\(\int \frac {-x^3+e^{2 x} (4 x^2-4 \log (2))+x \log (2)+e^{2 x} (-8-8 x+48 x^2) \log (-\frac {x}{4 e^{2 x}-x})+(4 e^{2 x}-x) \log ^2(-\frac {x}{4 e^{2 x}-x})}{4 e^{2 x} x^2-x^3} \, dx\) [7786]

Optimal result

Integrand size = 109, antiderivative size = 29 \[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=x+\left (3+\frac {1}{x}\right ) \left (\log (2)-\log ^2\left (\frac {x}{-4 e^{2 x}+x}\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=\int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx \]

[In]

Int[(-x^3 + E^(2*x)*(4*x^2 - 4*Log[2]) + x*Log[2] + E^(2*x)*(-8 - 8*x + 48*x^2)*Log[-(x/(4*E^(2*x) - x))] + (4
*E^(2*x) - x)*Log[-(x/(4*E^(2*x) - x))]^2)/(4*E^(2*x)*x^2 - x^3),x]

[Out]

x + Log[2]/x + (2*Log[-(x/(4*E^(2*x) - x))])/x + 12*x*Log[-(x/(4*E^(2*x) - x))] - 2*Log[-(x/(4*E^(2*x) - x))]*
Defer[Int][(4*E^(2*x) - x)^(-1), x] - 48*Defer[Int][E^(2*x)/(4*E^(2*x) - x), x] - 8*Defer[Int][E^(2*x)/((4*E^(
2*x) - x)*x^2), x] - 2*Log[-(x/(4*E^(2*x) - x))]*Defer[Int][1/((4*E^(2*x) - x)*x), x] + 16*Defer[Int][E^(2*x)/
((4*E^(2*x) - x)*x), x] + 12*Log[-(x/(4*E^(2*x) - x))]*Defer[Int][x/(4*E^(2*x) - x), x] + 96*Defer[Int][(E^(2*
x)*x)/(4*E^(2*x) - x), x] - 2*Defer[Int][Log[-(x/(4*E^(2*x) - x))]/x, x] + Defer[Int][Log[-(x/(4*E^(2*x) - x))
]^2/x^2, x] - 16*Defer[Int][(E^(2*x)*Defer[Int][(4*E^(2*x) - x)^(-1), x])/(4*E^(2*x) - x), x] + 8*Defer[Int][(
E^(2*x)*Defer[Int][(4*E^(2*x) - x)^(-1), x])/((4*E^(2*x) - x)*x), x] + 96*Defer[Int][(E^(2*x)*Defer[Int][x/(4*
E^(2*x) - x), x])/(4*E^(2*x) - x), x] - 48*Defer[Int][(E^(2*x)*Defer[Int][x/(4*E^(2*x) - x), x])/((4*E^(2*x) -
x)*x), x] - 16*Defer[Int][(E^(2*x)*Defer[Int][(4*E^(2*x)*x - x^2)^(-1), x])/(4*E^(2*x) - x), x] + 8*Defer[Int
][(E^(2*x)*Defer[Int][(4*E^(2*x)*x - x^2)^(-1), x])/((4*E^(2*x) - x)*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2-\log (2)+\frac {8 e^{2 x} \left (-1-x+6 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x}-x}+\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = \int \left (\frac {2 \left (-1-x+6 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{\left (4 e^{2 x}-x\right ) x}+\frac {x^2-\log (2)-2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )-2 x \log \left (-\frac {x}{4 e^{2 x}-x}\right )+12 x^2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {\left (-1-x+6 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{\left (4 e^{2 x}-x\right ) x} \, dx+\int \frac {x^2-\log (2)-2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )-2 x \log \left (-\frac {x}{4 e^{2 x}-x}\right )+12 x^2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = -\left (2 \int \frac {4 e^{2 x} (1-2 x) \left (-\int \frac {1}{4 e^{2 x}-x} \, dx+6 \int \frac {x}{4 e^{2 x}-x} \, dx-\int \frac {1}{4 e^{2 x} x-x^2} \, dx\right )}{\left (4 e^{2 x}-x\right ) x} \, dx\right )-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \left (\frac {x^2-\log (2)}{x^2}+\frac {2 \left (-1-x+6 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2}+\frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {\left (-1-x+6 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx-8 \int \frac {e^{2 x} (1-2 x) \left (-\int \frac {1}{4 e^{2 x}-x} \, dx+6 \int \frac {x}{4 e^{2 x}-x} \, dx-\int \frac {1}{4 e^{2 x} x-x^2} \, dx\right )}{\left (4 e^{2 x}-x\right ) x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {x^2-\log (2)}{x^2} \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = 2 \int \left (6 \log \left (-\frac {x}{4 e^{2 x}-x}\right )-\frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2}-\frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x}\right ) \, dx-8 \int \left (\frac {2 e^{2 x} \left (\int \frac {1}{4 e^{2 x}-x} \, dx-6 \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {1}{4 e^{2 x} x-x^2} \, dx\right )}{4 e^{2 x}-x}-\frac {e^{2 x} \left (\int \frac {1}{4 e^{2 x}-x} \, dx-6 \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {1}{4 e^{2 x} x-x^2} \, dx\right )}{\left (4 e^{2 x}-x\right ) x}\right ) \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \left (1-\frac {\log (2)}{x^2}\right ) \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = x+\frac {\log (2)}{x}-2 \int \frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx-2 \int \frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \, dx+8 \int \frac {e^{2 x} \left (\int \frac {1}{4 e^{2 x}-x} \, dx-6 \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {1}{4 e^{2 x} x-x^2} \, dx\right )}{\left (4 e^{2 x}-x\right ) x} \, dx+12 \int \log \left (-\frac {x}{4 e^{2 x}-x}\right ) \, dx-16 \int \frac {e^{2 x} \left (\int \frac {1}{4 e^{2 x}-x} \, dx-6 \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {1}{4 e^{2 x} x-x^2} \, dx\right )}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = x+\frac {\log (2)}{x}+\frac {2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x}+12 x \log \left (-\frac {x}{4 e^{2 x}-x}\right )-2 \int \frac {4 e^{2 x} (1-2 x)}{\left (4 e^{2 x}-x\right ) x^2} \, dx-2 \int \frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \, dx+8 \int \left (\frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x}-\frac {6 e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x}+\frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{\left (4 e^{2 x}-x\right ) x}\right ) \, dx-12 \int \frac {4 e^{2 x} (1-2 x)}{4 e^{2 x}-x} \, dx-16 \int \left (\frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x}-\frac {6 e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x}+\frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{4 e^{2 x}-x}\right ) \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = x+\frac {\log (2)}{x}+\frac {2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x}+12 x \log \left (-\frac {x}{4 e^{2 x}-x}\right )-2 \int \frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \, dx-8 \int \frac {e^{2 x} (1-2 x)}{\left (4 e^{2 x}-x\right ) x^2} \, dx+8 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+8 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx-16 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x} \, dx-16 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{4 e^{2 x}-x} \, dx-48 \int \frac {e^{2 x} (1-2 x)}{4 e^{2 x}-x} \, dx-48 \int \frac {e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+96 \int \frac {e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = x+\frac {\log (2)}{x}+\frac {2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x}+12 x \log \left (-\frac {x}{4 e^{2 x}-x}\right )-2 \int \frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \, dx-8 \int \left (\frac {e^{2 x}}{\left (4 e^{2 x}-x\right ) x^2}-\frac {2 e^{2 x}}{\left (4 e^{2 x}-x\right ) x}\right ) \, dx+8 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+8 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx-16 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x} \, dx-16 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{4 e^{2 x}-x} \, dx-48 \int \left (\frac {e^{2 x}}{4 e^{2 x}-x}-\frac {2 e^{2 x} x}{4 e^{2 x}-x}\right ) \, dx-48 \int \frac {e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+96 \int \frac {e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ & = x+\frac {\log (2)}{x}+\frac {2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x}+12 x \log \left (-\frac {x}{4 e^{2 x}-x}\right )-2 \int \frac {\log \left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \, dx-8 \int \frac {e^{2 x}}{\left (4 e^{2 x}-x\right ) x^2} \, dx+8 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+8 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+16 \int \frac {e^{2 x}}{\left (4 e^{2 x}-x\right ) x} \, dx-16 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x} \, dx-16 \int \frac {e^{2 x} \int \frac {1}{4 e^{2 x} x-x^2} \, dx}{4 e^{2 x}-x} \, dx-48 \int \frac {e^{2 x}}{4 e^{2 x}-x} \, dx-48 \int \frac {e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{\left (4 e^{2 x}-x\right ) x} \, dx+96 \int \frac {e^{2 x} x}{4 e^{2 x}-x} \, dx+96 \int \frac {e^{2 x} \int \frac {x}{4 e^{2 x}-x} \, dx}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{4 e^{2 x}-x} \, dx-\left (2 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {1}{\left (4 e^{2 x}-x\right ) x} \, dx+\left (12 \log \left (-\frac {x}{4 e^{2 x}-x}\right )\right ) \int \frac {x}{4 e^{2 x}-x} \, dx+\int \frac {\log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=x+\frac {\log (2)}{x}+\frac {(-1-3 x) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \]

[In]

Integrate[(-x^3 + E^(2*x)*(4*x^2 - 4*Log[2]) + x*Log[2] + E^(2*x)*(-8 - 8*x + 48*x^2)*Log[-(x/(4*E^(2*x) - x))
] + (4*E^(2*x) - x)*Log[-(x/(4*E^(2*x) - x))]^2)/(4*E^(2*x)*x^2 - x^3),x]

[Out]

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79

method	result	size
parallelrisch	\(\frac {-3 \ln \left (-\frac {x}{4 \,{\mathrm e}^{2 x}-x}\right )^{2} x +x^{2}-\ln \left (-\frac {x}{4 \,{\mathrm e}^{2 x}-x}\right )^{2}+\ln \left (2\right )}{x}\)	\(52\)
risch	\(\text {Expression too large to display}\)	\(934\)

[In]

int(((4*exp(2*x)-x)*ln(-x/(4*exp(2*x)-x))^2+(48*x^2-8*x-8)*exp(2*x)*ln(-x/(4*exp(2*x)-x))+(-4*ln(2)+4*x^2)*exp
(2*x)+x*ln(2)-x^3)/(4*exp(2*x)*x^2-x^3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=-\frac {{\left (3 \, x + 1\right )} \log \left (\frac {x}{x - 4 \, e^{\left (2 \, x\right )}}\right )^{2} - x^{2} - \log \left (2\right )}{x} \]

[In]

integrate(((4*exp(2*x)-x)*log(-x/(4*exp(2*x)-x))^2+(48*x^2-8*x-8)*exp(2*x)*log(-x/(4*exp(2*x)-x))+(-4*log(2)+4
*x^2)*exp(2*x)+x*log(2)-x^3)/(4*exp(2*x)*x^2-x^3),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=x + \frac {\left (- 3 x - 1\right ) \log {\left (- \frac {x}{- x + 4 e^{2 x}} \right )}^{2}}{x} + \frac {\log {\left (2 \right )}}{x} \]

[In]

integrate(((4*exp(2*x)-x)*ln(-x/(4*exp(2*x)-x))**2+(48*x**2-8*x-8)*exp(2*x)*ln(-x/(4*exp(2*x)-x))+(-4*ln(2)+4*
x**2)*exp(2*x)+x*ln(2)-x**3)/(4*exp(2*x)*x**2-x**3),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).

Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=-\frac {{\left (3 \, x + 1\right )} \log \left (x - 4 \, e^{\left (2 \, x\right )}\right )^{2} - 2 \, {\left (3 \, x + 1\right )} \log \left (x - 4 \, e^{\left (2 \, x\right )}\right ) \log \left (x\right ) + {\left (3 \, x + 1\right )} \log \left (x\right )^{2} - x^{2} - \log \left (2\right )}{x} \]

[In]

integrate(((4*exp(2*x)-x)*log(-x/(4*exp(2*x)-x))^2+(48*x^2-8*x-8)*exp(2*x)*log(-x/(4*exp(2*x)-x))+(-4*log(2)+4
*x^2)*exp(2*x)+x*log(2)-x^3)/(4*exp(2*x)*x^2-x^3),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=\int { \frac {x^{3} - 8 \, {\left (6 \, x^{2} - x - 1\right )} e^{\left (2 \, x\right )} \log \left (\frac {x}{x - 4 \, e^{\left (2 \, x\right )}}\right ) + {\left (x - 4 \, e^{\left (2 \, x\right )}\right )} \log \left (\frac {x}{x - 4 \, e^{\left (2 \, x\right )}}\right )^{2} - 4 \, {\left (x^{2} - \log \left (2\right )\right )} e^{\left (2 \, x\right )} - x \log \left (2\right )}{x^{3} - 4 \, x^{2} e^{\left (2 \, x\right )}} \,d x } \]

[In]

integrate(((4*exp(2*x)-x)*log(-x/(4*exp(2*x)-x))^2+(48*x^2-8*x-8)*exp(2*x)*log(-x/(4*exp(2*x)-x))+(-4*log(2)+4
*x^2)*exp(2*x)+x*log(2)-x^3)/(4*exp(2*x)*x^2-x^3),x, algorithm="giac")

[Out]

integrate((x^3 - 8*(6*x^2 - x - 1)*e^(2*x)*log(x/(x - 4*e^(2*x))) + (x - 4*e^(2*x))*log(x/(x - 4*e^(2*x)))^2 -
4*(x^2 - log(2))*e^(2*x) - x*log(2))/(x^3 - 4*x^2*e^(2*x)), x)

Mupad [B] (verification not implemented)

Time = 14.95 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-x^3+e^{2 x} \left (4 x^2-4 \log (2)\right )+x \log (2)+e^{2 x} \left (-8-8 x+48 x^2\right ) \log \left (-\frac {x}{4 e^{2 x}-x}\right )+\left (4 e^{2 x}-x\right ) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{4 e^{2 x} x^2-x^3} \, dx=x-{\ln \left (\frac {x}{x-4\,{\mathrm {e}}^{2\,x}}\right )}^2\,\left (\frac {1}{x}+3\right )+\frac {\ln \left (2\right )}{x} \]

[In]

int(-(log(x/(x - 4*exp(2*x)))^2*(x - 4*exp(2*x)) - x*log(2) + x^3 + exp(2*x)*(4*log(2) - 4*x^2) + log(x/(x - 4
*exp(2*x)))*exp(2*x)*(8*x - 48*x^2 + 8))/(4*x^2*exp(2*x) - x^3),x)

[Out]

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