Integrand size = 8, antiderivative size = 63 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {\log (x)}{3} \]
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Time = 0.05 (sec) , antiderivative size = 63, 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 = {5331, 3843, 4270, 4269, 3556} \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {\log (x)}{3} \]
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Rule 3556
Rule 3843
Rule 4269
Rule 4270
Rule 5331
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^2 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right ) \\ & = \frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{2} \text {Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right ) \\ & = \frac {x^2}{12}+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{3} \text {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(x)\right ) \\ & = \frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{3} \text {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(x)\right ) \\ & = \frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {\log (x)}{3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{12} \left (x^2+2 \sqrt {1-\frac {1}{x^2}} x \left (2+x^2\right ) \csc ^{-1}(x)+3 x^4 \csc ^{-1}(x)^2+4 \log (x)\right ) \]
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Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {x^{4} \operatorname {arccsc}\left (x \right )^{2}}{4}+\frac {x^{3} \operatorname {arccsc}\left (x \right ) \sqrt {\frac {x^{2}-1}{x^{2}}}}{6}+\frac {x^{2}}{12}+\frac {\operatorname {arccsc}\left (x \right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, x}{3}-\frac {\ln \left (\frac {1}{x}\right )}{3}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.56 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arccsc}\left (x\right )^{2} + \frac {1}{6} \, {\left (x^{2} + 2\right )} \sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right ) + \frac {1}{12} \, x^{2} + \frac {1}{3} \, \log \left (x\right ) \]
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\[ \int x^3 \csc ^{-1}(x)^2 \, dx=\int x^{3} \operatorname {acsc}^{2}{\left (x \right )}\, dx \]
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Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.51 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arccsc}\left (x\right )^{2} + \frac {2 \, x^{4} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + 2 \, x^{2} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + {\left (x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt {x + 1} \sqrt {x - 1} - 4 \, \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right )}{12 \, \sqrt {x + 1} \sqrt {x - 1}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (49) = 98\).
Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.68 \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\frac {1}{4} \, x^{4} \arcsin \left (\frac {1}{x}\right )^{2} + \frac {1}{12} \, x^{2} {\left (\frac {2}{x^{2}} + 1\right )} + \frac {1}{48} \, {\left (x^{3} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{3} + 9 \, x {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )} - \frac {9 \, x^{2} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{2} + 1}{x^{3} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{3}}\right )} \arcsin \left (\frac {1}{x}\right ) - \frac {1}{6} \, \log \left (\frac {1}{x^{2}}\right ) \]
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Timed out. \[ \int x^3 \csc ^{-1}(x)^2 \, dx=\int x^3\,{\mathrm {asin}\left (\frac {1}{x}\right )}^2 \,d x \]
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