Optimal. Leaf size=20 \[ -4+\left (4+4 x \left (1-\frac {4}{1+x}\right ) \log (x)\right )^2 \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.35, antiderivative size = 89, normalized size of antiderivative = 4.45, number of steps
used = 29, number of rules used = 17, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.218, Rules used = {6820, 12,
6874, 45, 2404, 2332, 2341, 2351, 31, 2354, 2438, 2333, 2342, 2356, 2389, 2379, 2355}
\begin {gather*} -512 \text {PolyLog}\left (2,-\frac {1}{x}\right )-512 \text {PolyLog}(2,-x)+16 x^2 \log ^2(x)-128 x \log ^2(x)+\frac {640 x \log ^2(x)}{x+1}+\frac {256 \log ^2(x)}{(x+1)^2}+32 x \log (x)+512 \log \left (\frac {1}{x}+1\right ) \log (x)-512 \log (x+1) \log (x)-\frac {128 x \log (x)}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 45
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
Rule 2404
Rule 2438
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 \left ((-3+x) (1+x)^2+\left (-3+8 x+6 x^2-4 x^3+x^4\right ) \log (x)+x \left (9-9 x-x^2+x^3\right ) \log ^2(x)\right )}{(1+x)^3} \, dx\\ &=32 \int \frac {(-3+x) (1+x)^2+\left (-3+8 x+6 x^2-4 x^3+x^4\right ) \log (x)+x \left (9-9 x-x^2+x^3\right ) \log ^2(x)}{(1+x)^3} \, dx\\ &=32 \int \left (\frac {-3+x}{1+x}+\frac {\left (-3+11 x-5 x^2+x^3\right ) \log (x)}{(1+x)^2}+\frac {(-3+x) (-1+x) x (3+x) \log ^2(x)}{(1+x)^3}\right ) \, dx\\ &=32 \int \frac {-3+x}{1+x} \, dx+32 \int \frac {\left (-3+11 x-5 x^2+x^3\right ) \log (x)}{(1+x)^2} \, dx+32 \int \frac {(-3+x) (-1+x) x (3+x) \log ^2(x)}{(1+x)^3} \, dx\\ &=32 \int \left (1-\frac {4}{1+x}\right ) \, dx+32 \int \left (-7 \log (x)+x \log (x)-\frac {20 \log (x)}{(1+x)^2}+\frac {24 \log (x)}{1+x}\right ) \, dx+32 \int \left (-4 \log ^2(x)+x \log ^2(x)-\frac {16 \log ^2(x)}{(1+x)^3}+\frac {20 \log ^2(x)}{(1+x)^2}\right ) \, dx\\ &=32 x-128 \log (1+x)+32 \int x \log (x) \, dx+32 \int x \log ^2(x) \, dx-128 \int \log ^2(x) \, dx-224 \int \log (x) \, dx-512 \int \frac {\log ^2(x)}{(1+x)^3} \, dx-640 \int \frac {\log (x)}{(1+x)^2} \, dx+640 \int \frac {\log ^2(x)}{(1+x)^2} \, dx+768 \int \frac {\log (x)}{1+x} \, dx\\ &=256 x-8 x^2-224 x \log (x)+16 x^2 \log (x)-\frac {640 x \log (x)}{1+x}-128 x \log ^2(x)+16 x^2 \log ^2(x)+\frac {256 \log ^2(x)}{(1+x)^2}+\frac {640 x \log ^2(x)}{1+x}-128 \log (1+x)+768 \log (x) \log (1+x)-32 \int x \log (x) \, dx+256 \int \log (x) \, dx-512 \int \frac {\log (x)}{x (1+x)^2} \, dx+640 \int \frac {1}{1+x} \, dx-768 \int \frac {\log (1+x)}{x} \, dx-1280 \int \frac {\log (x)}{1+x} \, dx\\ &=32 x \log (x)-\frac {640 x \log (x)}{1+x}-128 x \log ^2(x)+16 x^2 \log ^2(x)+\frac {256 \log ^2(x)}{(1+x)^2}+\frac {640 x \log ^2(x)}{1+x}+512 \log (1+x)-512 \log (x) \log (1+x)+768 \text {Li}_2(-x)+512 \int \frac {\log (x)}{(1+x)^2} \, dx-512 \int \frac {\log (x)}{x (1+x)} \, dx+1280 \int \frac {\log (1+x)}{x} \, dx\\ &=32 x \log (x)-\frac {128 x \log (x)}{1+x}-128 x \log ^2(x)+16 x^2 \log ^2(x)+\frac {256 \log ^2(x)}{(1+x)^2}+\frac {640 x \log ^2(x)}{1+x}+512 \log (1+x)-512 \log (x) \log (1+x)-512 \text {Li}_2(-x)-512 \int \frac {1}{1+x} \, dx-512 \int \frac {\log (x)}{x} \, dx+512 \int \frac {\log (x)}{1+x} \, dx\\ &=32 x \log (x)-\frac {128 x \log (x)}{1+x}-256 \log ^2(x)-128 x \log ^2(x)+16 x^2 \log ^2(x)+\frac {256 \log ^2(x)}{(1+x)^2}+\frac {640 x \log ^2(x)}{1+x}-512 \text {Li}_2(-x)-512 \int \frac {\log (1+x)}{x} \, dx\\ &=32 x \log (x)-\frac {128 x \log (x)}{1+x}-256 \log ^2(x)-128 x \log ^2(x)+16 x^2 \log ^2(x)+\frac {256 \log ^2(x)}{(1+x)^2}+\frac {640 x \log ^2(x)}{1+x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 26, normalized size = 1.30 \begin {gather*} \frac {16 (-3+x) x \log (x) (2 (1+x)+(-3+x) x \log (x))}{(1+x)^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. \(47\) vs.
\(2(20)=40\).
time = 0.46, size = 48, normalized size = 2.40
method | result | size |
risch | \(\frac {16 x^{2} \left (x^{2}-6 x +9\right ) \ln \left (x \right )^{2}}{x^{2}+2 x +1}+\frac {32 \left (x^{2}+x +4\right ) \ln \left (x \right )}{x +1}-128 \ln \left (x \right )\) | \(48\) |
norman | \(\frac {-96 x \ln \left (x \right )-64 x^{2} \ln \left (x \right )+144 x^{2} \ln \left (x \right )^{2}+32 x^{3} \ln \left (x \right )-96 x^{3} \ln \left (x \right )^{2}+16 x^{4} \ln \left (x \right )^{2}}{\left (x +1\right )^{2}}\) | \(54\) |
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. 173 vs.
\(2 (21) = 42\).
time = 0.33, size = 173, normalized size = 8.65 \begin {gather*} 32 \, x - \frac {128 \, {\left (2 \, x + 1\right )} \log \left (x\right )}{x^{2} + 2 \, x + 1} - \frac {16 \, {\left (2 \, x^{3} - {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2} + 4 \, x^{2} - {\left (2 \, x^{3} - 9 \, x^{2}\right )} \log \left (x\right ) - 9 \, x - 11\right )}}{x^{2} + 2 \, x + 1} - \frac {16 \, {\left (6 \, x + 5\right )}}{x^{2} + 2 \, x + 1} - \frac {16 \, {\left (4 \, x + 3\right )}}{x^{2} + 2 \, x + 1} + \frac {80 \, {\left (2 \, x + 1\right )}}{x^{2} + 2 \, x + 1} + \frac {48 \, \log \left (x\right )}{x^{2} + 2 \, x + 1} + \frac {48}{x^{2} + 2 \, x + 1} - \frac {176}{x + 1} + 80 \, \log \left (x\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. 48 vs.
\(2 (21) = 42\).
time = 0.37, size = 48, normalized size = 2.40 \begin {gather*} \frac {16 \, {\left ({\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{3} - 2 \, x^{2} - 3 \, x\right )} \log \left (x\right )\right )}}{x^{2} + 2 \, x + 1} \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. 49 vs.
\(2 (17) = 34\).
time = 0.13, size = 49, normalized size = 2.45 \begin {gather*} - 128 \log {\left (x \right )} + \frac {\left (16 x^{4} - 96 x^{3} + 144 x^{2}\right ) \log {\left (x \right )}^{2}}{x^{2} + 2 x + 1} + \frac {\left (32 x^{2} + 32 x + 128\right ) \log {\left (x \right )}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.50, size = 29, normalized size = 1.45 \begin {gather*} \frac {16\,x\,\ln \left (x\right )\,\left (x-3\right )\,\left (2\,x+x^2\,\ln \left (x\right )-3\,x\,\ln \left (x\right )+2\right )}{{\left (x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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