Integrand size = 33, antiderivative size = 88 \[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=-x-4 \sqrt {1-x^2}+5 \arcsin (x)+\frac {25 \arctan \left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}-\frac {25 \arctan \left (\frac {5 x}{2 \sqrt {6} \sqrt {1-x^2}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6874, 1605, 196, 45, 6872, 209, 399, 222, 385} \[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=5 \arcsin (x)-\frac {25 \arctan \left (\frac {5 x}{2 \sqrt {6} \sqrt {1-x^2}}\right )}{2 \sqrt {6}}+\frac {25 \arctan \left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}-4 \sqrt {1-x^2}+20 \log \left (\sqrt {1-x^2}+5\right )-x \]
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Rule 45
Rule 196
Rule 209
Rule 222
Rule 385
Rule 399
Rule 1605
Rule 6872
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 x}{5+\sqrt {1-x^2}}-\frac {\sqrt {1-x^2}}{5+\sqrt {1-x^2}}\right ) \, dx \\ & = 4 \int \frac {x}{5+\sqrt {1-x^2}} \, dx-\int \frac {\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{5+\sqrt {x}} \, dx,x,1-x^2\right )\right )-\int \left (1-\frac {5}{5+\sqrt {1-x^2}}\right ) \, dx \\ & = -x-4 \text {Subst}\left (\int \frac {x}{5+x} \, dx,x,\sqrt {1-x^2}\right )+5 \int \frac {1}{5+\sqrt {1-x^2}} \, dx \\ & = -x-4 \text {Subst}\left (\int \left (1-\frac {5}{5+x}\right ) \, dx,x,\sqrt {1-x^2}\right )+5 \int \left (\frac {5}{24+x^2}-\frac {\sqrt {1-x^2}}{24+x^2}\right ) \, dx \\ & = -x-4 \sqrt {1-x^2}+20 \log \left (5+\sqrt {1-x^2}\right )-5 \int \frac {\sqrt {1-x^2}}{24+x^2} \, dx+25 \int \frac {1}{24+x^2} \, dx \\ & = -x-4 \sqrt {1-x^2}+\frac {25 \arctan \left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right )+5 \int \frac {1}{\sqrt {1-x^2}} \, dx-125 \int \frac {1}{\sqrt {1-x^2} \left (24+x^2\right )} \, dx \\ & = -x-4 \sqrt {1-x^2}+5 \arcsin (x)+\frac {25 \arctan \left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right )-125 \text {Subst}\left (\int \frac {1}{24+25 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right ) \\ & = -x-4 \sqrt {1-x^2}+5 \arcsin (x)+\frac {25 \arctan \left (\frac {x}{2 \sqrt {6}}\right )}{2 \sqrt {6}}-\frac {25 \arctan \left (\frac {5 x}{2 \sqrt {6} \sqrt {1-x^2}}\right )}{2 \sqrt {6}}+20 \log \left (5+\sqrt {1-x^2}\right ) \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.23 \[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=-x-4 \sqrt {1-x^2}+10 \arctan \left (\frac {x}{-1+\sqrt {1-x^2}}\right )-\frac {25 \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{-1+\sqrt {1-x^2}}\right )}{\sqrt {6}}-20 \log \left (-1+\sqrt {1-x^2}\right )+20 \log \left (-4-x^2+4 \sqrt {1-x^2}\right ) \]
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Time = 1.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {25 \arctan \left (\frac {x \sqrt {6}}{12}\right ) \sqrt {6}}{12}+10 \ln \left (x^{2}+24\right )-x +5 \arcsin \left (x \right )+\frac {25 \sqrt {6}\, \arctan \left (\frac {5 \sqrt {6}\, \sqrt {-x^{2}+1}\, x}{12 \left (x^{2}-1\right )}\right )}{12}-4 \sqrt {-x^{2}+1}+20 \,\operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}}{5}\right )\) | \(82\) |
trager | \(\text {Expression too large to display}\) | \(883\) |
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (66) = 132\).
Time = 0.25 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.82 \[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=\frac {25}{12} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} x\right ) + \frac {25}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {-x^{2} + 1} - \sqrt {6}}{2 \, x}\right ) + \frac {25}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {-x^{2} + 1} - \sqrt {6}}{3 \, x}\right ) - x - 4 \, \sqrt {-x^{2} + 1} - 10 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + 10 \, \log \left (x^{2} + 24\right ) - 10 \, \log \left (-\frac {x^{2} + 6 \, \sqrt {-x^{2} + 1} - 6}{x^{2}}\right ) + 10 \, \log \left (\frac {x^{2} - 4 \, \sqrt {-x^{2} + 1} + 4}{x^{2}}\right ) \]
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\[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=\int \frac {4 x - \sqrt {1 - x^{2}}}{\sqrt {1 - x^{2}} + 5}\, dx \]
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\[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=\int { \frac {4 \, x - \sqrt {-x^{2} + 1}}{\sqrt {-x^{2} + 1} + 5} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (66) = 132\).
Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.53 \[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=\frac {25}{12} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} x\right ) - \frac {25}{12} \, \sqrt {6} \arctan \left (-\frac {\sqrt {6} {\left (\sqrt {-x^{2} + 1} - 1\right )}}{3 \, x}\right ) - \frac {25}{12} \, \sqrt {6} \arctan \left (-\frac {\sqrt {6} {\left (\sqrt {-x^{2} + 1} - 1\right )}}{2 \, x}\right ) - x - 4 \, \sqrt {-x^{2} + 1} + 5 \, \arcsin \left (x\right ) + 10 \, \log \left (x^{2} + 24\right ) - 10 \, \log \left (\frac {3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 2\right ) + 10 \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 3\right ) \]
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Time = 0.47 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.81 \[ \int \frac {4 x-\sqrt {1-x^2}}{5+\sqrt {1-x^2}} \, dx=5\,\mathrm {asin}\left (x\right )-x-4\,\sqrt {1-x^2}-\frac {\sqrt {24}\,\ln \left (\frac {\frac {2\,\sqrt {6}\,x}{5}+\sqrt {1-x^2}\,1{}\mathrm {i}+\frac {1}{5}{}\mathrm {i}}{x-\sqrt {6}\,2{}\mathrm {i}}\right )\,\left (125+\sqrt {24}\,100{}\mathrm {i}\right )\,1{}\mathrm {i}}{240}-\frac {\sqrt {24}\,\ln \left (\frac {-\frac {\sqrt {24}\,x}{5}+\sqrt {1-x^2}\,1{}\mathrm {i}+\frac {1}{5}{}\mathrm {i}}{x+\sqrt {24}\,1{}\mathrm {i}}\right )\,\left (-125+\sqrt {24}\,100{}\mathrm {i}\right )\,1{}\mathrm {i}}{240}-\frac {\sqrt {24}\,\ln \left (x-\sqrt {6}\,2{}\mathrm {i}\right )\,\left (25+\sqrt {24}\,20{}\mathrm {i}\right )\,1{}\mathrm {i}}{48}-\frac {\sqrt {24}\,\ln \left (x+\sqrt {24}\,1{}\mathrm {i}\right )\,\left (-25+\sqrt {24}\,20{}\mathrm {i}\right )\,1{}\mathrm {i}}{48} \]
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