Optimal. Leaf size=49 \[ a x+\frac {1}{12} (27 a+9 b+3 c+d) \log (3-x)-\frac {1}{3} d \log (x)-\frac {1}{4} (a-b+c-d) \log (1+x) \]
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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1626}
\begin {gather*} \frac {1}{12} \log (3-x) (27 a+9 b+3 c+d)-\frac {1}{4} \log (x+1) (a-b+c-d)+a x-\frac {1}{3} d \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 1626
Rubi steps
\begin {align*} \int \frac {d+c x+b x^2+a x^3}{(-3+x) x (1+x)} \, dx &=\int \left (a+\frac {27 a+9 b+3 c+d}{12 (-3+x)}-\frac {d}{3 x}+\frac {-a+b-c+d}{4 (1+x)}\right ) \, dx\\ &=a x+\frac {1}{12} (27 a+9 b+3 c+d) \log (3-x)-\frac {1}{3} d \log (x)-\frac {1}{4} (a-b+c-d) \log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} a x+\frac {1}{12} (27 a+9 b+3 c+d) \log (3-x)-\frac {1}{3} d \log (x)+\frac {1}{4} (-a+b-c+d) \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 65.47, size = 41, normalized size = 0.84 \begin {gather*} a x-\frac {d \text {Log}\left [x\right ]}{3}-\frac {\text {Log}\left [1+x\right ] \left (a-b+c-d\right )}{4}+\frac {\text {Log}\left [-3+x\right ] \left (27 a+9 b+3 c+d\right )}{12} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 46, normalized size = 0.94
method | result | size |
default | \(a x +\left (\frac {9 a}{4}+\frac {3 b}{4}+\frac {c}{4}+\frac {d}{12}\right ) \ln \left (-3+x \right )-\frac {d \ln \left (x \right )}{3}+\left (-\frac {a}{4}+\frac {b}{4}-\frac {c}{4}+\frac {d}{4}\right ) \ln \left (1+x \right )\) | \(46\) |
norman | \(a x +\left (\frac {9 a}{4}+\frac {3 b}{4}+\frac {c}{4}+\frac {d}{12}\right ) \ln \left (-3+x \right )-\frac {d \ln \left (x \right )}{3}+\left (-\frac {a}{4}+\frac {b}{4}-\frac {c}{4}+\frac {d}{4}\right ) \ln \left (1+x \right )\) | \(46\) |
risch | \(a x +\frac {9 \ln \left (3-x \right ) a}{4}+\frac {3 \ln \left (3-x \right ) b}{4}+\frac {\ln \left (3-x \right ) c}{4}+\frac {\ln \left (3-x \right ) d}{12}-\frac {d \ln \left (x \right )}{3}-\frac {\ln \left (-1-x \right ) a}{4}+\frac {\ln \left (-1-x \right ) b}{4}-\frac {\ln \left (-1-x \right ) c}{4}+\frac {\ln \left (-1-x \right ) d}{4}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 41, normalized size = 0.84 \begin {gather*} a x - \frac {1}{4} \, {\left (a - b + c - d\right )} \log \left (x + 1\right ) + \frac {1}{12} \, {\left (27 \, a + 9 \, b + 3 \, c + d\right )} \log \left (x - 3\right ) - \frac {1}{3} \, d \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 41, normalized size = 0.84 \begin {gather*} a x - \frac {1}{4} \, {\left (a - b + c - d\right )} \log \left (x + 1\right ) + \frac {1}{12} \, {\left (27 \, a + 9 \, b + 3 \, c + d\right )} \log \left (x - 3\right ) - \frac {1}{3} \, d \log \left (x\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. 762 vs.
\(2 (49) = 98\)
time = 55.71, size = 762, normalized size = 15.55 \begin {gather*} a x - \frac {d \log {\left (x \right )}}{3} - \frac {\left (a - b + c - d\right ) \log {\left (x + \frac {- 1512 a^{2} d + 1134 a^{2} \left (a - b + c - d\right ) - 864 a b d + 648 a b \left (a - b + c - d\right ) - 432 a c d + 324 a c \left (a - b + c - d\right ) - 144 a d^{2} + 81 a \left (a - b + c - d\right )^{2} - 216 b^{2} d + 162 b^{2} \left (a - b + c - d\right ) - 288 b d^{2} + 108 b d \left (a - b + c - d\right ) + 81 b \left (a - b + c - d\right )^{2} - 72 c^{2} d + 54 c^{2} \left (a - b + c - d\right ) + 144 c d^{2} - 72 c d \left (a - b + c - d\right ) - 27 c \left (a - b + c - d\right )^{2} - 136 d^{3} - 54 d^{2} \left (a - b + c - d\right ) + 117 d \left (a - b + c - d\right )^{2}}{1215 a^{3} - 567 a^{2} b + 1593 a^{2} c - 2691 a^{2} d - 567 a b^{2} + 378 a b c - 1638 a b d + 405 a c^{2} - 702 a c d - 351 a d^{2} - 81 b^{3} - 27 b^{2} c - 207 b^{2} d + 81 b c^{2} - 270 b c d - 27 b d^{2} + 27 c^{3} - 27 c^{2} d - 99 c d^{2} + 35 d^{3}} \right )}}{4} + \frac {\left (27 a + 9 b + 3 c + d\right ) \log {\left (x + \frac {- 1512 a^{2} d - 378 a^{2} \cdot \left (27 a + 9 b + 3 c + d\right ) - 864 a b d - 216 a b \left (27 a + 9 b + 3 c + d\right ) - 432 a c d - 108 a c \left (27 a + 9 b + 3 c + d\right ) - 144 a d^{2} + 9 a \left (27 a + 9 b + 3 c + d\right )^{2} - 216 b^{2} d - 54 b^{2} \cdot \left (27 a + 9 b + 3 c + d\right ) - 288 b d^{2} - 36 b d \left (27 a + 9 b + 3 c + d\right ) + 9 b \left (27 a + 9 b + 3 c + d\right )^{2} - 72 c^{2} d - 18 c^{2} \cdot \left (27 a + 9 b + 3 c + d\right ) + 144 c d^{2} + 24 c d \left (27 a + 9 b + 3 c + d\right ) - 3 c \left (27 a + 9 b + 3 c + d\right )^{2} - 136 d^{3} + 18 d^{2} \cdot \left (27 a + 9 b + 3 c + d\right ) + 13 d \left (27 a + 9 b + 3 c + d\right )^{2}}{1215 a^{3} - 567 a^{2} b + 1593 a^{2} c - 2691 a^{2} d - 567 a b^{2} + 378 a b c - 1638 a b d + 405 a c^{2} - 702 a c d - 351 a d^{2} - 81 b^{3} - 27 b^{2} c - 207 b^{2} d + 81 b c^{2} - 270 b c d - 27 b d^{2} + 27 c^{3} - 27 c^{2} d - 99 c d^{2} + 35 d^{3}} \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 46, normalized size = 0.94 \begin {gather*} -\frac {1}{3} d \ln \left |x\right |+\frac {1}{12} \left (27 a+9 b+3 c+d\right ) \ln \left |x-3\right |+\frac {1}{4} \left (-a+b-c+d\right ) \ln \left |x+1\right |+x a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 46, normalized size = 0.94 \begin {gather*} \ln \left (x-3\right )\,\left (\frac {9\,a}{4}+\frac {3\,b}{4}+\frac {c}{4}+\frac {d}{12}\right )-\ln \left (x+1\right )\,\left (\frac {a}{4}-\frac {b}{4}+\frac {c}{4}-\frac {d}{4}\right )+a\,x-\frac {d\,\ln \left (x\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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