Optimal. Leaf size=24 \[ \frac {4 \left (4-x^2 \log ^2(x)\right )}{\frac {4}{5}+e^{4 x}} \]
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Rubi [F] time = 1.25, 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 {-1600 e^{4 x}+\left (-160 x-200 e^{4 x} x\right ) \log (x)+\left (-160 x+e^{4 x} \left (-200 x+400 x^2\right )\right ) \log ^2(x)}{16+40 e^{4 x}+25 e^{8 x}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1600 e^{4 x}+\left (-160 x-200 e^{4 x} x\right ) \log (x)+\left (-160 x+e^{4 x} \left (-200 x+400 x^2\right )\right ) \log ^2(x)}{\left (4+5 e^{4 x}\right )^2} \, dx\\ &=\int \left (-\frac {320 \left (-4+x^2 \log ^2(x)\right )}{\left (4+5 e^{4 x}\right )^2}+\frac {40 \left (-8-x \log (x)-x \log ^2(x)+2 x^2 \log ^2(x)\right )}{4+5 e^{4 x}}\right ) \, dx\\ &=40 \int \frac {-8-x \log (x)-x \log ^2(x)+2 x^2 \log ^2(x)}{4+5 e^{4 x}} \, dx-320 \int \frac {-4+x^2 \log ^2(x)}{\left (4+5 e^{4 x}\right )^2} \, dx\\ &=40 \int \left (-\frac {8}{4+5 e^{4 x}}-\frac {x \log (x)}{4+5 e^{4 x}}-\frac {x \log ^2(x)}{4+5 e^{4 x}}+\frac {2 x^2 \log ^2(x)}{4+5 e^{4 x}}\right ) \, dx-320 \int \left (-\frac {4}{\left (4+5 e^{4 x}\right )^2}+\frac {x^2 \log ^2(x)}{\left (4+5 e^{4 x}\right )^2}\right ) \, dx\\ &=-\left (40 \int \frac {x \log (x)}{4+5 e^{4 x}} \, dx\right )-40 \int \frac {x \log ^2(x)}{4+5 e^{4 x}} \, dx+80 \int \frac {x^2 \log ^2(x)}{4+5 e^{4 x}} \, dx-320 \int \frac {1}{4+5 e^{4 x}} \, dx-320 \int \frac {x^2 \log ^2(x)}{\left (4+5 e^{4 x}\right )^2} \, dx+1280 \int \frac {1}{\left (4+5 e^{4 x}\right )^2} \, dx\\ &=-\left (40 \int \frac {x \log (x)}{4+5 e^{4 x}} \, dx\right )-40 \int \frac {x \log ^2(x)}{4+5 e^{4 x}} \, dx+80 \int \frac {x^2 \log ^2(x)}{4+5 e^{4 x}} \, dx-80 \operatorname {Subst}\left (\int \frac {1}{x (4+5 x)} \, dx,x,e^{4 x}\right )-320 \int \frac {x^2 \log ^2(x)}{\left (4+5 e^{4 x}\right )^2} \, dx+320 \operatorname {Subst}\left (\int \frac {1}{x (4+5 x)^2} \, dx,x,e^{4 x}\right )\\ &=-\left (20 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{4 x}\right )\right )-40 \int \frac {x \log (x)}{4+5 e^{4 x}} \, dx-40 \int \frac {x \log ^2(x)}{4+5 e^{4 x}} \, dx+80 \int \frac {x^2 \log ^2(x)}{4+5 e^{4 x}} \, dx+100 \operatorname {Subst}\left (\int \frac {1}{4+5 x} \, dx,x,e^{4 x}\right )-320 \int \frac {x^2 \log ^2(x)}{\left (4+5 e^{4 x}\right )^2} \, dx+320 \operatorname {Subst}\left (\int \left (\frac {1}{16 x}-\frac {5}{4 (4+5 x)^2}-\frac {5}{16 (4+5 x)}\right ) \, dx,x,e^{4 x}\right )\\ &=\frac {80}{4+5 e^{4 x}}-40 \int \frac {x \log (x)}{4+5 e^{4 x}} \, dx-40 \int \frac {x \log ^2(x)}{4+5 e^{4 x}} \, dx+80 \int \frac {x^2 \log ^2(x)}{4+5 e^{4 x}} \, dx-320 \int \frac {x^2 \log ^2(x)}{\left (4+5 e^{4 x}\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.25, size = 24, normalized size = 1.00 \begin {gather*} \frac {40 \left (4-x^2 \log ^2(x)\right )}{8+10 e^{4 x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 22, normalized size = 0.92 \begin {gather*} -\frac {20 \, {\left (x^{2} \log \relax (x)^{2} - 4\right )}}{5 \, e^{\left (4 \, x\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 22, normalized size = 0.92 \begin {gather*} -\frac {20 \, {\left (x^{2} \log \relax (x)^{2} - 4\right )}}{5 \, e^{\left (4 \, x\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 1.38
method | result | size |
risch | \(-\frac {20 x^{2} \ln \relax (x )^{2}}{5 \,{\mathrm e}^{4 x}+4}+\frac {80}{5 \,{\mathrm e}^{4 x}+4}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 32, normalized size = 1.33 \begin {gather*} -\frac {20 \, x^{2} \log \relax (x)^{2}}{5 \, e^{\left (4 \, x\right )} + 4} + \frac {80}{5 \, e^{\left (4 \, x\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 22, normalized size = 0.92 \begin {gather*} -\frac {20\,\left (x^2\,{\ln \relax (x)}^2-4\right )}{5\,{\mathrm {e}}^{4\,x}+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 19, normalized size = 0.79 \begin {gather*} \frac {- 4 x^{2} \log {\relax (x )}^{2} + 16}{e^{4 x} + \frac {4}{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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