3.64.14 \(\int \frac {e^{4 x} (6+8 x-2 x^2)+e^{2 x} (6 x+8 x^2-2 x^3)+(e^{4 x} (-8 x+4 x^2)+e^{2 x} (6 x-12 x^2-14 x^3+4 x^4)) \log (x)+(-12 x-4 x^3+e^{2 x} (-10 x-36 x^2+8 x^3)) \log ^2(x)}{(e^{4 x} x+2 e^{2 x} x^2+x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=35 \[ \frac {2 \left (-x+(-3+(-4+x) x) \left (-2+\frac {e^{2 x}}{\log (x)}\right )\right )}{e^{2 x}+x} \]

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Rubi [F]  time = 5.42, 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 {e^{4 x} \left (6+8 x-2 x^2\right )+e^{2 x} \left (6 x+8 x^2-2 x^3\right )+\left (e^{4 x} \left (-8 x+4 x^2\right )+e^{2 x} \left (6 x-12 x^2-14 x^3+4 x^4\right )\right ) \log (x)+\left (-12 x-4 x^3+e^{2 x} \left (-10 x-36 x^2+8 x^3\right )\right ) \log ^2(x)}{\left (e^{4 x} x+2 e^{2 x} x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4*x)*(6 + 8*x - 2*x^2) + E^(2*x)*(6*x + 8*x^2 - 2*x^3) + (E^(4*x)*(-8*x + 4*x^2) + E^(2*x)*(6*x - 12*x
^2 - 14*x^3 + 4*x^4))*Log[x] + (-12*x - 4*x^3 + E^(2*x)*(-10*x - 36*x^2 + 8*x^3))*Log[x]^2)/((E^(4*x)*x + 2*E^
(2*x)*x^2 + x^3)*Log[x]^2),x]

[Out]

4*ExpIntegralEi[2*Log[x]] - 8*LogIntegral[x] - 12*Defer[Int][(E^(2*x) + x)^(-2), x] + 10*Defer[Int][x/(E^(2*x)
+ x)^2, x] + 32*Defer[Int][x^2/(E^(2*x) + x)^2, x] - 8*Defer[Int][x^3/(E^(2*x) + x)^2, x] - 10*Defer[Int][(E^
(2*x) + x)^(-1), x] - 36*Defer[Int][x/(E^(2*x) + x), x] + 8*Defer[Int][x^2/(E^(2*x) + x), x] - 6*Defer[Int][1/
((E^(2*x) + x)*Log[x]^2), x] - 8*Defer[Int][x/((E^(2*x) + x)*Log[x]^2), x] + 2*Defer[Int][x^2/((E^(2*x) + x)*L
og[x]^2), x] + 2*Defer[Int][(3 + 4*x - x^2)/(x*Log[x]^2), x] - 6*Defer[Int][x/((E^(2*x) + x)^2*Log[x]), x] + 4
*Defer[Int][x^2/((E^(2*x) + x)^2*Log[x]), x] + 18*Defer[Int][x^3/((E^(2*x) + x)^2*Log[x]), x] - 4*Defer[Int][x
^4/((E^(2*x) + x)^2*Log[x]), x] + 6*Defer[Int][1/((E^(2*x) + x)*Log[x]), x] + 4*Defer[Int][x/((E^(2*x) + x)*Lo
g[x]), x] - 22*Defer[Int][x^2/((E^(2*x) + x)*Log[x]), x] + 4*Defer[Int][x^3/((E^(2*x) + x)*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-e^{2 x} \left (e^{2 x}+x\right ) \left (-3-4 x+x^2\right )+e^{2 x} x \left (3+2 e^{2 x} (-2+x)-6 x-7 x^2+2 x^3\right ) \log (x)+x \left (-2 \left (3+x^2\right )+e^{2 x} \left (-5-18 x+4 x^2\right )\right ) \log ^2(x)\right )}{x \left (e^{2 x}+x\right )^2 \log ^2(x)} \, dx\\ &=2 \int \frac {-e^{2 x} \left (e^{2 x}+x\right ) \left (-3-4 x+x^2\right )+e^{2 x} x \left (3+2 e^{2 x} (-2+x)-6 x-7 x^2+2 x^3\right ) \log (x)+x \left (-2 \left (3+x^2\right )+e^{2 x} \left (-5-18 x+4 x^2\right )\right ) \log ^2(x)}{x \left (e^{2 x}+x\right )^2 \log ^2(x)} \, dx\\ &=2 \int \left (-\frac {(-1+2 x) \left (-3 x-4 x^2+x^3-6 \log (x)-7 x \log (x)+2 x^2 \log (x)\right )}{\left (e^{2 x}+x\right )^2 \log (x)}+\frac {3+4 x-x^2-4 x \log (x)+2 x^2 \log (x)}{x \log ^2(x)}+\frac {-3-4 x+x^2+3 \log (x)+2 x \log (x)-11 x^2 \log (x)+2 x^3 \log (x)-5 \log ^2(x)-18 x \log ^2(x)+4 x^2 \log ^2(x)}{\left (e^{2 x}+x\right ) \log ^2(x)}\right ) \, dx\\ &=-\left (2 \int \frac {(-1+2 x) \left (-3 x-4 x^2+x^3-6 \log (x)-7 x \log (x)+2 x^2 \log (x)\right )}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx\right )+2 \int \frac {3+4 x-x^2-4 x \log (x)+2 x^2 \log (x)}{x \log ^2(x)} \, dx+2 \int \frac {-3-4 x+x^2+3 \log (x)+2 x \log (x)-11 x^2 \log (x)+2 x^3 \log (x)-5 \log ^2(x)-18 x \log ^2(x)+4 x^2 \log ^2(x)}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx\\ &=2 \int \left (\frac {3+4 x-x^2}{x \log ^2(x)}+\frac {2 (-2+x)}{\log (x)}\right ) \, dx+2 \int \left (-\frac {5}{e^{2 x}+x}-\frac {18 x}{e^{2 x}+x}+\frac {4 x^2}{e^{2 x}+x}-\frac {3}{\left (e^{2 x}+x\right ) \log ^2(x)}-\frac {4 x}{\left (e^{2 x}+x\right ) \log ^2(x)}+\frac {x^2}{\left (e^{2 x}+x\right ) \log ^2(x)}+\frac {3}{\left (e^{2 x}+x\right ) \log (x)}+\frac {2 x}{\left (e^{2 x}+x\right ) \log (x)}-\frac {11 x^2}{\left (e^{2 x}+x\right ) \log (x)}+\frac {2 x^3}{\left (e^{2 x}+x\right ) \log (x)}\right ) \, dx-2 \int \left (-\frac {-3 x-4 x^2+x^3-6 \log (x)-7 x \log (x)+2 x^2 \log (x)}{\left (e^{2 x}+x\right )^2 \log (x)}+\frac {2 x \left (-3 x-4 x^2+x^3-6 \log (x)-7 x \log (x)+2 x^2 \log (x)\right )}{\left (e^{2 x}+x\right )^2 \log (x)}\right ) \, dx\\ &=2 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+2 \int \frac {3+4 x-x^2}{x \log ^2(x)} \, dx+2 \int \frac {-3 x-4 x^2+x^3-6 \log (x)-7 x \log (x)+2 x^2 \log (x)}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {-2+x}{\log (x)} \, dx+4 \int \frac {x}{\left (e^{2 x}+x\right ) \log (x)} \, dx+4 \int \frac {x^3}{\left (e^{2 x}+x\right ) \log (x)} \, dx-4 \int \frac {x \left (-3 x-4 x^2+x^3-6 \log (x)-7 x \log (x)+2 x^2 \log (x)\right )}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-6 \int \frac {1}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+6 \int \frac {1}{\left (e^{2 x}+x\right ) \log (x)} \, dx+8 \int \frac {x^2}{e^{2 x}+x} \, dx-8 \int \frac {x}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-10 \int \frac {1}{e^{2 x}+x} \, dx-22 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log (x)} \, dx-36 \int \frac {x}{e^{2 x}+x} \, dx\\ &=2 \int \left (-\frac {6}{\left (e^{2 x}+x\right )^2}-\frac {7 x}{\left (e^{2 x}+x\right )^2}+\frac {2 x^2}{\left (e^{2 x}+x\right )^2}-\frac {3 x}{\left (e^{2 x}+x\right )^2 \log (x)}-\frac {4 x^2}{\left (e^{2 x}+x\right )^2 \log (x)}+\frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)}\right ) \, dx+2 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+2 \int \frac {3+4 x-x^2}{x \log ^2(x)} \, dx+4 \int \left (-\frac {2}{\log (x)}+\frac {x}{\log (x)}\right ) \, dx-4 \int \left (-\frac {6 x}{\left (e^{2 x}+x\right )^2}-\frac {7 x^2}{\left (e^{2 x}+x\right )^2}+\frac {2 x^3}{\left (e^{2 x}+x\right )^2}-\frac {3 x^2}{\left (e^{2 x}+x\right )^2 \log (x)}-\frac {4 x^3}{\left (e^{2 x}+x\right )^2 \log (x)}+\frac {x^4}{\left (e^{2 x}+x\right )^2 \log (x)}\right ) \, dx+4 \int \frac {x}{\left (e^{2 x}+x\right ) \log (x)} \, dx+4 \int \frac {x^3}{\left (e^{2 x}+x\right ) \log (x)} \, dx-6 \int \frac {1}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+6 \int \frac {1}{\left (e^{2 x}+x\right ) \log (x)} \, dx+8 \int \frac {x^2}{e^{2 x}+x} \, dx-8 \int \frac {x}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-10 \int \frac {1}{e^{2 x}+x} \, dx-22 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log (x)} \, dx-36 \int \frac {x}{e^{2 x}+x} \, dx\\ &=2 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+2 \int \frac {3+4 x-x^2}{x \log ^2(x)} \, dx+2 \int \frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {x^2}{\left (e^{2 x}+x\right )^2} \, dx+4 \int \frac {x}{\log (x)} \, dx-4 \int \frac {x^4}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {x}{\left (e^{2 x}+x\right ) \log (x)} \, dx+4 \int \frac {x^3}{\left (e^{2 x}+x\right ) \log (x)} \, dx-6 \int \frac {1}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-6 \int \frac {x}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+6 \int \frac {1}{\left (e^{2 x}+x\right ) \log (x)} \, dx-8 \int \frac {x^3}{\left (e^{2 x}+x\right )^2} \, dx+8 \int \frac {x^2}{e^{2 x}+x} \, dx-8 \int \frac {x}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\log (x)} \, dx-8 \int \frac {x^2}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-10 \int \frac {1}{e^{2 x}+x} \, dx-12 \int \frac {1}{\left (e^{2 x}+x\right )^2} \, dx+12 \int \frac {x^2}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-14 \int \frac {x}{\left (e^{2 x}+x\right )^2} \, dx+16 \int \frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-22 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log (x)} \, dx+24 \int \frac {x}{\left (e^{2 x}+x\right )^2} \, dx+28 \int \frac {x^2}{\left (e^{2 x}+x\right )^2} \, dx-36 \int \frac {x}{e^{2 x}+x} \, dx\\ &=-8 \text {li}(x)+2 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+2 \int \frac {3+4 x-x^2}{x \log ^2(x)} \, dx+2 \int \frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {x^2}{\left (e^{2 x}+x\right )^2} \, dx-4 \int \frac {x^4}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {x}{\left (e^{2 x}+x\right ) \log (x)} \, dx+4 \int \frac {x^3}{\left (e^{2 x}+x\right ) \log (x)} \, dx+4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-6 \int \frac {1}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-6 \int \frac {x}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+6 \int \frac {1}{\left (e^{2 x}+x\right ) \log (x)} \, dx-8 \int \frac {x^3}{\left (e^{2 x}+x\right )^2} \, dx+8 \int \frac {x^2}{e^{2 x}+x} \, dx-8 \int \frac {x}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-8 \int \frac {x^2}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-10 \int \frac {1}{e^{2 x}+x} \, dx-12 \int \frac {1}{\left (e^{2 x}+x\right )^2} \, dx+12 \int \frac {x^2}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-14 \int \frac {x}{\left (e^{2 x}+x\right )^2} \, dx+16 \int \frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-22 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log (x)} \, dx+24 \int \frac {x}{\left (e^{2 x}+x\right )^2} \, dx+28 \int \frac {x^2}{\left (e^{2 x}+x\right )^2} \, dx-36 \int \frac {x}{e^{2 x}+x} \, dx\\ &=4 \text {Ei}(2 \log (x))-8 \text {li}(x)+2 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx+2 \int \frac {3+4 x-x^2}{x \log ^2(x)} \, dx+2 \int \frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {x^2}{\left (e^{2 x}+x\right )^2} \, dx-4 \int \frac {x^4}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+4 \int \frac {x}{\left (e^{2 x}+x\right ) \log (x)} \, dx+4 \int \frac {x^3}{\left (e^{2 x}+x\right ) \log (x)} \, dx-6 \int \frac {1}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-6 \int \frac {x}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx+6 \int \frac {1}{\left (e^{2 x}+x\right ) \log (x)} \, dx-8 \int \frac {x^3}{\left (e^{2 x}+x\right )^2} \, dx+8 \int \frac {x^2}{e^{2 x}+x} \, dx-8 \int \frac {x}{\left (e^{2 x}+x\right ) \log ^2(x)} \, dx-8 \int \frac {x^2}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-10 \int \frac {1}{e^{2 x}+x} \, dx-12 \int \frac {1}{\left (e^{2 x}+x\right )^2} \, dx+12 \int \frac {x^2}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-14 \int \frac {x}{\left (e^{2 x}+x\right )^2} \, dx+16 \int \frac {x^3}{\left (e^{2 x}+x\right )^2 \log (x)} \, dx-22 \int \frac {x^2}{\left (e^{2 x}+x\right ) \log (x)} \, dx+24 \int \frac {x}{\left (e^{2 x}+x\right )^2} \, dx+28 \int \frac {x^2}{\left (e^{2 x}+x\right )^2} \, dx-36 \int \frac {x}{e^{2 x}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 43, normalized size = 1.23 \begin {gather*} \frac {2 \left (e^{2 x} \left (-3-4 x+x^2\right )+\left (6+7 x-2 x^2\right ) \log (x)\right )}{\left (e^{2 x}+x\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x)*(6 + 8*x - 2*x^2) + E^(2*x)*(6*x + 8*x^2 - 2*x^3) + (E^(4*x)*(-8*x + 4*x^2) + E^(2*x)*(6*x
- 12*x^2 - 14*x^3 + 4*x^4))*Log[x] + (-12*x - 4*x^3 + E^(2*x)*(-10*x - 36*x^2 + 8*x^3))*Log[x]^2)/((E^(4*x)*x
+ 2*E^(2*x)*x^2 + x^3)*Log[x]^2),x]

[Out]

(2*(E^(2*x)*(-3 - 4*x + x^2) + (6 + 7*x - 2*x^2)*Log[x]))/((E^(2*x) + x)*Log[x])

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fricas [A]  time = 0.83, size = 42, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left ({\left (x^{2} - 4 \, x - 3\right )} e^{\left (2 \, x\right )} - {\left (2 \, x^{2} - 7 \, x - 6\right )} \log \relax (x)\right )}}{{\left (x + e^{\left (2 \, x\right )}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3-36*x^2-10*x)*exp(x)^2-4*x^3-12*x)*log(x)^2+((4*x^2-8*x)*exp(x)^4+(4*x^4-14*x^3-12*x^2+6*x)*
exp(x)^2)*log(x)+(-2*x^2+8*x+6)*exp(x)^4+(-2*x^3+8*x^2+6*x)*exp(x)^2)/(x*exp(x)^4+2*exp(x)^2*x^2+x^3)/log(x)^2
,x, algorithm="fricas")

[Out]

2*((x^2 - 4*x - 3)*e^(2*x) - (2*x^2 - 7*x - 6)*log(x))/((x + e^(2*x))*log(x))

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giac [A]  time = 0.18, size = 54, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} \log \relax (x) - 4 \, x e^{\left (2 \, x\right )} + 7 \, x \log \relax (x) - 3 \, e^{\left (2 \, x\right )} + 6 \, \log \relax (x)\right )}}{x \log \relax (x) + e^{\left (2 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3-36*x^2-10*x)*exp(x)^2-4*x^3-12*x)*log(x)^2+((4*x^2-8*x)*exp(x)^4+(4*x^4-14*x^3-12*x^2+6*x)*
exp(x)^2)*log(x)+(-2*x^2+8*x+6)*exp(x)^4+(-2*x^3+8*x^2+6*x)*exp(x)^2)/(x*exp(x)^4+2*exp(x)^2*x^2+x^3)/log(x)^2
,x, algorithm="giac")

[Out]

2*(x^2*e^(2*x) - 2*x^2*log(x) - 4*x*e^(2*x) + 7*x*log(x) - 3*e^(2*x) + 6*log(x))/(x*log(x) + e^(2*x)*log(x))

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maple [A]  time = 0.06, size = 48, normalized size = 1.37

method	result	size
risch	\(-\frac {2 \left (2 x^{2}-7 x -6\right )}{{\mathrm e}^{2 x}+x}+\frac {2 \left (x^{2}-4 x -3\right ) {\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}+x \right ) \ln \relax (x )}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^3-36*x^2-10*x)*exp(x)^2-4*x^3-12*x)*ln(x)^2+((4*x^2-8*x)*exp(x)^4+(4*x^4-14*x^3-12*x^2+6*x)*exp(x)^
2)*ln(x)+(-2*x^2+8*x+6)*exp(x)^4+(-2*x^3+8*x^2+6*x)*exp(x)^2)/(x*exp(x)^4+2*exp(x)^2*x^2+x^3)/ln(x)^2,x,method
=_RETURNVERBOSE)

[Out]

-2*(2*x^2-7*x-6)/(exp(2*x)+x)+2*(x^2-4*x-3)*exp(2*x)/(exp(2*x)+x)/ln(x)

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maxima [A]  time = 0.44, size = 44, normalized size = 1.26 \begin {gather*} \frac {2 \, {\left ({\left (x^{2} - 4 \, x - 3\right )} e^{\left (2 \, x\right )} - {\left (2 \, x^{2} - 7 \, x - 6\right )} \log \relax (x)\right )}}{x \log \relax (x) + e^{\left (2 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3-36*x^2-10*x)*exp(x)^2-4*x^3-12*x)*log(x)^2+((4*x^2-8*x)*exp(x)^4+(4*x^4-14*x^3-12*x^2+6*x)*
exp(x)^2)*log(x)+(-2*x^2+8*x+6)*exp(x)^4+(-2*x^3+8*x^2+6*x)*exp(x)^2)/(x*exp(x)^4+2*exp(x)^2*x^2+x^3)/log(x)^2
,x, algorithm="maxima")

[Out]

2*((x^2 - 4*x - 3)*e^(2*x) - (2*x^2 - 7*x - 6)*log(x))/(x*log(x) + e^(2*x)*log(x))

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mupad [B]  time = 4.45, size = 179, normalized size = 5.11 \begin {gather*} 4\,x^2-\frac {\frac {2\,{\mathrm {e}}^{2\,x}\,\left (-x^2+4\,x+3\right )}{x+{\mathrm {e}}^{2\,x}}-\frac {2\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (x)\,\left (6\,x+4\,{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^{2\,x}+7\,x^2-2\,x^3-3\right )}{{\left (x+{\mathrm {e}}^{2\,x}\right )}^2}}{\ln \relax (x)}-8\,x-\frac {2\,\left (4\,x^6-20\,x^5+5\,x^4+8\,x^3-3\,x^2\right )}{\left (2\,x-1\right )\,\left ({\mathrm {e}}^{4\,x}+2\,x\,{\mathrm {e}}^{2\,x}+x^2\right )}+\frac {2\,\left (4\,x^5-24\,x^4+11\,x^3+20\,x^2+2\,x-6\right )}{\left (x+{\mathrm {e}}^{2\,x}\right )\,\left (2\,x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*(8*x - 2*x^2 + 6) - log(x)*(exp(4*x)*(8*x - 4*x^2) - exp(2*x)*(6*x - 12*x^2 - 14*x^3 + 4*x^4)) +
exp(2*x)*(6*x + 8*x^2 - 2*x^3) - log(x)^2*(12*x + exp(2*x)*(10*x + 36*x^2 - 8*x^3) + 4*x^3))/(log(x)^2*(x*exp
(4*x) + 2*x^2*exp(2*x) + x^3)),x)

[Out]

4*x^2 - ((2*exp(2*x)*(4*x - x^2 + 3))/(x + exp(2*x)) - (2*x*exp(2*x)*log(x)*(6*x + 4*exp(2*x) - 2*x*exp(2*x) +
7*x^2 - 2*x^3 - 3))/(x + exp(2*x))^2)/log(x) - 8*x - (2*(8*x^3 - 3*x^2 + 5*x^4 - 20*x^5 + 4*x^6))/((2*x - 1)*
(exp(4*x) + 2*x*exp(2*x) + x^2)) + (2*(2*x + 20*x^2 + 11*x^3 - 24*x^4 + 4*x^5 - 6))/((x + exp(2*x))*(2*x - 1))

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sympy [B]  time = 0.34, size = 60, normalized size = 1.71 \begin {gather*} \frac {2 x^{2} - 8 x - 6}{\log {\relax (x )}} + \frac {- 2 x^{3} - 4 x^{2} \log {\relax (x )} + 8 x^{2} + 14 x \log {\relax (x )} + 6 x + 12 \log {\relax (x )}}{x \log {\relax (x )} + e^{2 x} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**3-36*x**2-10*x)*exp(x)**2-4*x**3-12*x)*ln(x)**2+((4*x**2-8*x)*exp(x)**4+(4*x**4-14*x**3-12*x
**2+6*x)*exp(x)**2)*ln(x)+(-2*x**2+8*x+6)*exp(x)**4+(-2*x**3+8*x**2+6*x)*exp(x)**2)/(x*exp(x)**4+2*exp(x)**2*x
**2+x**3)/ln(x)**2,x)

[Out]

(2*x**2 - 8*x - 6)/log(x) + (-2*x**3 - 4*x**2*log(x) + 8*x**2 + 14*x*log(x) + 6*x + 12*log(x))/(x*log(x) + exp
(2*x)*log(x))

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