Optimal. Leaf size=29 \[ 1-\left (e^3+\left (-x+\frac {4}{-5+3 x^3}\right )^2\right ) \log (3 x) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.58, antiderivative size = 772, normalized size of antiderivative = 26.62, number of steps
used = 43, number of rules used = 25, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.197, Rules used = {6820, 6874,
1843, 1848, 1889, 31, 648, 631, 210, 642, 2404, 2341, 2376, 272, 46, 205, 206, 2367, 2353, 2352,
2351, 2354, 2438, 27, 12} \begin {gather*} -\frac {48 i \sqrt [6]{3} \text {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \text {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \text {PolyLog}\left (2,(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \text {PolyLog}\left (2,-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}-\frac {8 \text {ArcTan}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{5}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6} 5^{2/3}}-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {16}{15 \left (5-3 x^3\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {16}{75} \log \left (5-3 x^3\right )-x^2 \log (3 x)+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (3^{2/3} x^2+\sqrt [3]{15} x+5^{2/3}\right )-\frac {8 x \log (3 x)}{\sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{-1} 3^{2/3} x+\sqrt [3]{15}\right )}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)+\frac {16 \log (x)}{25}+\frac {8 \sqrt [3]{-1} 3^{2/3} \log \left (\sqrt [3]{-3} x+\sqrt [3]{5}\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {8 \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rule 46
Rule 205
Rule 206
Rule 210
Rule 272
Rule 631
Rule 642
Rule 648
Rule 1843
Rule 1848
Rule 1889
Rule 2341
Rule 2351
Rule 2352
Rule 2353
Rule 2354
Rule 2367
Rule 2376
Rule 2404
Rule 2438
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-5+3 x^3\right ) \left (e^3 \left (5-3 x^3\right )^2+\left (4+5 x-3 x^4\right )^2\right )-2 x \left (100+125 x+144 x^2+60 x^3-225 x^4-72 x^6+135 x^7-27 x^{10}\right ) \log (3 x)}{x \left (5-3 x^3\right )^3} \, dx\\ &=\int \left (\frac {-16 \left (1+\frac {25 e^3}{16}\right )-40 x-25 x^2+30 e^3 x^3+24 x^4+30 x^5-9 e^3 x^6-9 x^8}{x \left (5-3 x^3\right )^2}-\frac {2 \left (-4-5 x+3 x^4\right ) \left (25+36 x^2-30 x^3+9 x^6\right ) \log (3 x)}{\left (-5+3 x^3\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-4-5 x+3 x^4\right ) \left (25+36 x^2-30 x^3+9 x^6\right ) \log (3 x)}{\left (-5+3 x^3\right )^3} \, dx\right )+\int \frac {-16 \left (1+\frac {25 e^3}{16}\right )-40 x-25 x^2+30 e^3 x^3+24 x^4+30 x^5-9 e^3 x^6-9 x^8}{x \left (5-3 x^3\right )^2} \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {1}{405} \int \frac {-81 \left (16+25 e^3\right )-3240 x-2025 x^2+1215 e^3 x^3+1215 x^5}{x \left (5-3 x^3\right )} \, dx-2 \int \left (x \log (3 x)-\frac {144 x^2 \log (3 x)}{\left (-5+3 x^3\right )^3}+\frac {60 \log (3 x)}{\left (-5+3 x^3\right )^2}+\frac {8 \log (3 x)}{-5+3 x^3}\right ) \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {1}{405} \int \left (-\frac {81 \left (16+25 e^3\right )}{5 x}-405 x+\frac {648 \left (25+6 x^2\right )}{5 \left (-5+3 x^3\right )}\right ) \, dx-2 \int x \log (3 x) \, dx-16 \int \frac {\log (3 x)}{-5+3 x^3} \, dx-120 \int \frac {\log (3 x)}{\left (-5+3 x^3\right )^2} \, dx+288 \int \frac {x^2 \log (3 x)}{\left (-5+3 x^3\right )^3} \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}+\frac {8}{25} \int \frac {25+6 x^2}{-5+3 x^3} \, dx+16 \int \frac {1}{x \left (-5+3 x^3\right )^2} \, dx-16 \int \left (-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}+\sqrt [3]{-3} x\right )}-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}-(-1)^{2/3} \sqrt [3]{3} x\right )}\right ) \, dx-120 \int \left (\frac {2 \log (3 x)}{15 \sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}+\frac {3 (-1)^{2/3} \sqrt [3]{\frac {3}{5}} \log (3 x)}{5 \left (1+\sqrt [3]{-1}\right )^4 \left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )^2}+\frac {6 (-1)^{5/6} \sqrt [6]{3} \log (3 x)}{5\ 5^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )}+\frac {3 \sqrt [3]{\frac {3}{5}} \log (3 x)}{5 \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x\right )^2}+\frac {4 \log (3 x)}{15 \sqrt [3]{3} 5^{2/3} \left (2\ 3^{2/3} \sqrt [3]{5}+3 \left (1-i \sqrt {3}\right ) x\right )}-\frac {\log (3 x)}{15\ 3^{2/3} \sqrt [3]{5} \left (-\sqrt [3]{3} 5^{2/3}+2\ 3^{2/3} \sqrt [3]{5} x-3 x^2\right )}\right ) \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}+\frac {16}{3} \text {Subst}\left (\int \frac {1}{x (-5+3 x)^2} \, dx,x,x^3\right )-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )^2} \, dx-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x\right )^2} \, dx+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}+\sqrt [3]{-3} x} \, dx}{3\ 5^{2/3}}+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}-\sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}-(-1)^{2/3} \sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}-\frac {8 \int \frac {\sqrt [3]{\frac {5}{3}} \left (50-2 \sqrt [3]{3} 5^{2/3}\right )+\left (25-4 \sqrt [3]{3} 5^{2/3}\right ) x}{\left (\frac {5}{3}\right )^{2/3}+\sqrt [3]{\frac {5}{3}} x+x^2} \, dx}{75 \sqrt [3]{3} 5^{2/3}}-\frac {16 \int \frac {\log (3 x)}{3^{2/3} \sqrt [3]{5}-3 x} \, dx}{\sqrt [3]{3} 5^{2/3}}-\frac {32 \int \frac {\log (3 x)}{2\ 3^{2/3} \sqrt [3]{5}+3 \left (1-i \sqrt {3}\right ) x} \, dx}{\sqrt [3]{3} 5^{2/3}}+\frac {8 \int \frac {\log (3 x)}{-\sqrt [3]{3} 5^{2/3}+2\ 3^{2/3} \sqrt [3]{5} x-3 x^2} \, dx}{3^{2/3} \sqrt [3]{5}}-\frac {\left (144 (-1)^{5/6} \sqrt [6]{3}\right ) \int \frac {\log (3 x)}{3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x} \, dx}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {1}{225} \left (8 \left (6+5\ 3^{2/3} \sqrt [3]{5}\right )\right ) \int \frac {1}{\sqrt [3]{\frac {5}{3}}-x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 35, normalized size = 1.21 \begin {gather*} -e^3 \log (x)-\frac {\left (-4-5 x+3 x^4\right )^2 \log (3 x)}{\left (-5+3 x^3\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. \(65\) vs.
\(2(28)=56\).
time = 1.01, size = 66, normalized size = 2.28
method | result | size |
risch | \(-\frac {\left (9 x^{8}-30 x^{5}-24 x^{4}+25 x^{2}+40 x +16\right ) \ln \left (3 x \right )}{9 x^{6}-30 x^{3}+25}-\ln \left (x \right ) {\mathrm e}^{3}\) | \(53\) |
derivativedivides | \(-\ln \left (3 x \right ) {\mathrm e}^{3}-\frac {16 \ln \left (3 x \right )}{25}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(66\) |
default | \(-\ln \left (3 x \right ) {\mathrm e}^{3}-\frac {16 \ln \left (3 x \right )}{25}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(66\) |
norman | \(\frac {-16 \ln \left (3 x \right )-40 x \ln \left (3 x \right )-25 x^{2} \ln \left (3 x \right )+24 \ln \left (3 x \right ) x^{4}+30 \ln \left (3 x \right ) x^{5}-9 \ln \left (3 x \right ) x^{8}}{\left (3 x^{3}-5\right )^{2}}-\ln \left (x \right ) {\mathrm e}^{3}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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. 60 vs.
\(2 (26) = 52\).
time = 0.36, size = 60, normalized size = 2.07 \begin {gather*} -\frac {{\left (9 \, x^{8} - 30 \, x^{5} - 24 \, x^{4} + 25 \, x^{2} + {\left (9 \, x^{6} - 30 \, x^{3} + 25\right )} e^{3} + 40 \, x + 16\right )} \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \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. 48 vs.
\(2 (20) = 40\).
time = 0.16, size = 48, normalized size = 1.66 \begin {gather*} - e^{3} \log {\left (x \right )} + \frac {\left (- 9 x^{8} + 30 x^{5} + 24 x^{4} - 25 x^{2} - 40 x - 16\right ) \log {\left (3 x \right )}}{9 x^{6} - 30 x^{3} + 25} \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. 90 vs.
\(2 (26) = 52\).
time = 0.43, size = 90, normalized size = 3.10 \begin {gather*} -\frac {9 \, x^{8} \log \left (3 \, x\right ) + 9 \, x^{6} e^{3} \log \left (x\right ) - 30 \, x^{5} \log \left (3 \, x\right ) - 24 \, x^{4} \log \left (3 \, x\right ) - 30 \, x^{3} e^{3} \log \left (x\right ) + 25 \, x^{2} \log \left (3 \, x\right ) + 40 \, x \log \left (3 \, x\right ) + 25 \, e^{3} \log \left (x\right ) + 16 \, \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.65, size = 48, normalized size = 1.66 \begin {gather*} -{\mathrm {e}}^3\,\ln \left (x\right )-\frac {\ln \left (3\,x\right )\,\left (x^8-\frac {10\,x^5}{3}-\frac {8\,x^4}{3}+\frac {25\,x^2}{9}+\frac {40\,x}{9}+\frac {16}{9}\right )}{x^6-\frac {10\,x^3}{3}+\frac {25}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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