\(\int \frac {(7 x+14 e^{2 x} x) \log (-\frac {10}{x})+(-7 e^{2 x}-7 x+(-7 e^{2 x} x-7 x^2) \log (-\frac {10}{x})) \log (e^{2 x}+x)}{3 e^{3 x} x+3 e^x x^2} \, dx\) [7254]

Optimal result

Integrand size = 81, antiderivative size = 23 \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right ) \]

[Out]

7/3*ln(exp(x)^2+x)*ln(-10/x)/exp(x)

Rubi [F]

\[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx \]

[In]

Int[((7*x + 14*E^(2*x)*x)*Log[-10/x] + (-7*E^(2*x) - 7*x + (-7*E^(2*x)*x - 7*x^2)*Log[-10/x])*Log[E^(2*x) + x]
)/(3*E^(3*x)*x + 3*E^x*x^2),x]

[Out]

(-14*ExpIntegralEi[-x])/3 + (7*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -x])/3 - (14*Log[-10/x])/(3*E^x) - (7
*ExpIntegralEi[-x]*Log[-10/x])/3 - (7*EulerGamma*Log[x])/3 - (7*(ExpIntegralE[1, x] + ExpIntegralEi[-x])*Log[x
])/3 - (7*Log[x]^2)/6 + (7*Log[-10/x]*Log[E^(2*x) + x])/(3*E^x) + (7*Log[-10/x]*Defer[Int][1/(E^x*(E^(2*x) + x
)), x])/3 - (14*Log[-10/x]*Defer[Int][E^x/(E^(2*x) + x), x])/3 + (7*Log[-10/x]*Defer[Int][E^x/(x*(E^(2*x) + x)
), x])/3 - (14*Log[-10/x]*Defer[Int][x/(E^x*(E^(2*x) + x)), x])/3 + (7*Defer[Int][Defer[Int][1/(E^x*(E^(2*x) +
x)), x]/x, x])/3 - (14*Defer[Int][Defer[Int][E^x/(E^(2*x) + x), x]/x, x])/3 + (7*Defer[Int][Defer[Int][E^x/(x
*(E^(2*x) + x)), x]/x, x])/3 - (14*Defer[Int][Defer[Int][x/(E^x*(E^(2*x) + x)), x]/x, x])/3

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )\right )}{3 x \left (e^{2 x}+x\right )} \, dx \\ & = \frac {1}{3} \int \frac {e^{-x} \left (\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )\right )}{x \left (e^{2 x}+x\right )} \, dx \\ & = \frac {1}{3} \int \left (-\frac {7 e^{-x} (-1+2 x) \log \left (-\frac {10}{x}\right )}{e^{2 x}+x}-\frac {7 e^{-x} \left (-2 x \log \left (-\frac {10}{x}\right )+\log \left (e^{2 x}+x\right )+x \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )\right )}{x}\right ) \, dx \\ & = -\left (\frac {7}{3} \int \frac {e^{-x} (-1+2 x) \log \left (-\frac {10}{x}\right )}{e^{2 x}+x} \, dx\right )-\frac {7}{3} \int \frac {e^{-x} \left (-2 x \log \left (-\frac {10}{x}\right )+\log \left (e^{2 x}+x\right )+x \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )\right )}{x} \, dx \\ & = -\left (\frac {7}{3} \int \left (-2 e^{-x} \log \left (-\frac {10}{x}\right )+\frac {e^{-x} \left (1+x \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{x}\right ) \, dx\right )-\frac {7}{3} \int \frac {-\int \frac {e^{-x}}{e^{2 x}+x} \, dx+2 \int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\left (\frac {7}{3} \int \frac {e^{-x} \left (1+x \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{x} \, dx\right )-\frac {7}{3} \int \left (-\frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x}+\frac {2 \int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x}\right ) \, dx+\frac {14}{3} \int e^{-x} \log \left (-\frac {10}{x}\right ) \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \int \left (\frac {e^{-x} \log \left (e^{2 x}+x\right )}{x}+e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {14}{3} \int \frac {e^{-x}}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \int \frac {e^{-x} \log \left (e^{2 x}+x\right )}{x} \, dx-\frac {7}{3} \int e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (e^{2 x}+x\right )+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )+\frac {7}{3} \int \frac {\left (1+2 e^{2 x}\right ) \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x} \, dx-\frac {7}{3} \int \frac {e^{-x} \left (1+2 e^{2 x}\right ) \log \left (-\frac {10}{x}\right )}{e^{2 x}+x} \, dx+\frac {7}{3} \int \frac {e^{-x} \log \left (e^{2 x}+x\right )}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )-\frac {7}{3} \int \frac {\left (1+2 e^{2 x}\right ) \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x} \, dx+\frac {7}{3} \int \left (2 \operatorname {ExpIntegralEi}(-x)-\frac {(-1+2 x) \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x}\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {7}{3} \int \frac {\operatorname {ExpIntegralEi}(-x)+2 \int \frac {e^x}{e^{2 x}+x} \, dx-\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )-\frac {7}{3} \int \frac {(-1+2 x) \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x} \, dx-\frac {7}{3} \int \left (2 \operatorname {ExpIntegralEi}(-x)-\frac {(-1+2 x) \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x}\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {7}{3} \int \left (\frac {\operatorname {ExpIntegralEi}(-x)+2 \int \frac {e^x}{e^{2 x}+x} \, dx}{x}-\frac {\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x}\right ) \, dx+\frac {14 \int \operatorname {ExpIntegralEi}(-x) \, dx}{3}-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = \frac {14 e^{-x}}{3}-\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}+\frac {14 x \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )+\frac {7}{3} \int \frac {(-1+2 x) \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x} \, dx-\frac {7}{3} \int \left (-\frac {\operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x}+\frac {2 x \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x}\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {7}{3} \int \frac {\operatorname {ExpIntegralEi}(-x)+2 \int \frac {e^x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x} \, dx-\frac {14 \int \operatorname {ExpIntegralEi}(-x) \, dx}{3}-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )+\frac {7}{3} \int \frac {\operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x} \, dx+\frac {7}{3} \int \left (-\frac {\operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x}+\frac {2 x \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x}\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx-\frac {7}{3} \int \left (\frac {\operatorname {ExpIntegralEi}(-x)}{x}+\frac {2 \int \frac {e^x}{e^{2 x}+x} \, dx}{x}\right ) \, dx+\frac {7}{3} \int \frac {\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x} \, dx-\frac {14}{3} \int \frac {x \operatorname {ExpIntegralEi}(-x)}{e^{2 x}+x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )-\frac {7}{3} \int \frac {\operatorname {ExpIntegralEi}(-x)}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^x}{e^{2 x}+x} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )-\frac {7}{3} (\operatorname {ExpIntegralE}(1,x)+\operatorname {ExpIntegralEi}(-x)) \log (x)+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )+\frac {7}{3} \int \frac {\operatorname {ExpIntegralE}(1,x)}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^x}{e^{2 x}+x} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ & = -\frac {14 \operatorname {ExpIntegralEi}(-x)}{3}+\frac {7}{3} x \, _3F_3(1,1,1;2,2,2;-x)-\frac {14}{3} e^{-x} \log \left (-\frac {10}{x}\right )-\frac {7}{3} \operatorname {ExpIntegralEi}(-x) \log \left (-\frac {10}{x}\right )-\frac {7}{3} \gamma \log (x)-\frac {7}{3} (\operatorname {ExpIntegralE}(1,x)+\operatorname {ExpIntegralEi}(-x)) \log (x)-\frac {7 \log ^2(x)}{6}+\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right )+\frac {7}{3} \int \frac {\int \frac {e^{-x}}{e^{2 x}+x} \, dx}{x} \, dx+\frac {7}{3} \int \frac {\int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^x}{e^{2 x}+x} \, dx}{x} \, dx-\frac {14}{3} \int \frac {\int \frac {e^{-x} x}{e^{2 x}+x} \, dx}{x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x}}{e^{2 x}+x} \, dx+\frac {1}{3} \left (7 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{x \left (e^{2 x}+x\right )} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^x}{e^{2 x}+x} \, dx-\frac {1}{3} \left (14 \log \left (-\frac {10}{x}\right )\right ) \int \frac {e^{-x} x}{e^{2 x}+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (e^{2 x}+x\right ) \]

[In]

Integrate[((7*x + 14*E^(2*x)*x)*Log[-10/x] + (-7*E^(2*x) - 7*x + (-7*E^(2*x)*x - 7*x^2)*Log[-10/x])*Log[E^(2*x
) + x])/(3*E^(3*x)*x + 3*E^x*x^2),x]

[Out]

(7*Log[-10/x]*Log[E^(2*x) + x])/(3*E^x)

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
parallelrisch	\(\frac {7 \ln \left ({\mathrm e}^{2 x}+x \right ) \ln \left (-\frac {10}{x}\right ) {\mathrm e}^{-x}}{3}\)	\(20\)
risch	\(\frac {7 \left (-2 i \pi \operatorname {csgn}\left (\frac {i}{x}\right )^{2}+2 i \pi \operatorname {csgn}\left (\frac {i}{x}\right )^{3}+2 i \pi +2 \ln \left (5\right )+2 \ln \left (2\right )-2 \ln \left (x \right )\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{2 x}+x \right )}{6}\)	\(57\)

[In]

int((((-7*x*exp(x)^2-7*x^2)*ln(-10/x)-7*exp(x)^2-7*x)*ln(exp(x)^2+x)+(14*x*exp(x)^2+7*x)*ln(-10/x))/(3*x*exp(x
)^3+3*exp(x)*x^2),x,method=_RETURNVERBOSE)

[Out]

7/3*ln(exp(x)^2+x)*ln(-10/x)/exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\frac {7}{3} \, e^{\left (-x\right )} \log \left (x + e^{\left (2 \, x\right )}\right ) \log \left (-\frac {10}{x}\right ) \]

[In]

integrate((((-7*x*exp(x)^2-7*x^2)*log(-10/x)-7*exp(x)^2-7*x)*log(exp(x)^2+x)+(14*x*exp(x)^2+7*x)*log(-10/x))/(
3*x*exp(x)^3+3*exp(x)*x^2),x, algorithm="fricas")

[Out]

7/3*e^(-x)*log(x + e^(2*x))*log(-10/x)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\frac {7 e^{- x} \log {\left (- \frac {10}{x} \right )} \log {\left (x + e^{2 x} \right )}}{3} \]

[In]

integrate((((-7*x*exp(x)**2-7*x**2)*ln(-10/x)-7*exp(x)**2-7*x)*ln(exp(x)**2+x)+(14*x*exp(x)**2+7*x)*ln(-10/x))
/(3*x*exp(x)**3+3*exp(x)*x**2),x)

[Out]

7*exp(-x)*log(-10/x)*log(x + exp(2*x))/3

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\frac {7}{3} \, {\left (\log \left (5\right ) + \log \left (2\right ) - \log \left (-x\right )\right )} e^{\left (-x\right )} \log \left (x + e^{\left (2 \, x\right )}\right ) \]

[In]

integrate((((-7*x*exp(x)^2-7*x^2)*log(-10/x)-7*exp(x)^2-7*x)*log(exp(x)^2+x)+(14*x*exp(x)^2+7*x)*log(-10/x))/(
3*x*exp(x)^3+3*exp(x)*x^2),x, algorithm="maxima")

[Out]

7/3*(log(5) + log(2) - log(-x))*e^(-x)*log(x + e^(2*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (19) = 38\).

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\frac {7}{12} \, {\left (\pi ^{2} \mathrm {sgn}\left (x + e^{\left (2 \, x\right )}\right ) \mathrm {sgn}\left (x\right ) + \pi ^{2} \mathrm {sgn}\left (x + e^{\left (2 \, x\right )}\right ) - \pi ^{2} \mathrm {sgn}\left (x\right ) - \pi ^{2} + 4 \, \log \left (10\right ) \log \left ({\left | x + e^{\left (2 \, x\right )} \right |}\right ) - 4 \, \log \left ({\left | x + e^{\left (2 \, x\right )} \right |}\right ) \log \left ({\left | x \right |}\right )\right )} e^{\left (-x\right )} \]

[In]

integrate((((-7*x*exp(x)^2-7*x^2)*log(-10/x)-7*exp(x)^2-7*x)*log(exp(x)^2+x)+(14*x*exp(x)^2+7*x)*log(-10/x))/(
3*x*exp(x)^3+3*exp(x)*x^2),x, algorithm="giac")

[Out]

7/12*(pi^2*sgn(x + e^(2*x))*sgn(x) + pi^2*sgn(x + e^(2*x)) - pi^2*sgn(x) - pi^2 + 4*log(10)*log(abs(x + e^(2*x
))) - 4*log(abs(x + e^(2*x)))*log(abs(x)))*e^(-x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx=\int \frac {\ln \left (-\frac {10}{x}\right )\,\left (7\,x+14\,x\,{\mathrm {e}}^{2\,x}\right )-\ln \left (x+{\mathrm {e}}^{2\,x}\right )\,\left (7\,x+7\,{\mathrm {e}}^{2\,x}+\ln \left (-\frac {10}{x}\right )\,\left (7\,x\,{\mathrm {e}}^{2\,x}+7\,x^2\right )\right )}{3\,x\,{\mathrm {e}}^{3\,x}+3\,x^2\,{\mathrm {e}}^x} \,d x \]

[In]

int((log(-10/x)*(7*x + 14*x*exp(2*x)) - log(x + exp(2*x))*(7*x + 7*exp(2*x) + log(-10/x)*(7*x*exp(2*x) + 7*x^2
)))/(3*x*exp(3*x) + 3*x^2*exp(x)),x)

[Out]

int((log(-10/x)*(7*x + 14*x*exp(2*x)) - log(x + exp(2*x))*(7*x + 7*exp(2*x) + log(-10/x)*(7*x*exp(2*x) + 7*x^2
)))/(3*x*exp(3*x) + 3*x^2*exp(x)), x)