Optimal. Leaf size=14 \[ -64 \log ^2\left (7+(1-x)^2\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.31, antiderivative size = 249, normalized size of antiderivative = 17.79, number of steps
used = 18, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2608, 2604,
2465, 2437, 2338, 2441, 2440, 2438} \begin {gather*} 128 \text {PolyLog}\left (2,-\frac {-i x-\sqrt {7}+i}{2 \sqrt {7}}\right )+128 \text {PolyLog}\left (2,\frac {-i x+\sqrt {7}+i}{2 \sqrt {7}}\right )-128 \log \left (2 x-2 \left (1-i \sqrt {7}\right )\right ) \log \left (x^2-2 x+8\right )-128 \log \left (2 x-2 \left (1+i \sqrt {7}\right )\right ) \log \left (x^2-2 x+8\right )+64 \log ^2\left (-2 \left (-x-i \sqrt {7}+1\right )\right )+64 \log ^2\left (-2 \left (-x+i \sqrt {7}+1\right )\right )+128 \log \left (-\frac {i \left (-x+i \sqrt {7}+1\right )}{2 \sqrt {7}}\right ) \log \left (2 x-2 \left (1-i \sqrt {7}\right )\right )+128 \log \left (\frac {i \left (-x-i \sqrt {7}+1\right )}{2 \sqrt {7}}\right ) \log \left (2 x-2 \left (1+i \sqrt {7}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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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 {256 \log \left (8-2 x+x^2\right )}{-2-2 i \sqrt {7}+2 x}-\frac {256 \log \left (8-2 x+x^2\right )}{-2+2 i \sqrt {7}+2 x}\right ) \, dx\\ &=-\left (256 \int \frac {\log \left (8-2 x+x^2\right )}{-2-2 i \sqrt {7}+2 x} \, dx\right )-256 \int \frac {\log \left (8-2 x+x^2\right )}{-2+2 i \sqrt {7}+2 x} \, dx\\ &=-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \int \frac {(-2+2 x) \log \left (-2-2 i \sqrt {7}+2 x\right )}{8-2 x+x^2} \, dx+128 \int \frac {(-2+2 x) \log \left (-2+2 i \sqrt {7}+2 x\right )}{8-2 x+x^2} \, dx\\ &=-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \int \left (\frac {2 \log \left (-2-2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x}+\frac {2 \log \left (-2-2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x}\right ) \, dx+128 \int \left (\frac {2 \log \left (-2+2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x}+\frac {2 \log \left (-2+2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x}\right ) \, dx\\ &=-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+256 \int \frac {\log \left (-2-2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x} \, dx+256 \int \frac {\log \left (-2-2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x} \, dx+256 \int \frac {\log \left (-2+2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x} \, dx+256 \int \frac {\log \left (-2+2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x} \, dx\\ &=128 \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )+128 \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2-2 i \sqrt {7}+2 x\right )+128 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+2 i \sqrt {7}+2 x\right )-256 \int \frac {\log \left (\frac {2 \left (-2-2 i \sqrt {7}+2 x\right )}{2 \left (-2-2 i \sqrt {7}\right )-2 \left (-2+2 i \sqrt {7}\right )}\right )}{-2+2 i \sqrt {7}+2 x} \, dx-256 \int \frac {\log \left (\frac {2 \left (-2+2 i \sqrt {7}+2 x\right )}{-2 \left (-2-2 i \sqrt {7}\right )+2 \left (-2+2 i \sqrt {7}\right )}\right )}{-2-2 i \sqrt {7}+2 x} \, dx\\ &=64 \log ^2\left (-2 \left (1-i \sqrt {7}-x\right )\right )+64 \log ^2\left (-2 \left (1+i \sqrt {7}-x\right )\right )+128 \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )+128 \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (-2-2 i \sqrt {7}\right )-2 \left (-2+2 i \sqrt {7}\right )}\right )}{x} \, dx,x,-2+2 i \sqrt {7}+2 x\right )-128 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (-2-2 i \sqrt {7}\right )+2 \left (-2+2 i \sqrt {7}\right )}\right )}{x} \, dx,x,-2-2 i \sqrt {7}+2 x\right )\\ &=64 \log ^2\left (-2 \left (1-i \sqrt {7}-x\right )\right )+64 \log ^2\left (-2 \left (1+i \sqrt {7}-x\right )\right )+128 \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )+128 \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \text {Li}_2\left (-\frac {i-\sqrt {7}-i x}{2 \sqrt {7}}\right )+128 \text {Li}_2\left (\frac {i+\sqrt {7}-i x}{2 \sqrt {7}}\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 = 0.04, size = 265, normalized size = 18.93 \begin {gather*} -256 \left (-\frac {1}{2} \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )-\frac {1}{4} \log ^2\left (-2 \left (1-i \sqrt {7}\right )+2 x\right )-\frac {1}{2} \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-\frac {1}{4} \log ^2\left (-2 \left (1+i \sqrt {7}\right )+2 x\right )+\frac {1}{2} \log \left (-2-2 i \sqrt {7}+2 x\right ) \log \left (8-2 x+x^2\right )+\frac {1}{2} \log \left (-2+2 i \sqrt {7}+2 x\right ) \log \left (8-2 x+x^2\right )-\frac {1}{2} \text {PolyLog}\left (2,\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right )-\frac {1}{2} \text {PolyLog}\left (2,-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\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 = 7.71, size = 164, normalized size = 11.71
method | result | size |
norman | \(-64 \ln \left (x^{2}-2 x +8\right )^{2}\) | \(14\) |
risch | \(-64 \ln \left (x^{2}-2 x +8\right )^{2}\) | \(14\) |
default | \(-128 \ln \left (x -1-i \sqrt {7}\right ) \ln \left (x^{2}-2 x +8\right )+128 \dilog \left (-\frac {i \left (-1+i \sqrt {7}+x \right ) \sqrt {7}}{14}\right )+128 \ln \left (x -1-i \sqrt {7}\right ) \ln \left (-\frac {i \left (-1+i \sqrt {7}+x \right ) \sqrt {7}}{14}\right )+64 \ln \left (x -1-i \sqrt {7}\right )^{2}-128 \ln \left (-1+i \sqrt {7}+x \right ) \ln \left (x^{2}-2 x +8\right )+128 \dilog \left (\frac {i \left (x -1-i \sqrt {7}\right ) \sqrt {7}}{14}\right )+128 \ln \left (-1+i \sqrt {7}+x \right ) \ln \left (\frac {i \left (x -1-i \sqrt {7}\right ) \sqrt {7}}{14}\right )+64 \ln \left (-1+i \sqrt {7}+x \right )^{2}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 0.93 \begin {gather*} -64 \, \log \left (x^{2} - 2 \, x + 8\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 13, normalized size = 0.93 \begin {gather*} -64 \, \log \left (x^{2} - 2 \, x + 8\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} - 64 \log {\left (x^{2} - 2 x + 8 \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 13, normalized size = 0.93 \begin {gather*} -64 \, \log \left (x^{2} - 2 \, x + 8\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 13, normalized size = 0.93 \begin {gather*} -64\,{\ln \left (x^2-2\,x+8\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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