Integrand size = 20, antiderivative size = 67 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}-\text {arctanh}\left (\sqrt {1-x^2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1821, 849, 821, 272, 65, 212} \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=-\text {arctanh}\left (\sqrt {1-x^2}\right )-\frac {5 \sqrt {1-x^2}}{3 x}-\frac {\sqrt {1-x^2}}{x^2}-\frac {\sqrt {1-x^2}}{3 x^3} \]
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Rule 65
Rule 212
Rule 272
Rule 821
Rule 849
Rule 1821
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-x^2}}{3 x^3}-\frac {1}{3} \int \frac {-6-5 x}{x^3 \sqrt {1-x^2}} \, dx \\ & = -\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}+\frac {1}{6} \int \frac {10+6 x}{x^2 \sqrt {1-x^2}} \, dx \\ & = -\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}+\int \frac {1}{x \sqrt {1-x^2}} \, dx \\ & = -\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right ) \\ & = -\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=\frac {\left (-1-3 x-5 x^2\right ) \sqrt {1-x^2}}{3 x^3}-\log (x)+\log \left (-1+\sqrt {1-x^2}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66
method | result | size |
trager | \(-\frac {\left (5 x^{2}+3 x +1\right ) \sqrt {-x^{2}+1}}{3 x^{3}}-\ln \left (\frac {\sqrt {-x^{2}+1}+1}{x}\right )\) | \(44\) |
risch | \(\frac {5 x^{4}+3 x^{3}-4 x^{2}-3 x -1}{3 x^{3} \sqrt {-x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\) | \(48\) |
default | \(-\frac {5 \sqrt {-x^{2}+1}}{3 x}-\frac {\sqrt {-x^{2}+1}}{3 x^{3}}-\frac {\sqrt {-x^{2}+1}}{x^{2}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\) | \(56\) |
meijerg | \(-\frac {\left (2 x^{2}+1\right ) \sqrt {-x^{2}+1}}{3 x^{3}}-\frac {-\frac {\sqrt {\pi }\, \left (-4 x^{2}+8\right )}{8 x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-x^{2}+1}}{x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {-x^{2}+1}}{x}\) | \(118\) |
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=\frac {3 \, x^{3} \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - {\left (5 \, x^{2} + 3 \, x + 1\right )} \sqrt {-x^{2} + 1}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 3.38 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.91 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=\begin {cases} - \frac {\sqrt {1 - x^{2}}}{x} - \frac {\left (1 - x^{2}\right )^{\frac {3}{2}}}{3 x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases} + \begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases} + 2 \left (\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x} \right )}}{2} + \frac {1}{2 x \sqrt {-1 + \frac {1}{x^{2}}}} - \frac {1}{2 x^{3} \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x} \right )}}{2} - \frac {i \sqrt {1 - \frac {1}{x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=-\frac {5 \, \sqrt {-x^{2} + 1}}{3 \, x} - \frac {\sqrt {-x^{2} + 1}}{x^{2}} - \frac {\sqrt {-x^{2} + 1}}{3 \, x^{3}} - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (55) = 110\).
Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=-\frac {x^{3} {\left (\frac {6 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {21 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{24 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} - \frac {7 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{8 \, x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{24 \, x^{3}} + \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx=\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\sqrt {1-x^2}\,\left (\frac {2}{3\,x}+\frac {1}{3\,x^3}\right )-\frac {\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2}}{x^2} \]
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