Integrand size = 15, antiderivative size = 70 \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=-\frac {1}{\sqrt {x^2}}+\frac {\sqrt {x^2}}{6 \left (-1+x^2\right )}-\frac {11}{6} \coth ^{-1}\left (\sqrt {x^2}\right )+\frac {\left (3-12 x^2+8 x^4\right ) \csc ^{-1}(x)}{3 x \left (-1+x^2\right )^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.30, number of steps used = 5, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {277, 198, 197, 5347, 12, 1273, 464, 212} \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=-\frac {11 x \text {arctanh}(x)}{6 \sqrt {x^2}}-\frac {1}{\sqrt {x^2}}-\frac {\sqrt {x^2}}{6 \left (1-x^2\right )}+\frac {8 x \csc ^{-1}(x)}{3 \sqrt {x^2-1}}-\frac {4 x \csc ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}}+\frac {\csc ^{-1}(x)}{x \left (x^2-1\right )^{3/2}} \]
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Rule 12
Rule 197
Rule 198
Rule 212
Rule 277
Rule 464
Rule 1273
Rule 5347
Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac {4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {8 x \csc ^{-1}(x)}{3 \sqrt {-1+x^2}}+\frac {x \int \frac {3-12 x^2+8 x^4}{3 x^2 \left (1-x^2\right )^2} \, dx}{\sqrt {x^2}} \\ & = \frac {\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac {4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {8 x \csc ^{-1}(x)}{3 \sqrt {-1+x^2}}+\frac {x \int \frac {3-12 x^2+8 x^4}{x^2 \left (1-x^2\right )^2} \, dx}{3 \sqrt {x^2}} \\ & = -\frac {\sqrt {x^2}}{6 \left (1-x^2\right )}+\frac {\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac {4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {8 x \csc ^{-1}(x)}{3 \sqrt {-1+x^2}}-\frac {x \int \frac {-6+17 x^2}{x^2 \left (1-x^2\right )} \, dx}{6 \sqrt {x^2}} \\ & = -\frac {1}{\sqrt {x^2}}-\frac {\sqrt {x^2}}{6 \left (1-x^2\right )}+\frac {\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac {4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {8 x \csc ^{-1}(x)}{3 \sqrt {-1+x^2}}-\frac {(11 x) \int \frac {1}{1-x^2} \, dx}{6 \sqrt {x^2}} \\ & = -\frac {1}{\sqrt {x^2}}-\frac {\sqrt {x^2}}{6 \left (1-x^2\right )}+\frac {\csc ^{-1}(x)}{x \left (-1+x^2\right )^{3/2}}-\frac {4 x \csc ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {8 x \csc ^{-1}(x)}{3 \sqrt {-1+x^2}}-\frac {11 x \text {arctanh}(x)}{6 \sqrt {x^2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13 \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=\frac {4 \left (3-12 x^2+8 x^4\right ) \csc ^{-1}(x)+\sqrt {1-\frac {1}{x^2}} x \left (12-10 x^2+11 x \left (-1+x^2\right ) \log (1-x)-11 x \left (-1+x^2\right ) \log (1+x)\right )}{12 x \left (-1+x^2\right )^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.91
method | result | size |
default | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (\operatorname {arccsc}\left (x \right )+i\right ) \left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x +i\right )}{2 x}+\frac {\left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -i\right ) \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (\operatorname {arccsc}\left (x \right )-i\right )}{2 x}+\frac {\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{2} \operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (10 \,\operatorname {arccsc}\left (x \right ) x^{2}+\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -12 \,\operatorname {arccsc}\left (x \right )\right )}{6 \left (x^{2}-1\right )^{2}}-\frac {11 \,\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \ln \left (\frac {i}{x}+\sqrt {1-\frac {1}{x^{2}}}+i\right )}{6}+\frac {11 \,\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \ln \left (\frac {i}{x}+\sqrt {1-\frac {1}{x^{2}}}-i\right )}{6}\) | \(204\) |
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Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.16 \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=-\frac {10 \, x^{4} - 4 \, {\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right ) - 22 \, x^{2} + 11 \, {\left (x^{5} - 2 \, x^{3} + x\right )} \log \left (x + 1\right ) - 11 \, {\left (x^{5} - 2 \, x^{3} + x\right )} \log \left (x - 1\right ) + 12}{12 \, {\left (x^{5} - 2 \, x^{3} + x\right )}} \]
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Timed out. \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (56) = 112\).
Time = 0.55 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.76 \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=\frac {32 \, x^{4} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) - {\left (x^{3} - x\right )} \sqrt {x + 1} \sqrt {x - 1} {\left (\frac {2 \, {\left (5 \, x^{2} - 6\right )}}{x^{3} - x} + 11 \, \log \left (x + 1\right ) - 11 \, \log \left (x - 1\right )\right )} - 48 \, x^{2} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + 12 \, \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right )}{12 \, {\left (x^{3} - x\right )} \sqrt {x + 1} \sqrt {x - 1}} \]
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Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {{\left (5 \, x^{2} - 6\right )} x}{{\left (x^{2} - 1\right )}^{\frac {3}{2}}} + \frac {6}{{\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1}\right )} \arcsin \left (\frac {1}{x}\right ) + \frac {2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right )}{\mathrm {sgn}\left (x\right )} - \frac {11 \, \log \left ({\left | x + 1 \right |}\right )}{12 \, \mathrm {sgn}\left (x\right )} + \frac {11 \, \log \left ({\left | x - 1 \right |}\right )}{12 \, \mathrm {sgn}\left (x\right )} - \frac {5 \, x^{2} - 6}{6 \, {\left (x^{3} - x\right )} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {\csc ^{-1}(x)}{x^2 \left (-1+x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{x}\right )}{x^2\,{\left (x^2-1\right )}^{5/2}} \,d x \]
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