Optimal. Leaf size=20 \[ \log ^2\left (\frac {1}{\left (\frac {5 x^2}{3}+(1+x) (4+x)\right )^2}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.34, antiderivative size = 265, normalized size of antiderivative = 13.25, number of steps
used = 20, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {2608, 2604,
2080, 2465, 2437, 2338, 2441, 2440, 2438} \begin {gather*} -8 \text {Li}_2\left (-\frac {16 i x-\sqrt {159}+15 i}{2 \sqrt {159}}\right )-8 \text {Li}_2\left (\frac {16 i x+\sqrt {159}+15 i}{2 \sqrt {159}}\right )-4 \log \left (\frac {9}{64 x^4+240 x^3+417 x^2+360 x+144}\right ) \log \left (16 x-i \sqrt {159}+15\right )-4 \log \left (16 x+i \sqrt {159}+15\right ) \log \left (\frac {9}{64 x^4+240 x^3+417 x^2+360 x+144}\right )-4 \log ^2\left (16 x-i \sqrt {159}+15\right )-4 \log ^2\left (16 x+i \sqrt {159}+15\right )-8 \log \left (-\frac {i \left (16 x+i \sqrt {159}+15\right )}{2 \sqrt {159}}\right ) \log \left (16 x-i \sqrt {159}+15\right )-8 \log \left (\frac {i \left (16 x-i \sqrt {159}+15\right )}{2 \sqrt {159}}\right ) \log \left (16 x+i \sqrt {159}+15\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2080
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rule 2604
Rule 2608
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {64 \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15-i \sqrt {159}+16 x}-\frac {64 \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15+i \sqrt {159}+16 x}\right ) \, dx\\ &=-\left (64 \int \frac {\log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15-i \sqrt {159}+16 x} \, dx\right )-64 \int \frac {\log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )}{15+i \sqrt {159}+16 x} \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \int \frac {\left (360+834 x+720 x^2+256 x^3\right ) \log \left (15-i \sqrt {159}+16 x\right )}{144+360 x+417 x^2+240 x^3+64 x^4} \, dx-4 \int \frac {\left (360+834 x+720 x^2+256 x^3\right ) \log \left (15+i \sqrt {159}+16 x\right )}{144+360 x+417 x^2+240 x^3+64 x^4} \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \int \frac {(30+32 x) \log \left (15-i \sqrt {159}+16 x\right )}{12+15 x+8 x^2} \, dx-4 \int \frac {(30+32 x) \log \left (15+i \sqrt {159}+16 x\right )}{12+15 x+8 x^2} \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \int \left (\frac {32 \log \left (15-i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x}+\frac {32 \log \left (15-i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x}\right ) \, dx-4 \int \left (\frac {32 \log \left (15+i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x}+\frac {32 \log \left (15+i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x}\right ) \, dx\\ &=-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-128 \int \frac {\log \left (15-i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x} \, dx-128 \int \frac {\log \left (15-i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x} \, dx-128 \int \frac {\log \left (15+i \sqrt {159}+16 x\right )}{15-i \sqrt {159}+16 x} \, dx-128 \int \frac {\log \left (15+i \sqrt {159}+16 x\right )}{15+i \sqrt {159}+16 x} \, dx\\ &=-8 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )-8 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-8 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,15-i \sqrt {159}+16 x\right )-8 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,15+i \sqrt {159}+16 x\right )+128 \int \frac {\log \left (\frac {16 \left (15-i \sqrt {159}+16 x\right )}{16 \left (15-i \sqrt {159}\right )-16 \left (15+i \sqrt {159}\right )}\right )}{15+i \sqrt {159}+16 x} \, dx+128 \int \frac {\log \left (\frac {16 \left (15+i \sqrt {159}+16 x\right )}{-16 \left (15-i \sqrt {159}\right )+16 \left (15+i \sqrt {159}\right )}\right )}{15-i \sqrt {159}+16 x} \, dx\\ &=-4 \log ^2\left (15-i \sqrt {159}+16 x\right )-8 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )-4 \log ^2\left (15+i \sqrt {159}+16 x\right )-8 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )+8 \text {Subst}\left (\int \frac {\log \left (1+\frac {16 x}{16 \left (15-i \sqrt {159}\right )-16 \left (15+i \sqrt {159}\right )}\right )}{x} \, dx,x,15+i \sqrt {159}+16 x\right )+8 \text {Subst}\left (\int \frac {\log \left (1+\frac {16 x}{-16 \left (15-i \sqrt {159}\right )+16 \left (15+i \sqrt {159}\right )}\right )}{x} \, dx,x,15-i \sqrt {159}+16 x\right )\\ &=-4 \log ^2\left (15-i \sqrt {159}+16 x\right )-8 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )-4 \log ^2\left (15+i \sqrt {159}+16 x\right )-8 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )-4 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-4 \log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{144+360 x+417 x^2+240 x^3+64 x^4}\right )-8 \text {Li}_2\left (-\frac {15 i-\sqrt {159}+16 i x}{2 \sqrt {159}}\right )-8 \text {Li}_2\left (\frac {15 i+\sqrt {159}+16 i x}{2 \sqrt {159}}\right )\\ \end {aligned} \end {gather*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.05, size = 243, normalized size = 12.15 \begin {gather*} -4 \left (\log ^2\left (15-i \sqrt {159}+16 x\right )+2 \log \left (\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right ) \log \left (15+i \sqrt {159}+16 x\right )+\log ^2\left (15+i \sqrt {159}+16 x\right )+2 \log \left (15-i \sqrt {159}+16 x\right ) \log \left (-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )+\log \left (15-i \sqrt {159}+16 x\right ) \log \left (\frac {9}{\left (12+15 x+8 x^2\right )^2}\right )+\log \left (15+i \sqrt {159}+16 x\right ) \log \left (\frac {9}{\left (12+15 x+8 x^2\right )^2}\right )+2 \text {PolyLog}\left (2,\frac {i \left (15-i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )+2 \text {PolyLog}\left (2,-\frac {i \left (15+i \sqrt {159}+16 x\right )}{2 \sqrt {159}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.80, size = 215, normalized size = 10.75
method | result | size |
norman | \(\ln \left (\frac {9}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )^{2}\) | \(28\) |
default | \(-8 \ln \left (3\right ) \ln \left (8 x^{2}+15 x +12\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right )^{2}-8 \dilog \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right )^{2}-8 \dilog \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )\) | \(215\) |
risch | \(-8 \ln \left (3\right ) \ln \left (8 x^{2}+15 x +12\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-4 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right )^{2}-8 \dilog \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}-\frac {i \sqrt {159}}{16}\right ) \ln \left (-\frac {i \left (\frac {15}{2}+\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {1}{64 x^{4}+240 x^{3}+417 x^{2}+360 x +144}\right )-4 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right )^{2}-8 \dilog \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )-8 \ln \left (x +\frac {15}{16}+\frac {i \sqrt {159}}{16}\right ) \ln \left (\frac {i \left (\frac {15}{2}-\frac {i \sqrt {159}}{2}+8 x \right ) \sqrt {159}}{159}\right )\) | \(215\) |
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. 54 vs.
\(2 (21) = 42\).
time = 0.26, size = 54, normalized size = 2.70 \begin {gather*} -4 \, \log \left (8 \, x^{2} + 15 \, x + 12\right )^{2} - 4 \, \log \left (8 \, x^{2} + 15 \, x + 12\right ) \log \left (\frac {9}{64 \, x^{4} + 240 \, x^{3} + 417 \, x^{2} + 360 \, x + 144}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 27, normalized size = 1.35 \begin {gather*} \log \left (\frac {9}{64 \, x^{4} + 240 \, x^{3} + 417 \, x^{2} + 360 \, x + 144}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 24, normalized size = 1.20 \begin {gather*} \log {\left (\frac {9}{64 x^{4} + 240 x^{3} + 417 x^{2} + 360 x + 144} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 27, normalized size = 1.35 \begin {gather*} \log \left (\frac {9}{64 \, x^{4} + 240 \, x^{3} + 417 \, x^{2} + 360 \, x + 144}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 41, normalized size = 2.05 \begin {gather*} {\ln \left (64\,x^4+240\,x^3+417\,x^2+360\,x+144\right )}^2-4\,\ln \left ({\left (8\,x^2+15\,x+12\right )}^2\right )\,\ln \left (3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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