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

Optimal. Leaf size=29 \[ x+\left (3+\frac {1}{x}\right ) \left (\log (2)-\log ^2\left (\frac {x}{-4 e^{2 x}+x}\right )\right ) \]

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Rubi [F]  time = 3.76, 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 {-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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 36, normalized size = 1.24 \begin {gather*} x+\frac {\log (2)}{x}+\frac {(-1-3 x) \log ^2\left (-\frac {x}{4 e^{2 x}-x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.70, size = 36, normalized size = 1.24 \begin {gather*} -\frac {{\left (3 \, x + 1\right )} \log \left (\frac {x}{x - 4 \, e^{\left (2 \, x\right )}}\right )^{2} - x^{2} - \log \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \relax (2)\right )} e^{\left (2 \, x\right )} - x \log \relax (2)}{x^{3} - 4 \, x^{2} e^{\left (2 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.27, size = 934, normalized size = 32.21

method	result	size
risch	\(-\frac {\left (3 x +1\right ) \ln \left (x -4 \,{\mathrm e}^{2 x}\right )^{2}}{x}+\frac {\left (i \pi \,\mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (i x \right )-i \pi \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (i x \right )+6 x \ln \relax (x )+2 \ln \relax (x )\right ) \ln \left (x -4 \,{\mathrm e}^{2 x}\right )}{x}+\frac {4 \ln \relax (2)-4 \ln \relax (x )^{2}+4 x^{2}-12 x \ln \relax (x )^{2}-2 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{3} \mathrm {csgn}\left (i x \right )+4 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{4} \mathrm {csgn}\left (i x \right )+\pi ^{2} \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (i x \right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{3} \mathrm {csgn}\left (i x \right )^{2}+4 i \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{3}-12 i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} x -12 i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (i x \right ) x +4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (i x \right )+12 i \pi \ln \left (-\frac {x}{4}+{\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} x +12 i \pi \ln \left (-\frac {x}{4}+{\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (i x \right ) x +\pi ^{2} \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{6}-4 i \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (i x \right )-12 i \pi \ln \left (-\frac {x}{4}+{\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{3} x +12 i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{3} x -4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{2}+12 i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (i x \right ) x -12 i \pi \ln \left (-\frac {x}{4}+{\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (i x \right ) x +\pi ^{2} \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{4} \mathrm {csgn}\left (i x \right )^{2}+\pi ^{2} \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right )^{2} \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{4}-2 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x -4 \,{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{5}-2 \pi ^{2} \mathrm {csgn}\left (\frac {i x}{x -4 \,{\mathrm e}^{2 x}}\right )^{5} \mathrm {csgn}\left (i x \right )}{4 x}\)	\(934\)

Verification of antiderivative is not currently implemented for this CAS.

[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*x+1)/x*ln(x-4*exp(2*x))^2+(I*Pi*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x)))^2-I*Pi*csgn(I/(x-4*exp(2*x
)))*csgn(I*x/(x-4*exp(2*x)))*csgn(I*x)-I*Pi*csgn(I*x/(x-4*exp(2*x)))^3+I*Pi*csgn(I*x/(x-4*exp(2*x)))^2*csgn(I*
x)+6*x*ln(x)+2*ln(x))/x*ln(x-4*exp(2*x))+1/4*(4*ln(2)-4*ln(x)^2+4*x^2-12*x*ln(x)^2-2*Pi^2*csgn(I/(x-4*exp(2*x)
))^2*csgn(I*x/(x-4*exp(2*x)))^3*csgn(I*x)+4*Pi^2*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x)))^4*csgn(I*x)+P
i^2*csgn(I/(x-4*exp(2*x)))^2*csgn(I*x/(x-4*exp(2*x)))^2*csgn(I*x)^2-2*Pi^2*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-
4*exp(2*x)))^3*csgn(I*x)^2+4*I*ln(x)*Pi*csgn(I*x/(x-4*exp(2*x)))^3-4*I*ln(x)*Pi*csgn(I*x/(x-4*exp(2*x)))^2*csg
n(I*x)-12*I*Pi*ln(-1/4*x+exp(2*x))*csgn(I*x/(x-4*exp(2*x)))^3*x+Pi^2*csgn(I*x/(x-4*exp(2*x)))^6+12*I*Pi*ln(x)*
csgn(I*x/(x-4*exp(2*x)))^3*x-4*I*ln(x)*Pi*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x)))^2+Pi^2*csgn(I*x/(x-4
*exp(2*x)))^4*csgn(I*x)^2+Pi^2*csgn(I/(x-4*exp(2*x)))^2*csgn(I*x/(x-4*exp(2*x)))^4-2*Pi^2*csgn(I/(x-4*exp(2*x)
))*csgn(I*x/(x-4*exp(2*x)))^5-2*Pi^2*csgn(I*x/(x-4*exp(2*x)))^5*csgn(I*x)+12*I*Pi*ln(x)*csgn(I/(x-4*exp(2*x)))
*csgn(I*x/(x-4*exp(2*x)))*csgn(I*x)*x-12*I*Pi*ln(-1/4*x+exp(2*x))*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x
)))*csgn(I*x)*x-12*I*Pi*ln(x)*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x)))^2*x-12*I*Pi*ln(x)*csgn(I*x/(x-4*
exp(2*x)))^2*csgn(I*x)*x+4*I*ln(x)*Pi*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x)))*csgn(I*x)+12*I*Pi*ln(-1/
4*x+exp(2*x))*csgn(I/(x-4*exp(2*x)))*csgn(I*x/(x-4*exp(2*x)))^2*x+12*I*Pi*ln(-1/4*x+exp(2*x))*csgn(I*x/(x-4*ex
p(2*x)))^2*csgn(I*x)*x)/x

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maxima [B]  time = 0.49, size = 60, normalized size = 2.07 \begin {gather*} -\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 \relax (x) + {\left (3 \, x + 1\right )} \log \relax (x)^{2} - x^{2} - \log \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 5.62, size = 30, normalized size = 1.03 \begin {gather*} x-{\ln \left (\frac {x}{x-4\,{\mathrm {e}}^{2\,x}}\right )}^2\,\left (\frac {1}{x}+3\right )+\frac {\ln \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.48, size = 29, normalized size = 1.00 \begin {gather*} x + \frac {\left (- 3 x - 1\right ) \log {\left (- \frac {x}{- x + 4 e^{2 x}} \right )}^{2}}{x} + \frac {\log {\relax (2 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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