∫ e3x(2−6x) log(5)+ex(4−4x) log(5)−8e2xx2 log(5)____ e6x+16exx+16e3xx+4e5xx+16x2+e4x(4+4x2)+e2x(4+16x2) dx

Optimal. Leaf size=26 \[ \frac {\log (5)}{\left (2+e^{2 x}\right ) \left (1+\frac {e^x}{2 x}\right )} \]

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Rubi [F] time = 7.16, 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^{3 x} (2-6 x) \log (5)+e^x (4-4 x) \log (5)-8 e^{2 x} x^2 \log (5)}{e^{6 x}+16 e^x x+16 e^{3 x} x+4 e^{5 x} x+16 x^2+e^{4 x} \left (4+4 x^2\right )+e^{2 x} \left (4+16 x^2\right )} \, dx \end {gather*}

Int[(E^(3*x)*(2 - 6*x)*Log[5] + E^x*(4 - 4*x)*Log[5] - 8*E^(2*x)*x^2*Log[5])/(E^(6*x) + 16*E^x*x + 16*E^(3*x)*

x + 4*E^(5*x)*x + 16*x^2 + E^(4*x)*(4 + 4*x^2) + E^(2*x)*(4 + 16*x^2)),x]

Log[5]/(2 + E^(2*x)) - ((I/2)*Log[5])/((2 + E^(2*x))*(I - Sqrt[2]*x)) - ((I/2)*Log[5])/((2 + E^(2*x))*(I + Sqr

t[2]*x)) + (I*Log[5]*Defer[Int][1/((2 + E^(2*x))*(I - Sqrt[2]*x)^2), x])/Sqrt[2] - (I/4)*Log[5]*Defer[Int][1/(

(I*Sqrt[2] - E^x)*(I - Sqrt[2]*x)), x] + (Log[5]*Defer[Int][1/((I*Sqrt[2] - E^x)*(I - Sqrt[2]*x)), x])/(4*Sqrt

[2]) + (I/4)*Log[5]*Defer[Int][1/((I*Sqrt[2] + E^x)*(I - Sqrt[2]*x)), x] - (Log[5]*Defer[Int][1/((I*Sqrt[2] +

E^x)*(I - Sqrt[2]*x)), x])/(4*Sqrt[2]) + Sqrt[2]*Log[5]*Defer[Int][E^x/((2 + E^(2*x))^2*(I - Sqrt[2]*x)), x] +

(I/2)*Log[5]*Defer[Int][E^x/((E^x + 2*x)^2*(I - Sqrt[2]*x)), x] + (Log[5]*Defer[Int][E^x/((E^x + 2*x)^2*(I -

Sqrt[2]*x)), x])/(2*Sqrt[2]) - (I*Log[5]*Defer[Int][1/((2 + E^(2*x))*(I + Sqrt[2]*x)^2), x])/Sqrt[2] - (I/4)*L

og[5]*Defer[Int][1/((I*Sqrt[2] - E^x)*(I + Sqrt[2]*x)), x] - (Log[5]*Defer[Int][1/((I*Sqrt[2] - E^x)*(I + Sqrt

[2]*x)), x])/(4*Sqrt[2]) + (I/4)*Log[5]*Defer[Int][1/((I*Sqrt[2] + E^x)*(I + Sqrt[2]*x)), x] + (Log[5]*Defer[I

nt][1/((I*Sqrt[2] + E^x)*(I + Sqrt[2]*x)), x])/(4*Sqrt[2]) - Sqrt[2]*Log[5]*Defer[Int][E^x/((2 + E^(2*x))^2*(I

+ Sqrt[2]*x)), x] + (I/2)*Log[5]*Defer[Int][E^x/((E^x + 2*x)^2*(I + Sqrt[2]*x)), x] - (Log[5]*Defer[Int][E^x/

((E^x + 2*x)^2*(I + Sqrt[2]*x)), x])/(2*Sqrt[2]) - 2*Log[5]*Defer[Int][E^x/((2 + E^(2*x))*(1 + 2*x^2)^2), x] -

2*Log[5]*Defer[Int][(E^(2*x)*x)/((2 + E^(2*x))*(1 + 2*x^2)^2), x] + 2*Log[5]*Defer[Int][(E^x*x)/((E^x + 2*x)*

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^x \left (-2 (-1+x)-4 e^x x^2-e^{2 x} (-1+3 x)\right ) \log (5)}{\left (2+e^{2 x}\right )^2 \left (e^x+2 x\right )^2} \, dx\\ &=(2 \log (5)) \int \frac {e^x \left (-2 (-1+x)-4 e^x x^2-e^{2 x} (-1+3 x)\right )}{\left (2+e^{2 x}\right )^2 \left (e^x+2 x\right )^2} \, dx\\ &=(2 \log (5)) \int \left (\frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2}-\frac {e^x (-1+x)}{2 \left (e^x+2 x\right )^2 \left (1+2 x^2\right )}-\frac {2 e^x x \left (1+e^x x\right )}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )}-\frac {e^x \left (1-x+2 e^x x-2 x^2-2 x^3\right )}{2 \left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}\right ) \, dx\\ &=-\left (\log (5) \int \frac {e^x (-1+x)}{\left (e^x+2 x\right )^2 \left (1+2 x^2\right )} \, dx\right )-\log (5) \int \frac {e^x \left (1-x+2 e^x x-2 x^2-2 x^3\right )}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx-(4 \log (5)) \int \frac {e^x x \left (1+e^x x\right )}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )} \, dx\\ &=-\left (\log (5) \int \left (\frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}-\frac {e^x x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}+\frac {2 e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}-\frac {2 e^x x^2}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}-\frac {2 e^x x^3}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}\right ) \, dx\right )-\log (5) \int \left (-\frac {e^x}{\left (e^x+2 x\right )^2 \left (1+2 x^2\right )}+\frac {e^x x}{\left (e^x+2 x\right )^2 \left (1+2 x^2\right )}\right ) \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx-(4 \log (5)) \int \left (\frac {e^x x}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )}+\frac {e^{2 x} x^2}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )}\right ) \, dx\\ &=-\left (\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx\right )+\log (5) \int \frac {e^x x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (1+2 x^2\right )} \, dx-\log (5) \int \frac {e^x x}{\left (e^x+2 x\right )^2 \left (1+2 x^2\right )} \, dx-(2 \log (5)) \int \frac {e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x^2}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x^3}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx-(4 \log (5)) \int \frac {e^x x}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )} \, dx-(4 \log (5)) \int \frac {e^{2 x} x^2}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )} \, dx\\ &=-\left (\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx\right )+\log (5) \int \frac {e^x x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+\log (5) \int \left (\frac {i e^x}{2 \left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )}+\frac {i e^x}{2 \left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )}\right ) \, dx-\log (5) \int \left (-\frac {e^x}{2 \sqrt {2} \left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )}+\frac {e^x}{2 \sqrt {2} \left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )}\right ) \, dx-(2 \log (5)) \int \frac {e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \left (-\frac {e^x}{2 \left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}+\frac {e^x}{2 \left (2+e^{2 x}\right ) \left (1+2 x^2\right )}\right ) \, dx+(2 \log (5)) \int \left (-\frac {e^x x}{2 \left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2}+\frac {e^x x}{2 \left (2+e^{2 x}\right ) \left (1+2 x^2\right )}\right ) \, dx-(4 \log (5)) \int \left (-\frac {e^x}{2 \sqrt {2} \left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )}+\frac {e^x}{2 \sqrt {2} \left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )}\right ) \, dx-(4 \log (5)) \int \left (\frac {e^{2 x}}{2 \left (2+e^{2 x}\right )^2}-\frac {e^{2 x}}{2 \left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )}\right ) \, dx\\ &=\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx-2 \left (\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx\right )+\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )} \, dx+\log (5) \int \frac {e^x x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )} \, dx-(2 \log (5)) \int \frac {e^{2 x}}{\left (2+e^{2 x}\right )^2} \, dx-(2 \log (5)) \int \frac {e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^{2 x}}{\left (2+e^{2 x}\right )^2 \left (1+2 x^2\right )} \, dx+\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}-\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}+\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )} \, dx-\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )} \, dx\\ &=\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx-2 \left (\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx\right )+\log (5) \int \left (\frac {i e^x}{2 \left (2+e^{2 x}\right ) \left (i-\sqrt {2} x\right )}+\frac {i e^x}{2 \left (2+e^{2 x}\right ) \left (i+\sqrt {2} x\right )}\right ) \, dx+\log (5) \int \left (-\frac {e^x}{2 \sqrt {2} \left (2+e^{2 x}\right ) \left (i-\sqrt {2} x\right )}+\frac {e^x}{2 \sqrt {2} \left (2+e^{2 x}\right ) \left (i+\sqrt {2} x\right )}\right ) \, dx-\log (5) \operatorname {Subst}\left (\int \frac {1}{(2+x)^2} \, dx,x,e^{2 x}\right )-(2 \log (5)) \int \frac {e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \left (\frac {i e^{2 x}}{2 \left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )}+\frac {i e^{2 x}}{2 \left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )}\right ) \, dx+\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}-\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}+\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )} \, dx-\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )} \, dx\\ &=\frac {\log (5)}{2+e^{2 x}}+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (i-\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (i+\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx+(i \log (5)) \int \frac {e^{2 x}}{\left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )} \, dx+(i \log (5)) \int \frac {e^{2 x}}{\left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )} \, dx-2 \left (\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx\right )-(2 \log (5)) \int \frac {e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx-\frac {\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (i-\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}+\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}+\frac {\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (i+\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}-\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}+\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )} \, dx-\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )} \, dx\\ &=\frac {\log (5)}{2+e^{2 x}}-\frac {i \log (5)}{2 \left (2+e^{2 x}\right ) \left (i-\sqrt {2} x\right )}-\frac {i \log (5)}{2 \left (2+e^{2 x}\right ) \left (i+\sqrt {2} x\right )}+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx+\frac {1}{2} (i \log (5)) \int \left (-\frac {1}{2 \left (i \sqrt {2}-e^x\right ) \left (i-\sqrt {2} x\right )}+\frac {1}{2 \left (i \sqrt {2}+e^x\right ) \left (i-\sqrt {2} x\right )}\right ) \, dx+\frac {1}{2} (i \log (5)) \int \left (-\frac {1}{2 \left (i \sqrt {2}-e^x\right ) \left (i+\sqrt {2} x\right )}+\frac {1}{2 \left (i \sqrt {2}+e^x\right ) \left (i+\sqrt {2} x\right )}\right ) \, dx-2 \left (\log (5) \int \frac {e^x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx\right )-(2 \log (5)) \int \frac {e^{2 x} x}{\left (2+e^{2 x}\right ) \left (1+2 x^2\right )^2} \, dx+(2 \log (5)) \int \frac {e^x x}{\left (e^x+2 x\right ) \left (1+2 x^2\right )^2} \, dx+\frac {(i \log (5)) \int \frac {1}{\left (2+e^{2 x}\right ) \left (i-\sqrt {2} x\right )^2} \, dx}{\sqrt {2}}-\frac {(i \log (5)) \int \frac {1}{\left (2+e^{2 x}\right ) \left (i+\sqrt {2} x\right )^2} \, dx}{\sqrt {2}}+\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i-\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}-\frac {\log (5) \int \frac {e^x}{\left (e^x+2 x\right )^2 \left (i+\sqrt {2} x\right )} \, dx}{2 \sqrt {2}}-\frac {\log (5) \int \left (-\frac {1}{2 \left (i \sqrt {2}-e^x\right ) \left (i-\sqrt {2} x\right )}+\frac {1}{2 \left (i \sqrt {2}+e^x\right ) \left (i-\sqrt {2} x\right )}\right ) \, dx}{2 \sqrt {2}}+\frac {\log (5) \int \left (-\frac {1}{2 \left (i \sqrt {2}-e^x\right ) \left (i+\sqrt {2} x\right )}+\frac {1}{2 \left (i \sqrt {2}+e^x\right ) \left (i+\sqrt {2} x\right )}\right ) \, dx}{2 \sqrt {2}}+\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i-\sqrt {2} x\right )} \, dx-\left (\sqrt {2} \log (5)\right ) \int \frac {e^x}{\left (2+e^{2 x}\right )^2 \left (i+\sqrt {2} x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 1.38, size = 23, normalized size = 0.88 \begin {gather*} \frac {2 x \log (5)}{\left (2+e^{2 x}\right ) \left (e^x+2 x\right )} \end {gather*}

Integrate[(E^(3*x)*(2 - 6*x)*Log[5] + E^x*(4 - 4*x)*Log[5] - 8*E^(2*x)*x^2*Log[5])/(E^(6*x) + 16*E^x*x + 16*E^

(3*x)*x + 4*E^(5*x)*x + 16*x^2 + E^(4*x)*(4 + 4*x^2) + E^(2*x)*(4 + 16*x^2)),x]

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fricas [A] time = 0.61, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \, x \log \relax (5)}{2 \, x e^{\left (2 \, x\right )} + 4 \, x + e^{\left (3 \, x\right )} + 2 \, e^{x}} \end {gather*}

integrate(((-6*x+2)*log(5)*exp(x)^3-8*x^2*log(5)*exp(x)^2+(-4*x+4)*log(5)*exp(x))/(exp(x)^6+4*x*exp(x)^5+(4*x^

2+4)*exp(x)^4+16*x*exp(x)^3+(16*x^2+4)*exp(x)^2+16*exp(x)*x+16*x^2),x, algorithm="fricas")

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giac [B] time = 0.71, size = 182, normalized size = 7.00 \begin {gather*} \frac {32 \, x^{5} \log \relax (5) + 4 \, x^{4} e^{x} \log \relax (5) + 2 \, x^{3} e^{\left (2 \, x\right )} \log \relax (5) - 4 \, x^{3} e^{x} \log \relax (5) + 40 \, x^{3} \log \relax (5) - 2 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5) + 2 \, x^{2} e^{x} \log \relax (5) - 8 \, x^{2} \log \relax (5) - 3 \, x e^{\left (2 \, x\right )} \log \relax (5) - 2 \, x e^{x} \log \relax (5) + 4 \, x \log \relax (5) + e^{\left (2 \, x\right )} \log \relax (5)}{2 \, {\left (8 \, x^{5} e^{\left (2 \, x\right )} + 16 \, x^{5} + 4 \, x^{4} e^{\left (3 \, x\right )} + 8 \, x^{4} e^{x} + 8 \, x^{3} e^{\left (2 \, x\right )} + 16 \, x^{3} + 4 \, x^{2} e^{\left (3 \, x\right )} + 8 \, x^{2} e^{x} + 2 \, x e^{\left (2 \, x\right )} + 4 \, x + e^{\left (3 \, x\right )} + 2 \, e^{x}\right )}} \end {gather*}

integrate(((-6*x+2)*log(5)*exp(x)^3-8*x^2*log(5)*exp(x)^2+(-4*x+4)*log(5)*exp(x))/(exp(x)^6+4*x*exp(x)^5+(4*x^

2+4)*exp(x)^4+16*x*exp(x)^3+(16*x^2+4)*exp(x)^2+16*exp(x)*x+16*x^2),x, algorithm="giac")

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

)*log(5) + 2*x^2*e^x*log(5) - 8*x^2*log(5) - 3*x*e^(2*x)*log(5) - 2*x*e^x*log(5) + 4*x*log(5) + e^(2*x)*log(5)

)/(8*x^5*e^(2*x) + 16*x^5 + 4*x^4*e^(3*x) + 8*x^4*e^x + 8*x^3*e^(2*x) + 16*x^3 + 4*x^2*e^(3*x) + 8*x^2*e^x + 2

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method	result	size

norman	\(\frac {2 x \ln \relax (5)}{\left (2+{\mathrm e}^{2 x}\right ) \left ({\mathrm e}^{x}+2 x \right )}\)	\(22\)
risch	\(\frac {2 \ln \relax (5) x}{{\mathrm e}^{3 x}+2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+4 x}\)	\(27\)

int(((-6*x+2)*ln(5)*exp(x)^3-8*x^2*ln(5)*exp(x)^2+(-4*x+4)*ln(5)*exp(x))/(exp(x)^6+4*x*exp(x)^5+(4*x^2+4)*exp(

x)^4+16*x*exp(x)^3+(16*x^2+4)*exp(x)^2+16*exp(x)*x+16*x^2),x,method=_RETURNVERBOSE)

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maxima [A] time = 0.50, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \, x \log \relax (5)}{2 \, x e^{\left (2 \, x\right )} + 4 \, x + e^{\left (3 \, x\right )} + 2 \, e^{x}} \end {gather*}

integrate(((-6*x+2)*log(5)*exp(x)^3-8*x^2*log(5)*exp(x)^2+(-4*x+4)*log(5)*exp(x))/(exp(x)^6+4*x*exp(x)^5+(4*x^

2+4)*exp(x)^4+16*x*exp(x)^3+(16*x^2+4)*exp(x)^2+16*exp(x)*x+16*x^2),x, algorithm="maxima")

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mupad [B] time = 3.33, size = 26, normalized size = 1.00 \begin {gather*} \frac {2\,x\,\ln \relax (5)}{4\,x+{\mathrm {e}}^{3\,x}+2\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^{2\,x}} \end {gather*}

int(-(exp(3*x)*log(5)*(6*x - 2) + exp(x)*log(5)*(4*x - 4) + 8*x^2*exp(2*x)*log(5))/(exp(6*x) + 16*x*exp(3*x) +

4*x*exp(5*x) + exp(4*x)*(4*x^2 + 4) + exp(2*x)*(16*x^2 + 4) + 16*x*exp(x) + 16*x^2),x)

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

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sympy [A] time = 0.18, size = 27, normalized size = 1.04 \begin {gather*} \frac {2 x \log {\relax (5 )}}{2 x e^{2 x} + 4 x + e^{3 x} + 2 e^{x}} \end {gather*}

integrate(((-6*x+2)*ln(5)*exp(x)**3-8*x**2*ln(5)*exp(x)**2+(-4*x+4)*ln(5)*exp(x))/(exp(x)**6+4*x*exp(x)**5+(4*

x**2+4)*exp(x)**4+16*x*exp(x)**3+(16*x**2+4)*exp(x)**2+16*exp(x)*x+16*x**2),x)

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

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3.47.58 \(\int \frac {e^{3 x} (2-6 x) \log (5)+e^x (4-4 x) \log (5)-8 e^{2 x} x^2 \log (5)}{e^{6 x}+16 e^x x+16 e^{3 x} x+4 e^{5 x} x+16 x^2+e^{4 x} (4+4 x^2)+e^{2 x} (4+16 x^2)} \, dx\)