Integrand size = 21, antiderivative size = 172 \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {683 \sqrt {-x+x^2}}{4096}+\frac {149 (1-2 x) \sqrt {-x+x^2}}{2048}-\frac {1}{12} \left (-x+x^2\right )^{3/2}-\frac {1}{32} x \left (-x+x^2\right )^{3/2}+\frac {\text {arctanh}\left (\frac {1-10 x}{6 \sqrt {-x+x^2}}\right )}{32768}-\frac {1537 \text {arctanh}\left (\frac {x}{\sqrt {-x+x^2}}\right )}{16384}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2617, 2615, 6874, 654, 634, 212, 626, 748, 857, 738, 684} \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {\text {arctanh}\left (\frac {1-10 x}{6 \sqrt {x^2-x}}\right )}{32768}-\frac {1537 \text {arctanh}\left (\frac {x}{\sqrt {x^2-x}}\right )}{16384}-\frac {x^4}{32}+\frac {x^3}{192}-\frac {x^2}{1024}-\frac {1}{32} \left (x^2-x\right )^{3/2} x-\frac {1}{12} \left (x^2-x\right )^{3/2}+\frac {149 (1-2 x) \sqrt {x^2-x}}{2048}-\frac {683 \sqrt {x^2-x}}{4096}+\frac {1}{4} x^4 \log \left (4 \sqrt {x^2-x}+4 x-1\right )+\frac {x}{4096}-\frac {\log (8 x+1)}{32768} \]
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Rule 212
Rule 626
Rule 634
Rule 654
Rule 684
Rule 738
Rule 748
Rule 857
Rule 2615
Rule 2617
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \, dx \\ & = \frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+2 \int \frac {x^4}{-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )} \, dx \\ & = \frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+2 \int \left (\frac {1}{8192}-\frac {x}{1024}+\frac {x^2}{128}-\frac {x^3}{16}-\frac {1}{8192 (1+8 x)}-\frac {x}{12 \sqrt {-x+x^2}}-\frac {85 \sqrt {-x+x^2}}{1024}+\frac {\sqrt {-x+x^2}}{3072 (-1-8 x)}-\frac {11}{128} x \sqrt {-x+x^2}-\frac {1}{16} x^2 \sqrt {-x+x^2}\right ) \, dx \\ & = \frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {\int \frac {\sqrt {-x+x^2}}{-1-8 x} \, dx}{1536}-\frac {1}{8} \int x^2 \sqrt {-x+x^2} \, dx-\frac {85}{512} \int \sqrt {-x+x^2} \, dx-\frac {1}{6} \int \frac {x}{\sqrt {-x+x^2}} \, dx-\frac {11}{64} \int x \sqrt {-x+x^2} \, dx \\ & = \frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {683 \sqrt {-x+x^2}}{4096}+\frac {85 (1-2 x) \sqrt {-x+x^2}}{2048}-\frac {11}{192} \left (-x+x^2\right )^{3/2}-\frac {1}{32} x \left (-x+x^2\right )^{3/2}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {\int \frac {1-10 x}{(-1-8 x) \sqrt {-x+x^2}} \, dx}{24576}+\frac {85 \int \frac {1}{\sqrt {-x+x^2}} \, dx}{4096}-\frac {5}{64} \int x \sqrt {-x+x^2} \, dx-\frac {1}{12} \int \frac {1}{\sqrt {-x+x^2}} \, dx-\frac {11}{128} \int \sqrt {-x+x^2} \, dx \\ & = \frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {683 \sqrt {-x+x^2}}{4096}+\frac {129 (1-2 x) \sqrt {-x+x^2}}{2048}-\frac {1}{12} \left (-x+x^2\right )^{3/2}-\frac {1}{32} x \left (-x+x^2\right )^{3/2}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {5 \int \frac {1}{\sqrt {-x+x^2}} \, dx}{98304}+\frac {3 \int \frac {1}{(-1-8 x) \sqrt {-x+x^2}} \, dx}{32768}+\frac {11 \int \frac {1}{\sqrt {-x+x^2}} \, dx}{1024}-\frac {5}{128} \int \sqrt {-x+x^2} \, dx+\frac {85 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )}{2048}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right ) \\ & = \frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {683 \sqrt {-x+x^2}}{4096}+\frac {149 (1-2 x) \sqrt {-x+x^2}}{2048}-\frac {1}{12} \left (-x+x^2\right )^{3/2}-\frac {1}{32} x \left (-x+x^2\right )^{3/2}-\frac {769 \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )}{6144}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )}{49152}-\frac {3 \text {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {-1+10 x}{\sqrt {-x+x^2}}\right )}{16384}+\frac {5 \int \frac {1}{\sqrt {-x+x^2}} \, dx}{1024}+\frac {11}{512} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right ) \\ & = \frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {683 \sqrt {-x+x^2}}{4096}+\frac {149 (1-2 x) \sqrt {-x+x^2}}{2048}-\frac {1}{12} \left (-x+x^2\right )^{3/2}-\frac {1}{32} x \left (-x+x^2\right )^{3/2}+\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {-x+x^2}}\right )}{32768}-\frac {1697 \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )}{16384}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {5}{512} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right ) \\ & = \frac {x}{4096}-\frac {x^2}{1024}+\frac {x^3}{192}-\frac {x^4}{32}-\frac {683 \sqrt {-x+x^2}}{4096}+\frac {149 (1-2 x) \sqrt {-x+x^2}}{2048}-\frac {1}{12} \left (-x+x^2\right )^{3/2}-\frac {1}{32} x \left (-x+x^2\right )^{3/2}+\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {-x+x^2}}\right )}{32768}-\frac {1537 \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )}{16384}-\frac {\log (1+8 x)}{32768}+\frac {1}{4} x^4 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.59 \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {24 \sqrt {1-x} x^{3/2}-96 \sqrt {1-x} x^{5/2}+512 \sqrt {1-x} x^{7/2}-3072 \sqrt {1-x} x^{9/2}-6112 \sqrt {1-x} x^{3/2} \sqrt {(-1+x) x}-5120 \sqrt {1-x} x^{5/2} \sqrt {(-1+x) x}-3072 \sqrt {1-x} x^{7/2} \sqrt {(-1+x) x}-9240 \sqrt {-(-1+x)^2 x^2}-9222 \sqrt {(-1+x) x} \arcsin \left (\sqrt {1-x}\right )+3 \sqrt {-((-1+x) x)} \text {arctanh}\left (\frac {1-10 x}{6 \sqrt {(-1+x) x}}\right )-3 \sqrt {-((-1+x) x)} \log (1+8 x)+24576 \sqrt {1-x} x^{9/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{98304 \sqrt {-((-1+x) x)}} \]
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Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.39
method | result | size |
parts | \(\frac {\ln \left (-1+4 x +4 \sqrt {\left (-1+x \right ) x}\right ) x^{4}}{4}+\frac {x}{4096}+\frac {x^{3}}{192}-\frac {x^{4}}{32}+\frac {\sqrt {64 \left (x +\frac {1}{8}\right )^{2}-80 x -1}}{65536}-\frac {5 \ln \left (-\frac {1}{2}+x +\sqrt {\left (x +\frac {1}{8}\right )^{2}-\frac {5 x}{4}-\frac {1}{64}}\right )}{65536}-\frac {41 x^{2} \sqrt {x^{2}-x}}{960}-\frac {283 x \sqrt {x^{2}-x}}{6144}-\frac {x^{2}}{1024}-\frac {3069 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x}\right )}{65536}+\frac {\operatorname {arctanh}\left (\frac {\frac {4}{3}-\frac {40 x}{3}}{\sqrt {64 \left (x +\frac {1}{8}\right )^{2}-80 x -1}}\right )}{32768}+\frac {\left (x^{2}-x \right )^{\frac {3}{2}}}{16}-\frac {581 \sqrt {x^{2}-x}}{8192}-\frac {\ln \left (1+8 x \right )}{32768}+\frac {95 \left (2 x -1\right ) \sqrt {x^{2}-x}}{4096}+\frac {x^{2} \left (x^{2}-x \right )^{\frac {3}{2}}}{10}-\frac {x^{4} \sqrt {x^{2}-x}}{10}-\frac {x^{3} \sqrt {x^{2}-x}}{320}+\frac {23 x \left (x^{2}-x \right )^{\frac {3}{2}}}{320}\) | \(239\) |
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Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.78 \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=-\frac {1}{32} \, x^{4} + \frac {1}{192} \, x^{3} - \frac {1}{1024} \, x^{2} + \frac {1}{4} \, {\left (x^{4} - 1\right )} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - \frac {1}{12288} \, {\left (384 \, x^{3} + 640 \, x^{2} + 764 \, x + 1155\right )} \sqrt {x^{2} - x} + \frac {1}{4096} \, x + \frac {4095}{32768} \, \log \left (8 \, x + 1\right ) - \frac {2559}{32768} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) + \frac {4095}{32768} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) - \frac {4095}{32768} \, \log \left (-4 \, x + 4 \, \sqrt {x^{2} - x} + 1\right ) \]
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Timed out. \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\text {Timed out} \]
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\[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\int { x^{3} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.78 \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{32} \, x^{4} + \frac {1}{192} \, x^{3} - \frac {1}{1024} \, x^{2} - \frac {1}{12288} \, {\left (4 \, {\left (32 \, {\left (3 \, x + 5\right )} x + 191\right )} x + 1155\right )} \sqrt {x^{2} - x} + \frac {1}{4096} \, x - \frac {1}{32768} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac {1537}{32768} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) - \frac {1}{32768} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} - 1 \right |}\right ) + \frac {1}{32768} \, \log \left ({\left | -4 \, x + 4 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \]
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Timed out. \[ \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\int x^3\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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