Optimal. Leaf size=172 \[ -\frac{x^4}{32}+\frac{x^3}{192}-\frac{x^2}{1024}-\frac{1}{32} \left (x^2-x\right )^{3/2} x-\frac{1}{12} \left (x^2-x\right )^{3/2}+\frac{149 (1-2 x) \sqrt{x^2-x}}{2048}-\frac{683 \sqrt{x^2-x}}{4096}+\frac{1}{4} x^4 \log \left (4 \sqrt{x^2-x}+4 x-1\right )+\frac{\tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )}{32768}-\frac{1537 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-x}}\right )}{16384}+\frac{x}{4096}-\frac{\log (8 x+1)}{32768} \]
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Rubi [A] time = 0.380669, antiderivative size = 172, normalized size of antiderivative = 1., 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 2537
Rule 2535
Rule 6742
Rule 640
Rule 620
Rule 206
Rule 612
Rule 734
Rule 843
Rule 724
Rule 670
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*}
Mathematica [A] time = 0.495725, size = 117, normalized size = 0.68 \[ \frac{-3072 x^4+512 x^3-96 x^2-8 \sqrt{(x-1) x} \left (384 x^3+640 x^2+764 x+1155\right )+24576 x^4 \log \left (4 x+4 \sqrt{(x-1) x}-1\right )+24 x-6 \log (8 x+1)-4611 \log \left (-2 x-2 \sqrt{(x-1) x}+1\right )+3 \log \left (-10 x+6 \sqrt{(x-1) x}+1\right )}{98304} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51507, size = 424, normalized size = 2.47 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35666, size = 181, normalized size = 1.05 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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