3.41.48 \(\int \frac {2 \log (2)-4 e^{3+x} \log (2)-2 e^{3+x} x \log (2) \log (x^2)+(\log (2)-2 e^{3+x} \log (2)) \log (x^2) \log ((1-2 e^{3+x}) \log (x^2))}{(-x^2+2 e^{3+x} x^2) \log (x^2) \log ^2((1-2 e^{3+x}) \log (x^2))} \, dx\)

Optimal. Leaf size=24 \[ \frac {\log (2)}{x \log \left (-\left (\left (-1+2 e^{3+x}\right ) \log \left (x^2\right )\right )\right )} \]

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Rubi [F]  time = 1.74, 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 {2 \log (2)-4 e^{3+x} \log (2)-2 e^{3+x} x \log (2) \log \left (x^2\right )+\left (\log (2)-2 e^{3+x} \log (2)\right ) \log \left (x^2\right ) \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}{\left (-x^2+2 e^{3+x} x^2\right ) \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-(Log[2]*Defer[Int][1/(x*Log[(1 - 2*E^(3 + x))*Log[x^2]]^2), x]) - Log[2]*Defer[Int][1/((-1 + 2*E^(3 + x))*x*L
og[(1 - 2*E^(3 + x))*Log[x^2]]^2), x] - 2*Log[2]*Defer[Int][1/(x^2*Log[x^2]*Log[(1 - 2*E^(3 + x))*Log[x^2]]^2)
, x] - Log[2]*Defer[Int][1/(x^2*Log[(1 - 2*E^(3 + x))*Log[x^2]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {\log (2)}{\left (-1+2 e^{3+x}\right ) x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}-\frac {\log (2) \left (2+x \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}\right ) \, dx\\ &=-\left (\log (2) \int \frac {1}{\left (-1+2 e^{3+x}\right ) x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\right )-\log (2) \int \frac {2+x \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}{x^2 \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\\ &=-\left (\log (2) \int \left (\frac {2+x \log \left (x^2\right )}{x^2 \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}+\frac {1}{x^2 \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}\right ) \, dx\right )-\log (2) \int \frac {1}{\left (-1+2 e^{3+x}\right ) x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\\ &=-\left (\log (2) \int \frac {1}{\left (-1+2 e^{3+x}\right ) x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\right )-\log (2) \int \frac {2+x \log \left (x^2\right )}{x^2 \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx-\log (2) \int \frac {1}{x^2 \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\\ &=-\left (\log (2) \int \left (\frac {1}{x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}+\frac {2}{x^2 \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )}\right ) \, dx\right )-\log (2) \int \frac {1}{\left (-1+2 e^{3+x}\right ) x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx-\log (2) \int \frac {1}{x^2 \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\\ &=-\left (\log (2) \int \frac {1}{x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\right )-\log (2) \int \frac {1}{\left (-1+2 e^{3+x}\right ) x \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx-\log (2) \int \frac {1}{x^2 \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx-(2 \log (2)) \int \frac {1}{x^2 \log \left (x^2\right ) \log ^2\left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 23, normalized size = 0.96 \begin {gather*} \frac {\log (2)}{x \log \left (\left (1-2 e^{3+x}\right ) \log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 23, normalized size = 0.96 \begin {gather*} \frac {\log \relax (2)}{x \log \left (-{\left (2 \, e^{\left (x + 3\right )} - 1\right )} \log \left (x^{2}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(2)*exp(3+x)+log(2))*log(x^2)*log((-2*exp(3+x)+1)*log(x^2))-2*x*log(2)*exp(3+x)*log(x^2)-4*l
og(2)*exp(3+x)+2*log(2))/(2*x^2*exp(3+x)-x^2)/log(x^2)/log((-2*exp(3+x)+1)*log(x^2))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.60, size = 24, normalized size = 1.00 \begin {gather*} \frac {\log \relax (2)}{x \log \left (-2 \, e^{\left (x + 3\right )} \log \left (x^{2}\right ) + \log \left (x^{2}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(2)*exp(3+x)+log(2))*log(x^2)*log((-2*exp(3+x)+1)*log(x^2))-2*x*log(2)*exp(3+x)*log(x^2)-4*l
og(2)*exp(3+x)+2*log(2))/(2*x^2*exp(3+x)-x^2)/log(x^2)/log((-2*exp(3+x)+1)*log(x^2))^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.26, size = 831, normalized size = 34.62




method result size



risch \(\frac {2 i \ln \relax (2)}{x \left (\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )+\pi \mathrm {csgn}\left (\left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (i \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{2}+\pi \mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{3}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{2}-\pi \mathrm {csgn}\left (\left ({\mathrm e}^{3+x}-\frac {1}{2}\right ) \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )^{3}-\pi +2 i \ln \left ({\mathrm e}^{3+x}-\frac {1}{2}\right )+2 i \ln \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right )}\) \(831\)



Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*ln(2)*exp(3+x)+ln(2))*ln(x^2)*ln((-2*exp(3+x)+1)*ln(x^2))-2*x*ln(2)*exp(3+x)*ln(x^2)-4*ln(2)*exp(3+x)
+2*ln(2))/(2*x^2*exp(3+x)-x^2)/ln(x^2)/ln((-2*exp(3+x)+1)*ln(x^2))^2,x,method=_RETURNVERBOSE)

[Out]

2*I*ln(2)/x/(Pi*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*cs
gn(I*x^2)^2))*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I
*x^2)^2))+Pi*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*
x^2)^2))^2+Pi*csgn(I*(exp(3+x)-1/2))*csgn(I*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I
*x)*csgn(I*x^2)^2))*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x
)*csgn(I*x^2)^2))-Pi*csgn(I*(exp(3+x)-1/2))*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*c
sgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2-Pi*csgn(I*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2
*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*P
i*csgn(I*x)*csgn(I*x^2)^2))^2+Pi*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-
2*Pi*csgn(I*x)*csgn(I*x^2)^2))^3-Pi*csgn(I*(exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^
2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2
*Pi*csgn(I*x)*csgn(I*x^2)^2))^2-Pi*csgn((exp(3+x)-1/2)*(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-
2*Pi*csgn(I*x)*csgn(I*x^2)^2))^3-Pi+2*I*ln(exp(3+x)-1/2)+2*I*ln(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn
(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))

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maxima [C]  time = 0.51, size = 30, normalized size = 1.25 \begin {gather*} \frac {\log \relax (2)}{{\left (i \, \pi + \log \relax (2)\right )} x + x \log \left (2 \, e^{\left (x + 3\right )} - 1\right ) + x \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(2)*exp(3+x)+log(2))*log(x^2)*log((-2*exp(3+x)+1)*log(x^2))-2*x*log(2)*exp(3+x)*log(x^2)-4*l
og(2)*exp(3+x)+2*log(2))/(2*x^2*exp(3+x)-x^2)/log(x^2)/log((-2*exp(3+x)+1)*log(x^2))^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,\ln \relax (2)-4\,{\mathrm {e}}^{x+3}\,\ln \relax (2)+\ln \left (-\ln \left (x^2\right )\,\left (2\,{\mathrm {e}}^{x+3}-1\right )\right )\,\ln \left (x^2\right )\,\left (\ln \relax (2)-2\,{\mathrm {e}}^{x+3}\,\ln \relax (2)\right )-2\,x\,\ln \left (x^2\right )\,{\mathrm {e}}^{x+3}\,\ln \relax (2)}{{\ln \left (-\ln \left (x^2\right )\,\left (2\,{\mathrm {e}}^{x+3}-1\right )\right )}^2\,\ln \left (x^2\right )\,\left (2\,x^2\,{\mathrm {e}}^{x+3}-x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.56, size = 19, normalized size = 0.79 \begin {gather*} \frac {\log {\relax (2 )}}{x \log {\left (\left (1 - 2 e^{x + 3}\right ) \log {\left (x^{2} \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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