Optimal. Leaf size=29 \[ \frac {2 x}{5 \log \left (\frac {1}{4} e^{9+\frac {e^3 x}{16 (4+x)}}\right )} \]
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Rubi [F]
time = 0.90, 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 {-e^3 x+\left (64+32 x+4 x^2\right ) \log \left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )}{\left (160+80 x+10 x^2\right ) \log ^2\left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^3 x+\left (64+32 x+4 x^2\right ) \log \left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )}{10 (4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )} \, dx\\ &=\frac {1}{10} \int \frac {-e^3 x+\left (64+32 x+4 x^2\right ) \log \left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )}{(4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )} \, dx\\ &=\frac {1}{10} \int \frac {-e^3 x+\left (64+32 x+4 x^2\right ) \log \left (\frac {1}{4} e^{\frac {576+144 x+e^3 x}{64+16 x}}\right )}{(4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{64+16 x}}\right )} \, dx\\ &=\frac {1}{10} \int \left (-\frac {e^3 x}{(4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {64}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {32 x}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {4 x^2}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}\right ) \, dx\\ &=\frac {2}{5} \int \frac {x^2}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx+\frac {16}{5} \int \frac {x}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx+\frac {32}{5} \int \frac {1}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx-\frac {1}{10} e^3 \int \frac {x}{(4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx\\ &=\frac {128 \log \left (\log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )\right )}{5 e^3}+\frac {2}{5} \int \left (\frac {1}{\log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {16}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}-\frac {8}{(4+x) \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}\right ) \, dx+\frac {16}{5} \int \left (-\frac {4}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {1}{(4+x) \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}\right ) \, dx-\frac {1}{10} e^3 \int \left (-\frac {4}{(4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {1}{(4+x) \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}\right ) \, dx\\ &=\frac {128 \log \left (\log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )\right )}{5 e^3}+\frac {2}{5} \int \frac {1}{\log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx+\frac {32}{5} \int \frac {1}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx-\frac {64}{5} \int \frac {1}{(4+x)^2 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx-\frac {1}{10} e^3 \int \frac {1}{(4+x) \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx+\frac {1}{5} \left (2 e^3\right ) \int \frac {1}{(4+x)^2 \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx\\ &=-\frac {8}{5 \log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )}+\frac {2}{5} \int \frac {1}{\log \left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx-\frac {1}{10} e^3 \int \frac {1}{(4+x) \log ^2\left (\frac {1}{4} e^{\frac {576+\left (144+e^3\right ) x}{16 (4+x)}}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(29)=58\).
time = 0.10, size = 127, normalized size = 4.38 \begin {gather*} \frac {2 \left (e^6 x+4 e^3 \left (-16+x^2\right ) \log \left (\frac {1}{4} e^{9+\frac {e^3 x}{64+16 x}}\right )+16 x (4+x)^2 \log ^2\left (\frac {1}{4} e^{9+\frac {e^3 x}{64+16 x}}\right )\right )}{5 \log \left (\frac {1}{4} e^{9+\frac {e^3 x}{64+16 x}}\right ) \left (e^3+4 (4+x) \log \left (\frac {1}{4} e^{9+\frac {e^3 x}{64+16 x}}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1295\) vs.
\(2(22)=44\).
time = 2.18, size = 1296, normalized size = 44.69
method | result | size |
risch | \(\frac {4 i x}{5 \left (-4 i \ln \left (2\right )+2 i \ln \left ({\mathrm e}^{\frac {x \,{\mathrm e}^{3}+144 x +576}{16 x +64}}\right )\right )}\) | \(34\) |
norman | \(\frac {\frac {8}{5} x +\frac {2}{5} x^{2}}{\left (4+x \right ) \ln \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{3}+144 x +576}{16 x +64}}}{4}\right )}\) | \(39\) |
default | \(\text {Expression too large to display}\) | \(1296\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1184 vs.
\(2 (21) = 42\).
time = 0.57, size = 1184, normalized size = 40.83 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs.
\(2 (21) = 42\).
time = 0.38, size = 150, normalized size = 5.17 \begin {gather*} -\frac {32 \, {\left (x^{2} e^{6} + 1024 \, {\left (x^{2} + 4 \, x\right )} \log \left (2\right )^{2} + 20736 \, x^{2} + 288 \, {\left (x^{2} + 2 \, x - 8\right )} e^{3} - 64 \, {\left (144 \, x^{2} + {\left (x^{2} + 2 \, x - 8\right )} e^{3} + 576 \, x\right )} \log \left (2\right ) + 82944 \, x\right )}}{5 \, {\left (32768 \, {\left (x + 4\right )} \log \left (2\right )^{3} - 1024 \, {\left ({\left (3 \, x + 8\right )} e^{3} + 432 \, x + 1728\right )} \log \left (2\right )^{2} - x e^{9} - 144 \, {\left (3 \, x + 4\right )} e^{6} - 20736 \, {\left (3 \, x + 8\right )} e^{3} + 32 \, {\left ({\left (3 \, x + 4\right )} e^{6} + 288 \, {\left (3 \, x + 8\right )} e^{3} + 62208 \, x + 248832\right )} \log \left (2\right ) - 2985984 \, x - 11943936\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs.
\(2 (20) = 40\).
time = 0.60, size = 150, normalized size = 5.17 \begin {gather*} \frac {32 x}{- 160 \log {\left (2 \right )} + 5 e^{3} + 720} + \frac {- 73728 e^{3} + 16384 e^{3} \log {\left (2 \right )}}{x \left (- 9953280 \log {\left (2 \right )} - 138240 e^{3} \log {\left (2 \right )} - 480 e^{6} \log {\left (2 \right )} - 163840 \log {\left (2 \right )}^{3} + 5 e^{9} + 15360 e^{3} \log {\left (2 \right )}^{2} + 2160 e^{6} + 2211840 \log {\left (2 \right )}^{2} + 311040 e^{3} + 14929920\right ) - 39813120 \log {\left (2 \right )} - 368640 e^{3} \log {\left (2 \right )} - 655360 \log {\left (2 \right )}^{3} - 640 e^{6} \log {\left (2 \right )} + 40960 e^{3} \log {\left (2 \right )}^{2} + 2880 e^{6} + 8847360 \log {\left (2 \right )}^{2} + 829440 e^{3} + 59719680} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (21) = 42\).
time = 0.39, size = 87, normalized size = 3.00 \begin {gather*} -\frac {32 \, {\left (x e^{3} - 32 \, x \log \left (2\right ) + 144 \, x\right )}}{5 \, {\left (64 \, e^{3} \log \left (2\right ) - 1024 \, \log \left (2\right )^{2} - e^{6} - 288 \, e^{3} + 9216 \, \log \left (2\right ) - 20736\right )}} + \frac {8192 \, {\left (2 \, e^{3} \log \left (2\right ) - 9 \, e^{3}\right )}}{5 \, {\left (x e^{3} - 32 \, x \log \left (2\right ) + 144 \, x - 128 \, \log \left (2\right ) + 576\right )} {\left (e^{3} - 32 \, \log \left (2\right ) + 144\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.82, size = 113, normalized size = 3.90 \begin {gather*} \frac {32\,\left (82944\,x-2304\,{\mathrm {e}}^3+1024\,x^2\,{\ln \left (2\right )}^2+512\,{\mathrm {e}}^3\,\ln \left (2\right )+576\,x\,{\mathrm {e}}^3-36864\,x\,\ln \left (2\right )+288\,x^2\,{\mathrm {e}}^3+x^2\,{\mathrm {e}}^6+4096\,x\,{\ln \left (2\right )}^2-9216\,x^2\,\ln \left (2\right )+20736\,x^2-64\,x^2\,{\mathrm {e}}^3\,\ln \left (2\right )-128\,x\,{\mathrm {e}}^3\,\ln \left (2\right )\right )}{5\,{\left ({\mathrm {e}}^3-32\,\ln \left (2\right )+144\right )}^2\,\left (144\,x-128\,\ln \left (2\right )+x\,{\mathrm {e}}^3-32\,x\,\ln \left (2\right )+576\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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