2 + Sqrt[3] - Log[x])) - ((2 + Sqrt[3])*x)/(6*(2 + Sqrt[3
] - Log[x])) + E^5/(1 - 4*Log[x] + Log[x]^2) - (5 - E^5)*Defer[Int][1/((1 + E^x)*(1 - 4*Log[x] + Log[x]^2)^2),
x] - 4*E^5*Defer[Int][1/((1 + E^x)*x*(1 - 4*Log[x] + Log[x]^2)^2), x] + Defer[Int][x/((1 + E^x)*(1 - 4*Log[x]
+ Log[x]^2)^2), x] + 2*(3 - 2*E^5)*Defer[Int][Log[x]/((1 + E^x)*(1 - 4*Log[x] + Log[x]^2)^2), x] + 2*E^5*Defe
r[Int][Log[x]/((1 + E^x)*x*(1 - 4*Log[x] + Log[x]^2)^2), x] - 4*Defer[Int][(x*Log[x])/((1 + E^x)*(1 - 4*Log[x]
+ Log[x]^2)^2), x] - (1 - E^5)*Defer[Int][Log[x]^2/((1 + E^x)*(1 - 4*Log[x] + Log[x]^2)^2), x] + Defer[Int][(
x*Log[x]^2)/((1 + E^x)*(1 - 4*Log[x] + Log[x]^2)^2), x] - E^5*Defer[Int][1/((1 + E^x)^2*(1 - 4*Log[x] + Log[x]
^2)), x] - Defer[Int][x/((1 + E^x)^2*(1 - 4*Log[x] + Log[x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{5+2 x}-3 x+2 e^{2 x} x+e^x (-1+x) x+e^{5+x} (4+x)-2 \left (e^{5+2 x}-12 x-9 e^{2 x} x+e^x x (-21+2 x)+e^{5+x} (1+2 x)\right ) \log (x)+\left (-54-53 e^{2 x}+e^{5+x}+e^x (-107+x)\right ) x \log ^2(x)+24 \left (1+e^x\right )^2 x \log ^3(x)-3 \left (1+e^x\right )^2 x \log ^4(x)}{\left (1+e^x\right )^2 x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\int \left (-\frac {e^5+x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )}+\frac {-4 e^5-5 \left (1-\frac {e^5}{5}\right ) x+x^2+2 e^5 \log (x)+6 \left (1-\frac {2 e^5}{3}\right ) x \log (x)-4 x^2 \log (x)-\left (1-e^5\right ) x \log ^2(x)+x^2 \log ^2(x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {4 e^5+2 x-2 e^5 \log (x)+18 x \log (x)-53 x \log ^2(x)+24 x \log ^3(x)-3 x \log ^4(x)}{x \left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=-\int \frac {e^5+x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \frac {-4 e^5-5 \left (1-\frac {e^5}{5}\right ) x+x^2+2 e^5 \log (x)+6 \left (1-\frac {2 e^5}{3}\right ) x \log (x)-4 x^2 \log (x)-\left (1-e^5\right ) x \log ^2(x)+x^2 \log ^2(x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {4 e^5+2 x-2 e^5 \log (x)+18 x \log (x)-53 x \log ^2(x)+24 x \log ^3(x)-3 x \log ^4(x)}{x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\int \frac {e^5 (-4+x)+(-5+x) x+\left (e^5 (2-4 x)+2 (3-2 x) x\right ) \log (x)+x \left (-1+e^5+x\right ) \log ^2(x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \left (-3-\frac {2 \left (e^5+x\right ) (-2+\log (x))}{x \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {1}{1-4 \log (x)+\log ^2(x)}\right ) \, dx-\int \left (\frac {e^5}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )}+\frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )}\right ) \, dx\\ &=-3 x-2 \int \frac {\left (e^5+x\right ) (-2+\log (x))}{x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \frac {1}{1-4 \log (x)+\log ^2(x)} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \left (-\frac {5 \left (1-\frac {e^5}{5}\right )}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}-\frac {4 e^5}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {6 \left (1-\frac {2 e^5}{3}\right ) \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {2 e^5 \log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2}-\frac {4 x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}-\frac {\left (1-e^5\right ) \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=-3 x-2 \int \left (\frac {-2+\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {e^5 (-2+\log (x))}{x \left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \left (-\frac {1}{\sqrt {3} \left (4+2 \sqrt {3}-2 \log (x)\right )}-\frac {1}{\sqrt {3} \left (-4+2 \sqrt {3}+2 \log (x)\right )}\right ) \, dx\\ &=-3 x-2 \int \frac {-2+\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {\int \frac {1}{4+2 \sqrt {3}-2 \log (x)} \, dx}{\sqrt {3}}-\frac {\int \frac {1}{-4+2 \sqrt {3}+2 \log (x)} \, dx}{\sqrt {3}}-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx-\left (2 e^5\right ) \int \frac {-2+\log (x)}{x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x-2 \int \left (-\frac {2}{\left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {\operatorname {Subst}\left (\int \frac {e^x}{4+2 \sqrt {3}-2 x} \, dx,x,\log (x)\right )}{\sqrt {3}}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{-4+2 \sqrt {3}+2 x} \, dx,x,\log (x)\right )}{\sqrt {3}}-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (2 e^5\right ) \operatorname {Subst}\left (\int \frac {-2+x}{\left (1-4 x+x^2\right )^2} \, dx,x,\log (x)\right )-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-2 \int \frac {\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+4 \int \frac {1}{\left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-2 \int \left (\frac {4+2 \sqrt {3}}{6 \left (4+2 \sqrt {3}-2 \log (x)\right )^2}+\frac {1}{3 \sqrt {3} \left (4+2 \sqrt {3}-2 \log (x)\right )}+\frac {4-2 \sqrt {3}}{6 \left (-4+2 \sqrt {3}+2 \log (x)\right )^2}+\frac {1}{3 \sqrt {3} \left (-4+2 \sqrt {3}+2 \log (x)\right )}\right ) \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+4 \int \left (\frac {1}{3 \left (4+2 \sqrt {3}-2 \log (x)\right )^2}+\frac {1}{6 \sqrt {3} \left (4+2 \sqrt {3}-2 \log (x)\right )}+\frac {1}{3 \left (-4+2 \sqrt {3}+2 \log (x)\right )^2}+\frac {1}{6 \sqrt {3} \left (-4+2 \sqrt {3}+2 \log (x)\right )}\right ) \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}+\frac {4}{3} \int \frac {1}{\left (4+2 \sqrt {3}-2 \log (x)\right )^2} \, dx+\frac {4}{3} \int \frac {1}{\left (-4+2 \sqrt {3}+2 \log (x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {1}{3} \left (2 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{\left (-4+2 \sqrt {3}+2 \log (x)\right )^2} \, dx-\frac {1}{3} \left (2 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {3}-2 \log (x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {x}{3 \left (2-\sqrt {3}-\log (x)\right )}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (2-\sqrt {3}-\log (x)\right )}+\frac {x}{3 \left (2+\sqrt {3}-\log (x)\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (2+\sqrt {3}-\log (x)\right )}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-\frac {2}{3} \int \frac {1}{4+2 \sqrt {3}-2 \log (x)} \, dx+\frac {2}{3} \int \frac {1}{-4+2 \sqrt {3}+2 \log (x)} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {1}{3} \left (-2-\sqrt {3}\right ) \int \frac {1}{4+2 \sqrt {3}-2 \log (x)} \, dx-\frac {1}{3} \left (2-\sqrt {3}\right ) \int \frac {1}{-4+2 \sqrt {3}+2 \log (x)} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {x}{3 \left (2-\sqrt {3}-\log (x)\right )}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (2-\sqrt {3}-\log (x)\right )}+\frac {x}{3 \left (2+\sqrt {3}-\log (x)\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (2+\sqrt {3}-\log (x)\right )}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {e^x}{4+2 \sqrt {3}-2 x} \, dx,x,\log (x)\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {e^x}{-4+2 \sqrt {3}+2 x} \, dx,x,\log (x)\right )-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {1}{3} \left (-2-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {e^x}{4+2 \sqrt {3}-2 x} \, dx,x,\log (x)\right )-\frac {1}{3} \left (2-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {e^x}{-4+2 \sqrt {3}+2 x} \, dx,x,\log (x)\right )-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {1}{3} e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {1}{6} \left (2+\sqrt {3}\right ) e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )+\frac {1}{3} e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {1}{6} \left (2-\sqrt {3}\right ) e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )+\frac {x}{3 \left (2-\sqrt {3}-\log (x)\right )}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (2-\sqrt {3}-\log (x)\right )}+\frac {x}{3 \left (2+\sqrt {3}-\log (x)\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (2+\sqrt {3}-\log (x)\right )}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 32, normalized size = 1.07 \begin {gather*} -3 x+\frac {e^x \left (e^5+x\right )}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x + E^(2*x)*(4*E^5 + 2*x) + E^x*(-x + x^2 + E^5*(4 + x)) + (24*x + E^(2*x)*(-2*E^5 + 18*x) + E^x
*(E^5*(-2 - 4*x) + 42*x - 4*x^2))*Log[x] + (-54*x - 53*E^(2*x)*x + E^x*(-107*x + E^5*x + x^2))*Log[x]^2 + (24*
x + 48*E^x*x + 24*E^(2*x)*x)*Log[x]^3 + (-3*x - 6*E^x*x - 3*E^(2*x)*x)*Log[x]^4)/(x + 2*E^x*x + E^(2*x)*x + (-
8*x - 16*E^x*x - 8*E^(2*x)*x)*Log[x] + (18*x + 36*E^x*x + 18*E^(2*x)*x)*Log[x]^2 + (-8*x - 16*E^x*x - 8*E^(2*x
)*x)*Log[x]^3 + (x + 2*E^x*x + E^(2*x)*x)*Log[x]^4),x]

[Out]

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

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fricas [B]  time = 0.65, size = 62, normalized size = 2.07 \begin {gather*} -\frac {3 \, {\left (x e^{x} + x\right )} \log \relax (x)^{2} + {\left (2 \, x - e^{5}\right )} e^{x} - 12 \, {\left (x e^{x} + x\right )} \log \relax (x) + 3 \, x}{{\left (e^{x} + 1\right )} \log \relax (x)^{2} - 4 \, {\left (e^{x} + 1\right )} \log \relax (x) + e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x)*x+24*x)*log(x)^3+(-53*x*exp(x)^2+(
x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)
*log(x)+(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^2+2*exp(x)*x+x)*log(x)^4+(-8*x*exp
(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*exp(x)^2+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(
x)+x*exp(x)^2+2*exp(x)*x+x),x, algorithm="fricas")

[Out]

-(3*(x*e^x + x)*log(x)^2 + (2*x - e^5)*e^x - 12*(x*e^x + x)*log(x) + 3*x)/((e^x + 1)*log(x)^2 - 4*(e^x + 1)*lo
g(x) + e^x + 1)

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giac [B]  time = 0.22, size = 71, normalized size = 2.37 \begin {gather*} -\frac {3 \, x e^{x} \log \relax (x)^{2} - 12 \, x e^{x} \log \relax (x) + 3 \, x \log \relax (x)^{2} + x e^{x} - 12 \, x \log \relax (x) + 3 \, x - 2 \, e^{\left (x + 5\right )}}{e^{x} \log \relax (x)^{2} - 4 \, e^{x} \log \relax (x) + \log \relax (x)^{2} + e^{x} - 4 \, \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x)*x+24*x)*log(x)^3+(-53*x*exp(x)^2+(
x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)
*log(x)+(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^2+2*exp(x)*x+x)*log(x)^4+(-8*x*exp
(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*exp(x)^2+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(
x)+x*exp(x)^2+2*exp(x)*x+x),x, algorithm="giac")

[Out]

-(3*x*e^x*log(x)^2 - 12*x*e^x*log(x) + 3*x*log(x)^2 + x*e^x - 12*x*log(x) + 3*x - 2*e^(x + 5))/(e^x*log(x)^2 -
4*e^x*log(x) + log(x)^2 + e^x - 4*log(x) + 1)

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maple [A]  time = 0.07, size = 30, normalized size = 1.00

method	result	size
risch	\(-3 x +\frac {\left ({\mathrm e}^{5}+x \right ) {\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right ) \left (\ln \relax (x )^{2}-4 \ln \relax (x )+1\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*ln(x)^4+(24*x*exp(x)^2+48*exp(x)*x+24*x)*ln(x)^3+(-53*x*exp(x)^2+(x*exp(5)
+x^2-107*x)*exp(x)-54*x)*ln(x)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)*ln(x)+(4
*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^2+2*exp(x)*x+x)*ln(x)^4+(-8*x*exp(x)^2-16*ex
p(x)*x-8*x)*ln(x)^3+(18*x*exp(x)^2+36*exp(x)*x+18*x)*ln(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*ln(x)+x*exp(x)^2+
2*exp(x)*x+x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.51, size = 66, normalized size = 2.20 \begin {gather*} -\frac {3 \, x \log \relax (x)^{2} + {\left (3 \, x \log \relax (x)^{2} - 12 \, x \log \relax (x) + 2 \, x - e^{5}\right )} e^{x} - 12 \, x \log \relax (x) + 3 \, x}{{\left (\log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} e^{x} + \log \relax (x)^{2} - 4 \, \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x)*x+24*x)*log(x)^3+(-53*x*exp(x)^2+(
x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)
*log(x)+(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^2+2*exp(x)*x+x)*log(x)^4+(-8*x*exp
(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*exp(x)^2+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(
x)+x*exp(x)^2+2*exp(x)*x+x),x, algorithm="maxima")

[Out]

-(3*x*log(x)^2 + (3*x*log(x)^2 - 12*x*log(x) + 2*x - e^5)*e^x - 12*x*log(x) + 3*x)/((log(x)^2 - 4*log(x) + 1)*
e^x + log(x)^2 - 4*log(x) + 1)

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mupad [B]  time = 8.92, size = 44, normalized size = 1.47 \begin {gather*} \frac {{\mathrm {e}}^{x+5}+{\mathrm {e}}^{2\,x+5}+x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^x+1\right )}^2\,\left ({\ln \relax (x)}^2-4\,\ln \relax (x)+1\right )}-3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(2*x + 4*exp(5)) - log(x)^2*(54*x + 53*x*exp(2*x) - exp(x)*(x*exp(5) - 107*x + x^2)) - 3*x - log
(x)^4*(3*x + 3*x*exp(2*x) + 6*x*exp(x)) + log(x)^3*(24*x + 24*x*exp(2*x) + 48*x*exp(x)) + log(x)*(24*x + exp(2
*x)*(18*x - 2*exp(5)) - exp(x)*(4*x^2 - 42*x + exp(5)*(4*x + 2))) + exp(x)*(exp(5)*(x + 4) - x + x^2))/(x + x*
exp(2*x) - log(x)*(8*x + 8*x*exp(2*x) + 16*x*exp(x)) + log(x)^4*(x + x*exp(2*x) + 2*x*exp(x)) - log(x)^3*(8*x
+ 8*x*exp(2*x) + 16*x*exp(x)) + log(x)^2*(18*x + 18*x*exp(2*x) + 36*x*exp(x)) + 2*x*exp(x)),x)

[Out]

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

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sympy [B]  time = 0.45, size = 53, normalized size = 1.77 \begin {gather*} - 3 x + \frac {- x - e^{5}}{\left (\log {\relax (x )}^{2} - 4 \log {\relax (x )} + 1\right ) e^{x} + \log {\relax (x )}^{2} - 4 \log {\relax (x )} + 1} + \frac {x + e^{5}}{\log {\relax (x )}^{2} - 4 \log {\relax (x )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x*exp(x)**2-6*exp(x)*x-3*x)*ln(x)**4+(24*x*exp(x)**2+48*exp(x)*x+24*x)*ln(x)**3+(-53*x*exp(x)**
2+(x*exp(5)+x**2-107*x)*exp(x)-54*x)*ln(x)**2+((-2*exp(5)+18*x)*exp(x)**2+((-4*x-2)*exp(5)-4*x**2+42*x)*exp(x)
+24*x)*ln(x)+(4*exp(5)+2*x)*exp(x)**2+((4+x)*exp(5)+x**2-x)*exp(x)-3*x)/((x*exp(x)**2+2*exp(x)*x+x)*ln(x)**4+(
-8*x*exp(x)**2-16*exp(x)*x-8*x)*ln(x)**3+(18*x*exp(x)**2+36*exp(x)*x+18*x)*ln(x)**2+(-8*x*exp(x)**2-16*exp(x)*
x-8*x)*ln(x)+x*exp(x)**2+2*exp(x)*x+x),x)

[Out]

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

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