Optimal. Leaf size=28 \[ 1-\left (2+(5+x)^2+\log \left (\frac {e^x (3-x)}{3+x}\right )\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(28)=56\).
time = 0.37, antiderivative size = 107, normalized size of antiderivative = 3.82, number of steps
used = 32, number of rules used = 18, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {6857, 213,
266, 327, 272, 45, 308, 1824, 2628, 455, 2631, 470, 212, 2632, 12, 396, 6021, 6095}
\begin {gather*} -x^4-20 x^3-153 x^2-2 x^2 \log \left (\frac {e^x (3-x)}{x+3}\right )-594 x-22 x \log \left (\frac {e^x (3-x)}{x+3}\right )-4 x \tanh ^{-1}\left (\frac {x}{3}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right )^2+108 \tanh ^{-1}\left (\frac {x}{3}\right )+4 \log \left (\frac {e^x (3-x)}{x+3}\right ) \tanh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 212
Rule 213
Rule 266
Rule 272
Rule 308
Rule 327
Rule 396
Rule 455
Rule 470
Rule 1824
Rule 2628
Rule 2631
Rule 2632
Rule 6021
Rule 6095
Rule 6857
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5022}{-9+x^2}+\frac {2832 x}{-9+x^2}-\frac {48 x^2}{-9+x^2}-\frac {292 x^3}{-9+x^2}-\frac {62 x^4}{-9+x^2}-\frac {4 x^5}{-9+x^2}-\frac {2 \left (-93-18 x+11 x^2+2 x^3\right ) \log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-93-18 x+11 x^2+2 x^3\right ) \log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2} \, dx\right )-4 \int \frac {x^5}{-9+x^2} \, dx-48 \int \frac {x^2}{-9+x^2} \, dx-62 \int \frac {x^4}{-9+x^2} \, dx-292 \int \frac {x^3}{-9+x^2} \, dx+2832 \int \frac {x}{-9+x^2} \, dx+5022 \int \frac {1}{-9+x^2} \, dx\\ &=-48 x-1674 \tanh ^{-1}\left (\frac {x}{3}\right )+1416 \log \left (9-x^2\right )-2 \int \left (11 \log \left (-\frac {e^x (-3+x)}{3+x}\right )+2 x \log \left (-\frac {e^x (-3+x)}{3+x}\right )+\frac {6 \log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2}\right ) \, dx-2 \text {Subst}\left (\int \frac {x^2}{-9+x} \, dx,x,x^2\right )-62 \int \left (9+x^2+\frac {81}{-9+x^2}\right ) \, dx-146 \text {Subst}\left (\int \frac {x}{-9+x} \, dx,x,x^2\right )-432 \int \frac {1}{-9+x^2} \, dx\\ &=-606 x-\frac {62 x^3}{3}-1530 \tanh ^{-1}\left (\frac {x}{3}\right )+1416 \log \left (9-x^2\right )-2 \text {Subst}\left (\int \left (9+\frac {81}{-9+x}+x\right ) \, dx,x,x^2\right )-4 \int x \log \left (-\frac {e^x (-3+x)}{3+x}\right ) \, dx-12 \int \frac {\log \left (-\frac {e^x (-3+x)}{3+x}\right )}{-9+x^2} \, dx-22 \int \log \left (-\frac {e^x (-3+x)}{3+x}\right ) \, dx-146 \text {Subst}\left (\int \left (1+\frac {9}{-9+x}\right ) \, dx,x,x^2\right )-5022 \int \frac {1}{-9+x^2} \, dx\\ &=-606 x-164 x^2-\frac {62 x^3}{3}-x^4+144 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )-60 \log \left (9-x^2\right )+2 \int \frac {x^2 \left (3-x^2\right )}{9-x^2} \, dx+12 \int \frac {\left (-3+x^2\right ) \tanh ^{-1}\left (\frac {x}{3}\right )}{3 \left (9-x^2\right )} \, dx+22 \int \frac {x \left (3-x^2\right )}{9-x^2} \, dx\\ &=-606 x-164 x^2-20 x^3-x^4+144 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )-60 \log \left (9-x^2\right )+4 \int \frac {\left (-3+x^2\right ) \tanh ^{-1}\left (\frac {x}{3}\right )}{9-x^2} \, dx+11 \text {Subst}\left (\int \frac {3-x}{9-x} \, dx,x,x^2\right )-12 \int \frac {x^2}{9-x^2} \, dx\\ &=-594 x-164 x^2-20 x^3-x^4+144 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )-60 \log \left (9-x^2\right )+4 \int \left (-\tanh ^{-1}\left (\frac {x}{3}\right )-\frac {6 \tanh ^{-1}\left (\frac {x}{3}\right )}{-9+x^2}\right ) \, dx+11 \text {Subst}\left (\int \left (1+\frac {6}{-9+x}\right ) \, dx,x,x^2\right )-108 \int \frac {1}{9-x^2} \, dx\\ &=-594 x-153 x^2-20 x^3-x^4+108 \tanh ^{-1}\left (\frac {x}{3}\right )-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )+6 \log \left (9-x^2\right )-4 \int \tanh ^{-1}\left (\frac {x}{3}\right ) \, dx-24 \int \frac {\tanh ^{-1}\left (\frac {x}{3}\right )}{-9+x^2} \, dx\\ &=-594 x-153 x^2-20 x^3-x^4+108 \tanh ^{-1}\left (\frac {x}{3}\right )-4 x \tanh ^{-1}\left (\frac {x}{3}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right )^2-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )+6 \log \left (9-x^2\right )+\frac {4}{3} \int \frac {x}{1-\frac {x^2}{9}} \, dx\\ &=-594 x-153 x^2-20 x^3-x^4+108 \tanh ^{-1}\left (\frac {x}{3}\right )-4 x \tanh ^{-1}\left (\frac {x}{3}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right )^2-22 x \log \left (\frac {e^x (3-x)}{3+x}\right )-2 x^2 \log \left (\frac {e^x (3-x)}{3+x}\right )+4 \tanh ^{-1}\left (\frac {x}{3}\right ) \log \left (\frac {e^x (3-x)}{3+x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(130\) vs. \(2(28)=56\).
time = 0.10, size = 130, normalized size = 4.64 \begin {gather*} -594 x-153 x^2-20 x^3-x^4+84 \log (3-x)-\log ^2\left (\frac {-3+x}{3+x}\right )-4 \tanh ^{-1}\left (\frac {x}{3}\right ) \left (-36+x+\log \left (\frac {-3+x}{3+x}\right )-\log \left (-\frac {e^x (-3+x)}{3+x}\right )\right )-22 x \log \left (-\frac {e^x (-3+x)}{3+x}\right )-2 x^2 \log \left (-\frac {e^x (-3+x)}{3+x}\right )+48 \log (3+x)-66 \log \left (9-x^2\right ) \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. \(171\) vs.
\(2(27)=54\).
time = 1.72, size = 172, normalized size = 6.14
method | result | size |
default | \(-x^{4}-20 x^{3}-153 x^{2}-594 x -48 \ln \left (x -3\right )+60 \ln \left (3+x \right )-2 \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right ) x^{2}+2 \ln \left (3+x \right ) \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right )-2 \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right ) \ln \left (x -3\right )-22 \ln \left (\frac {\left (3-x \right ) {\mathrm e}^{x}}{3+x}\right ) x +2 \left (x -3\right ) \ln \left (x -3\right )+12-2 \ln \left (x -3\right ) \ln \left (\frac {x}{6}+\frac {1}{2}\right )+\ln \left (x -3\right )^{2}-2 \left (3+x \right ) \ln \left (3+x \right )-2 \left (\ln \left (3+x \right )-\ln \left (\frac {x}{6}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x}{6}+\frac {1}{2}\right )+\ln \left (3+x \right )^{2}\) | \(172\) |
risch | \(\text {Expression too large to display}\) | \(1195\) |
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. 160 vs.
\(2 (26) = 52\).
time = 0.45, size = 160, normalized size = 5.71 \begin {gather*} -x^{4} - 22 \, x^{3} - 175 \, x^{2} + {\left (2 \, x^{2} + 22 \, x - 51\right )} \log \left (x + 3\right ) + 31 \, {\left (x + \log \left (x - 3\right ) + 3\right )} \log \left (x + 3\right ) - 32 \, \log \left (x + 3\right )^{2} - 31 \, {\left (x - 3\right )} \log \left (x - 3\right ) - \frac {31}{2} \, \log \left (x - 3\right )^{2} - {\left (2 \, x^{2} + 22 \, x - 33 \, \log \left (x + 3\right ) + 15\right )} \log \left (-x + 3\right ) - \frac {33}{2} \, \log \left (-x + 3\right )^{2} - 31 \, {\left (\log \left (x + 3\right ) - \log \left (x - 3\right )\right )} \log \left (-\frac {x e^{x}}{x + 3} + \frac {3 \, e^{x}}{x + 3}\right ) - 594 \, x - 60 \, \log \left (x^{2} - 9\right ) + 72 \, \log \left (x + 3\right ) - 72 \, \log \left (x - 3\right ) \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. 59 vs.
\(2 (26) = 52\).
time = 0.35, size = 59, normalized size = 2.11 \begin {gather*} -x^{4} - 20 \, x^{3} - 154 \, x^{2} - 2 \, {\left (x^{2} + 10 \, x + 27\right )} \log \left (-\frac {{\left (x - 3\right )} e^{x}}{x + 3}\right ) - \log \left (-\frac {{\left (x - 3\right )} e^{x}}{x + 3}\right )^{2} - 540 \, x \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. 65 vs.
\(2 (20) = 40\).
time = 0.13, size = 65, normalized size = 2.32 \begin {gather*} - x^{4} - 20 x^{3} - 154 x^{2} - 594 x + \left (- 2 x^{2} - 20 x\right ) \log {\left (\frac {\left (3 - x\right ) e^{x}}{x + 3} \right )} - \log {\left (\frac {\left (3 - x\right ) e^{x}}{x + 3} \right )}^{2} - 54 \log {\left (x - 3 \right )} + 54 \log {\left (x + 3 \right )} \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. 66 vs.
\(2 (26) = 52\).
time = 0.39, size = 66, normalized size = 2.36 \begin {gather*} -x^{4} - 22 \, x^{3} - 175 \, x^{2} - 2 \, {\left (x^{2} + 11 \, x\right )} \log \left (-\frac {x - 3}{x + 3}\right ) - \log \left (-\frac {x - 3}{x + 3}\right )^{2} - 594 \, x + 54 \, \log \left (x + 3\right ) - 54 \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.41, size = 72, normalized size = 2.57 \begin {gather*} -594\,x-2\,x^2\,\ln \left (-\frac {x-3}{x+3}\right )-{\ln \left (-\frac {x-3}{x+3}\right )}^2-22\,x\,\ln \left (-\frac {x-3}{x+3}\right )-175\,x^2-22\,x^3-x^4-\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{3}\right )\,108{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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