Optimal. Leaf size=20 \[ (4-x) \log ^2\left ((-4+x) \left (28+16 x^2\right )\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.38, antiderivative size = 1101, normalized size of antiderivative = 55.05, number of
steps used = 86, number of rules used = 14, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used =
{6857, 2608, 2603, 2100, 209, 2604, 2465, 2441, 2440, 2438, 266, 2463, 2437, 2338}
\begin {gather*} -\frac {1}{2} \left (8-i \sqrt {7}\right ) \log ^2\left (i \sqrt {7}-2 x\right )-\frac {1}{2} i \sqrt {7} \log ^2\left (i \sqrt {7}-2 x\right )-\left (8-i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-i \sqrt {7} \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-\left (8-i \sqrt {7}\right ) \log \left (-\frac {i \left (2 x+i \sqrt {7}\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-i \sqrt {7} \log \left (-\frac {i \left (2 x+i \sqrt {7}\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )+\left (8-i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right ) \log \left (i \sqrt {7}-2 x\right )+i \sqrt {7} \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right ) \log \left (i \sqrt {7}-2 x\right )-4 \log ^2(x-4)-\frac {1}{2} \left (8+i \sqrt {7}\right ) \log ^2\left (2 x+i \sqrt {7}\right )+\frac {1}{2} i \sqrt {7} \log ^2\left (2 x+i \sqrt {7}\right )-x \log ^2\left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-8 \log \left (-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right ) \log (x-4)-\left (8+i \sqrt {7}\right ) \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )+i \sqrt {7} \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )+i \sqrt {7} \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )-8 \log (x-4) \log \left (\frac {2 x+i \sqrt {7}}{8+i \sqrt {7}}\right )+8 \log (x-4) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (2 x+i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-i \sqrt {7} \log \left (2 x+i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-8 \text {PolyLog}\left (2,\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \text {PolyLog}\left (2,\frac {2 (4-x)}{8+i \sqrt {7}}\right )-\left (8+i \sqrt {7}\right ) \text {PolyLog}\left (2,-\frac {\sqrt {7}-2 i x}{8 i-\sqrt {7}}\right )+i \sqrt {7} \text {PolyLog}\left (2,-\frac {\sqrt {7}-2 i x}{8 i-\sqrt {7}}\right )-\left (8-i \sqrt {7}\right ) \text {PolyLog}\left (2,\frac {2 i x+\sqrt {7}}{8 i+\sqrt {7}}\right )-i \sqrt {7} \text {PolyLog}\left (2,\frac {2 i x+\sqrt {7}}{8 i+\sqrt {7}}\right )-\left (8+i \sqrt {7}\right ) \text {PolyLog}\left (2,\frac {1}{2}-\frac {i x}{\sqrt {7}}\right )+i \sqrt {7} \text {PolyLog}\left (2,\frac {1}{2}-\frac {i x}{\sqrt {7}}\right )-\left (8-i \sqrt {7}\right ) \text {PolyLog}\left (2,\frac {i x}{\sqrt {7}}+\frac {1}{2}\right )-i \sqrt {7} \text {PolyLog}\left (2,\frac {i x}{\sqrt {7}}+\frac {1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 2100
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2465
Rule 2603
Rule 2604
Rule 2608
Rule 6857
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (7-32 x+12 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2}-\log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )\right ) \, dx\\ &=-\left (2 \int \frac {\left (7-32 x+12 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2} \, dx\right )-\int \log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right ) \, dx\\ &=-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+2 \int \frac {x \left (7-32 x+12 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-28+7 x-16 x^2+4 x^3} \, dx-2 \int \left (3 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )-\frac {2 (7+16 x) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2}\right ) \, dx\\ &=-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+2 \int \left (3 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )+\frac {2 \left (42-7 x+8 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-28+7 x-16 x^2+4 x^3}\right ) \, dx+4 \int \frac {(7+16 x) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2} \, dx-6 \int \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right ) \, dx\\ &=-6 x \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+4 \int \frac {\left (42-7 x+8 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-28+7 x-16 x^2+4 x^3} \, dx+4 \int \left (\frac {\left (-56+7 i \sqrt {7}\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{14 \left (i \sqrt {7}-2 x\right )}+\frac {\left (56+7 i \sqrt {7}\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{14 \left (i \sqrt {7}+2 x\right )}\right ) \, dx+6 \int \frac {x \left (7-32 x+12 x^2\right )}{-28+7 x-16 x^2+4 x^3} \, dx+6 \int \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right ) \, dx\\ &=-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+4 \int \left (\frac {2 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-4+x}-\frac {7 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2}\right ) \, dx-6 \int \frac {x \left (7-32 x+12 x^2\right )}{-28+7 x-16 x^2+4 x^3} \, dx+6 \int \left (3+\frac {4}{-4+x}-\frac {14}{7+4 x^2}\right ) \, dx-\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}-2 x} \, dx+\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}+2 x} \, dx\\ &=18 x+24 \log (4-x)+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-6 \int \left (3+\frac {4}{-4+x}-\frac {14}{7+4 x^2}\right ) \, dx+8 \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-4+x} \, dx-28 \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2} \, dx-84 \int \frac {1}{7+4 x^2} \, dx+\left (-8-i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}+2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx-\left (8-i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}-2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx\\ &=-6 \sqrt {7} \tan ^{-1}\left (\frac {2 x}{\sqrt {7}}\right )+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \int \frac {\left (7-32 x+12 x^2\right ) \log (-4+x)}{-28+7 x-16 x^2+4 x^3} \, dx-28 \int \left (\frac {i \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{2 \sqrt {7} \left (i \sqrt {7}-2 x\right )}+\frac {i \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{2 \sqrt {7} \left (i \sqrt {7}+2 x\right )}\right ) \, dx+84 \int \frac {1}{7+4 x^2} \, dx+\left (-8-i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}+2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}+2 x\right )}{7+4 x^2}\right ) \, dx-\left (8-i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}-2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}-2 x\right )}{7+4 x^2}\right ) \, dx\\ &=\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \int \left (\frac {\log (-4+x)}{-4+x}+\frac {8 x \log (-4+x)}{7+4 x^2}\right ) \, dx-\left (2 i \sqrt {7}\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}-2 x} \, dx-\left (2 i \sqrt {7}\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}+2 x} \, dx+\left (-8-i \sqrt {7}\right ) \int \frac {\log \left (i \sqrt {7}+2 x\right )}{-4+x} \, dx-\left (8-i \sqrt {7}\right ) \int \frac {\log \left (i \sqrt {7}-2 x\right )}{-4+x} \, dx-\left (8 \left (8-i \sqrt {7}\right )\right ) \int \frac {x \log \left (i \sqrt {7}-2 x\right )}{7+4 x^2} \, dx-\left (8 \left (8+i \sqrt {7}\right )\right ) \int \frac {x \log \left (i \sqrt {7}+2 x\right )}{7+4 x^2} \, dx\\ &=-\left (\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )+i \sqrt {7} \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-i \sqrt {7} \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \int \frac {\log (-4+x)}{-4+x} \, dx-64 \int \frac {x \log (-4+x)}{7+4 x^2} \, dx-\left (i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}-2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx+\left (i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}+2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx-\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (-\frac {2 (-4+x)}{8-i \sqrt {7}}\right )}{i \sqrt {7}-2 x} \, dx-\left (8 \left (8-i \sqrt {7}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {7}-2 x\right )}{4 \left (i \sqrt {7}-2 x\right )}+\frac {\log \left (i \sqrt {7}-2 x\right )}{4 \left (i \sqrt {7}+2 x\right )}\right ) \, dx+\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (\frac {2 (-4+x)}{-8-i \sqrt {7}}\right )}{i \sqrt {7}+2 x} \, dx-\left (8 \left (8+i \sqrt {7}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {7}+2 x\right )}{4 \left (i \sqrt {7}-2 x\right )}+\frac {\log \left (i \sqrt {7}+2 x\right )}{4 \left (i \sqrt {7}+2 x\right )}\right ) \, dx\\ &=-\left (\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )+i \sqrt {7} \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-i \sqrt {7} \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-4+x\right )-64 \int \left (-\frac {\log (-4+x)}{4 \left (i \sqrt {7}-2 x\right )}+\frac {\log (-4+x)}{4 \left (i \sqrt {7}+2 x\right )}\right ) \, dx-\left (i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}-2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}-2 x\right )}{7+4 x^2}\right ) \, dx+\left (i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}+2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}+2 x\right )}{7+4 x^2}\right ) \, dx+\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}-2 x\right )}{i \sqrt {7}-2 x} \, dx-\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}-2 x\right )}{i \sqrt {7}+2 x} \, dx-\left (-8+i \sqrt {7}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{8-i \sqrt {7}}\right )}{x} \, dx,x,i \sqrt {7}-2 x\right )+\left (8+i \sqrt {7}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-8-i \sqrt {7}}\right )}{x} \, dx,x,i \sqrt {7}+2 x\right )+\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}+2 x\right )}{i \sqrt {7}-2 x} \, dx-\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}+2 x\right )}{i \sqrt {7}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 25, normalized size = 1.25 \begin {gather*} -\left ((-4+x) \log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 25, normalized size = 1.25
method | result | size |
risch | \(\left (-x +4\right ) \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\) | \(25\) |
norman | \(4 \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}-x \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\) | \(43\) |
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. 77 vs.
\(2 (20) = 40\).
time = 0.84, size = 77, normalized size = 3.85 \begin {gather*} -4 \, x \log \left (2\right )^{2} - {\left (x - 4\right )} \log \left (4 \, x^{2} + 7\right )^{2} - {\left (x - 4\right )} \log \left (x - 4\right )^{2} - 2 \, {\left (2 \, x \log \left (2\right ) + {\left (x - 4\right )} \log \left (x - 4\right ) - 8 \, \log \left (2\right )\right )} \log \left (4 \, x^{2} + 7\right ) - 4 \, {\left (x \log \left (2\right ) - 4 \, \log \left (2\right )\right )} \log \left (x - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 23, normalized size = 1.15 \begin {gather*} -{\left (x - 4\right )} \log \left (16 \, x^{3} - 64 \, x^{2} + 28 \, x - 112\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 20, normalized size = 1.00 \begin {gather*} \left (4 - x\right ) \log {\left (16 x^{3} - 64 x^{2} + 28 x - 112 \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 23, normalized size = 1.15 \begin {gather*} -{\left (x - 4\right )} \log \left (16 \, x^{3} - 64 \, x^{2} + 28 \, x - 112\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.21, size = 23, normalized size = 1.15 \begin {gather*} -{\ln \left (16\,x^3-64\,x^2+28\,x-112\right )}^2\,\left (x-4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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