Integrand size = 8, antiderivative size = 65 \[ \int \frac {\arccos (x)^2}{x^5} \, dx=-\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \arccos (x)}{6 x^3}+\frac {\sqrt {1-x^2} \arccos (x)}{3 x}-\frac {\arccos (x)^2}{4 x^4}+\frac {\log (x)}{3} \]
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Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4724, 4790, 4772, 29, 30} \[ \int \frac {\arccos (x)^2}{x^5} \, dx=-\frac {\arccos (x)^2}{4 x^4}+\frac {\sqrt {1-x^2} \arccos (x)}{3 x}+\frac {\sqrt {1-x^2} \arccos (x)}{6 x^3}-\frac {1}{12 x^2}+\frac {\log (x)}{3} \]
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Rule 29
Rule 30
Rule 4724
Rule 4772
Rule 4790
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (x)^2}{4 x^4}-\frac {1}{2} \int \frac {\arccos (x)}{x^4 \sqrt {1-x^2}} \, dx \\ & = \frac {\sqrt {1-x^2} \arccos (x)}{6 x^3}-\frac {\arccos (x)^2}{4 x^4}+\frac {1}{6} \int \frac {1}{x^3} \, dx-\frac {1}{3} \int \frac {\arccos (x)}{x^2 \sqrt {1-x^2}} \, dx \\ & = -\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \arccos (x)}{6 x^3}+\frac {\sqrt {1-x^2} \arccos (x)}{3 x}-\frac {\arccos (x)^2}{4 x^4}+\frac {1}{3} \int \frac {1}{x} \, dx \\ & = -\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \arccos (x)}{6 x^3}+\frac {\sqrt {1-x^2} \arccos (x)}{3 x}-\frac {\arccos (x)^2}{4 x^4}+\frac {\log (x)}{3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {\arccos (x)^2}{x^5} \, dx=-\frac {1}{12 x^2}+\frac {\sqrt {1-x^2} \left (1+2 x^2\right ) \arccos (x)}{6 x^3}-\frac {\arccos (x)^2}{4 x^4}+\frac {\log (x)}{3} \]
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Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {1}{12 x^{2}}-\frac {\arccos \left (x \right )^{2}}{4 x^{4}}+\frac {\ln \left (x \right )}{3}+\frac {\arccos \left (x \right ) \sqrt {-x^{2}+1}}{6 x^{3}}+\frac {\arccos \left (x \right ) \sqrt {-x^{2}+1}}{3 x}\) | \(52\) |
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Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68 \[ \int \frac {\arccos (x)^2}{x^5} \, dx=\frac {4 \, x^{4} \log \left (x\right ) + 2 \, {\left (2 \, x^{3} + x\right )} \sqrt {-x^{2} + 1} \arccos \left (x\right ) - x^{2} - 3 \, \arccos \left (x\right )^{2}}{12 \, x^{4}} \]
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\[ \int \frac {\arccos (x)^2}{x^5} \, dx=\int \frac {\operatorname {acos}^{2}{\left (x \right )}}{x^{5}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \frac {\arccos (x)^2}{x^5} \, dx=\frac {1}{6} \, {\left (\frac {2 \, \sqrt {-x^{2} + 1}}{x} + \frac {\sqrt {-x^{2} + 1}}{x^{3}}\right )} \arccos \left (x\right ) - \frac {1}{12 \, x^{2}} - \frac {\arccos \left (x\right )^{2}}{4 \, x^{4}} + \frac {1}{3} \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (51) = 102\).
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60 \[ \int \frac {\arccos (x)^2}{x^5} \, dx=-\frac {1}{48} \, {\left (\frac {x^{3} {\left (\frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} - \frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}}\right )} \arccos \left (x\right ) - \frac {2 \, x^{2} + 1}{12 \, x^{2}} - \frac {\arccos \left (x\right )^{2}}{4 \, x^{4}} + \frac {1}{6} \, \log \left (x^{2}\right ) \]
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Timed out. \[ \int \frac {\arccos (x)^2}{x^5} \, dx=\int \frac {{\mathrm {acos}\left (x\right )}^2}{x^5} \,d x \]
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