Optimal. Leaf size=172 \[ -\frac {x^4}{32}+\frac {x^3}{192}-\frac {x^2}{1024}-\frac {1}{32} \left (x^2-x\right )^{3/2} x-\frac {1}{12} \left (x^2-x\right )^{3/2}+\frac {149 (1-2 x) \sqrt {x^2-x}}{2048}-\frac {683 \sqrt {x^2-x}}{4096}+\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {x^2-x}}\right )}{32768}-\frac {1537 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right )}{16384}+\frac {1}{4} x^4 \log \left (4 \sqrt {x^2-x}+4 x-1\right )+\frac {x}{4096}-\frac {\log (8 x+1)}{32768} \]
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Rubi [A] time = 0.38, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2537, 2535, 6742, 640, 620, 206, 612, 734, 843, 724, 670} \[ -\frac {x^4}{32}+\frac {x^3}{192}-\frac {x^2}{1024}-\frac {1}{32} \left (x^2-x\right )^{3/2} x-\frac {1}{12} \left (x^2-x\right )^{3/2}+\frac {149 (1-2 x) \sqrt {x^2-x}}{2048}-\frac {683 \sqrt {x^2-x}}{4096}+\frac {1}{4} x^4 \log \left (4 \sqrt {x^2-x}+4 x-1\right )+\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {x^2-x}}\right )}{32768}-\frac {1537 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right )}{16384}+\frac {x}{4096}-\frac {\log (8 x+1)}{32768} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 670
Rule 724
Rule 734
Rule 843
Rule 2535
Rule 2537
Rule 6742
Rubi steps
\begin {align*} \int x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx &=\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 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )}{2048}-\frac {1}{6} \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )}{49152}-\frac {3 \operatorname {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} \operatorname {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} \operatorname {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*}
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Mathematica [A] time = 0.51, size = 117, normalized size = 0.68 \[ \frac {-3072 x^4+24576 x^4 \log \left (4 x+4 \sqrt {(x-1) x}-1\right )+512 x^3-96 x^2-8 \sqrt {(x-1) x} \left (384 x^3+640 x^2+764 x+1155\right )+24 x-6 \log (8 x+1)-4611 \log \left (-2 x-2 \sqrt {(x-1) x}+1\right )+3 \log \left (-10 x+6 \sqrt {(x-1) x}+1\right )}{98304} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 134, normalized size = 0.78 \[ -\frac {1}{32} \, x^{4} + \frac {1}{192} \, x^{3} - \frac {1}{1024} \, x^{2} + \frac {1}{4} \, {\left (x^{4} - 1\right )} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - \frac {1}{12288} \, {\left (384 \, x^{3} + 640 \, x^{2} + 764 \, x + 1155\right )} \sqrt {x^{2} - x} + \frac {1}{4096} \, x + \frac {4095}{32768} \, \log \left (8 \, x + 1\right ) - \frac {2559}{32768} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) + \frac {4095}{32768} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) - \frac {4095}{32768} \, \log \left (-4 \, x + 4 \, \sqrt {x^{2} - x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 134, normalized size = 0.78 \[ \frac {1}{4} \, x^{4} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{32} \, x^{4} + \frac {1}{192} \, x^{3} - \frac {1}{1024} \, x^{2} - \frac {1}{12288} \, {\left (4 \, {\left (32 \, {\left (3 \, x + 5\right )} x + 191\right )} x + 1155\right )} \sqrt {x^{2} - x} + \frac {1}{4096} \, x - \frac {1}{32768} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac {1537}{32768} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) - \frac {1}{32768} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} - 1 \right |}\right ) + \frac {1}{32768} \, \log \left ({\left | -4 \, x + 4 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{3} \ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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