3.100.80 \(\int \frac {-4 x \log (x)+(e^2+x) \log (\frac {3}{e^4+2 e^2 x+x^2})}{(e^2 x+x^2) \log (x) \log (\frac {3}{e^4+2 e^2 x+x^2}) \log (\frac {1}{5} \log (x) \log ^2(\frac {3}{e^4+2 e^2 x+x^2}))} \, dx\)

Optimal. Leaf size=24 \[ 3+\log \left (2 \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )\right ) \]

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Rubi [F]  time = 3.17, 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 {-4 x \log (x)+\left (e^2+x\right ) \log \left (\frac {3}{e^4+2 e^2 x+x^2}\right )}{\left (e^2 x+x^2\right ) \log (x) \log \left (\frac {3}{e^4+2 e^2 x+x^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{e^4+2 e^2 x+x^2}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][1/(x*Log[x]*Log[(Log[x]*Log[3/(E^2 + x)^2]^2)/5]), x] - 4*Defer[Int][1/((E^2 + x)*Log[3/(E^2 + x)^2
]*Log[(Log[x]*Log[3/(E^2 + x)^2]^2)/5]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x \log (x)+\left (e^2+x\right ) \log \left (\frac {3}{e^4+2 e^2 x+x^2}\right )}{x \left (e^2+x\right ) \log (x) \log \left (\frac {3}{e^4+2 e^2 x+x^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{e^4+2 e^2 x+x^2}\right )\right )} \, dx\\ &=\int \frac {-4 x \log (x)+\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{x \left (e^2+x\right ) \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx\\ &=\int \left (\frac {4 x \log (x)-e^2 \log \left (\frac {3}{\left (e^2+x\right )^2}\right )-x \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{e^2 \left (e^2+x\right ) \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}+\frac {-4 x \log (x)+e^2 \log \left (\frac {3}{\left (e^2+x\right )^2}\right )+x \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{e^2 x \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}\right ) \, dx\\ &=\frac {\int \frac {4 x \log (x)-e^2 \log \left (\frac {3}{\left (e^2+x\right )^2}\right )-x \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{\left (e^2+x\right ) \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}+\frac {\int \frac {-4 x \log (x)+e^2 \log \left (\frac {3}{\left (e^2+x\right )^2}\right )+x \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{x \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}\\ &=\frac {\int \frac {4 x \log (x)-\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{\left (e^2+x\right ) \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}+\frac {\int \frac {-4 x \log (x)+\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right )}{x \log (x) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}\\ &=\frac {\int \left (\frac {1}{\log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}+\frac {e^2}{x \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}-\frac {4}{\log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}\right ) \, dx}{e^2}+\frac {\int \left (-\frac {e^2}{\left (e^2+x\right ) \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}-\frac {x}{\left (e^2+x\right ) \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}+\frac {4 x}{\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}\right ) \, dx}{e^2}\\ &=\frac {\int \frac {1}{\log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}-\frac {\int \frac {x}{\left (e^2+x\right ) \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}-\frac {4 \int \frac {1}{\log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}+\frac {4 \int \frac {x}{\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}+\int \frac {1}{x \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx-\int \frac {1}{\left (e^2+x\right ) \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx\\ &=-\frac {\int \left (\frac {1}{\log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}-\frac {e^2}{\left (e^2+x\right ) \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}\right ) \, dx}{e^2}+\frac {\int \frac {1}{\log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}+\frac {4 \int \left (\frac {1}{\log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}-\frac {e^2}{\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )}\right ) \, dx}{e^2}-\frac {4 \int \frac {1}{\log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx}{e^2}+\int \frac {1}{x \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx-\int \frac {1}{\left (e^2+x\right ) \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx\\ &=-\left (4 \int \frac {1}{\left (e^2+x\right ) \log \left (\frac {3}{\left (e^2+x\right )^2}\right ) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx\right )+\int \frac {1}{x \log (x) \log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 20, normalized size = 0.83 \begin {gather*} \log \left (\log \left (\frac {1}{5} \log (x) \log ^2\left (\frac {3}{\left (e^2+x\right )^2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[(Log[x]*Log[3/(E^2 + x)^2]^2)/5]]

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fricas [A]  time = 1.04, size = 24, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\frac {1}{5} \, \log \relax (x) \log \left (\frac {3}{x^{2} + 2 \, x e^{2} + e^{4}}\right )^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)+(x+exp(2))*log(3/(exp(2)^2+2*exp(2)*x+x^2)))/(exp(2)*x+x^2)/log(3/(exp(2)^2+2*exp(2)*x+
x^2))/log(x)/log(1/5*log(3/(exp(2)^2+2*exp(2)*x+x^2))^2*log(x)),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 3.44, size = 50, normalized size = 2.08 \begin {gather*} \log \left (-\log \relax (5) + \log \left (\log \relax (3)^{2} \log \relax (x) - 2 \, \log \relax (3) \log \left (x^{2} + 2 \, x e^{2} + e^{4}\right ) \log \relax (x) + \log \left (x^{2} + 2 \, x e^{2} + e^{4}\right )^{2} \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)+(x+exp(2))*log(3/(exp(2)^2+2*exp(2)*x+x^2)))/(exp(2)*x+x^2)/log(3/(exp(2)^2+2*exp(2)*x+
x^2))/log(x)/log(1/5*log(3/(exp(2)^2+2*exp(2)*x+x^2))^2*log(x)),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.90, size = 1127, normalized size = 46.96

method	result	size
risch	\(\ln \left (\ln \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )-\pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )^{2}+2 \pi \mathrm {csgn}\left (i \ln \relax (x ) \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \relax (x ) \left (-4 i \ln \left (x +{\mathrm e}^{2}\right )+2 i \ln \relax (3)-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{2}\right )^{2}\right )^{2}\right )^{2}\right )^{3}-4 i \ln \relax (2)-2 i \ln \relax (5)+2 i \ln \left (\ln \relax (x )\right )-2 \pi \right )}{4}\right )\)	\(1127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*ln(x)+(x+exp(2))*ln(3/(exp(2)^2+2*exp(2)*x+x^2)))/(exp(2)*x+x^2)/ln(3/(exp(2)^2+2*exp(2)*x+x^2))/ln(
x)/ln(1/5*ln(3/(exp(2)^2+2*exp(2)*x+x^2))^2*ln(x)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*
csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)-1/4*I*(Pi*csgn(I*ln(x))*csgn(I*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn
(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2
)^2)*csgn(I*ln(x)*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp
(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)-Pi*csgn(I*ln(x))*csgn(I*ln(x)*(-4*I*ln(x+exp(2))+2*
I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*
(x+exp(2))^2)^2)^2)^2+2*Pi*csgn(I*ln(x)*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+ex
p(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)^2+Pi*csgn(I*(-4*I*ln(x+exp(2)
)+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csg
n(I*(x+exp(2))^2)^2))^2*csgn(I*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*
csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)-2*Pi*csgn(I*(-4*I*ln(x+exp(2))+2*I*ln(
3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+ex
p(2))^2)^2))*csgn(I*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+e
xp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)^2+Pi*csgn(I*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(
I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)
^2)^3-Pi*csgn(I*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2
))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)*csgn(I*ln(x)*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(
x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+exp(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)
^2-Pi*csgn(I*ln(x)*(-4*I*ln(x+exp(2))+2*I*ln(3)-Pi*csgn(I*(x+exp(2))^2)^3-Pi*csgn(I*(x+exp(2)))^2*csgn(I*(x+ex
p(2))^2)+2*Pi*csgn(I*(x+exp(2)))*csgn(I*(x+exp(2))^2)^2)^2)^3-4*I*ln(2)-2*I*ln(5)+2*I*ln(ln(x))-2*Pi))

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maxima [A]  time = 0.53, size = 24, normalized size = 1.00 \begin {gather*} \log \left (-\frac {1}{2} \, \log \relax (5) + \log \left (-\log \relax (3) + 2 \, \log \left (x + e^{2}\right )\right ) + \frac {1}{2} \, \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)+(x+exp(2))*log(3/(exp(2)^2+2*exp(2)*x+x^2)))/(exp(2)*x+x^2)/log(3/(exp(2)^2+2*exp(2)*x+
x^2))/log(x)/log(1/5*log(3/(exp(2)^2+2*exp(2)*x+x^2))^2*log(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 10.52, size = 24, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (\frac {{\ln \left (\frac {3}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}\right )}^2\,\ln \relax (x)}{5}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3/(exp(4) + 2*x*exp(2) + x^2))*(x + exp(2)) - 4*x*log(x))/(log(3/(exp(4) + 2*x*exp(2) + x^2))*log((lo
g(3/(exp(4) + 2*x*exp(2) + x^2))^2*log(x))/5)*log(x)*(x*exp(2) + x^2)),x)

[Out]

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

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sympy [A]  time = 0.75, size = 26, normalized size = 1.08 \begin {gather*} \log {\left (\log {\left (\frac {\log {\relax (x )} \log {\left (\frac {3}{x^{2} + 2 x e^{2} + e^{4}} \right )}^{2}}{5} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*ln(x)+(x+exp(2))*ln(3/(exp(2)**2+2*exp(2)*x+x**2)))/(exp(2)*x+x**2)/ln(3/(exp(2)**2+2*exp(2)*x
+x**2))/ln(x)/ln(1/5*ln(3/(exp(2)**2+2*exp(2)*x+x**2))**2*ln(x)),x)

[Out]

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

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