Integrand size = 17, antiderivative size = 74 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {24 \sqrt {-1+x^2}}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}}-\frac {12 \sqrt {-1+x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {-1+x^2} \csc ^{-1}(x)^4}{x} \]
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Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5351, 4767, 4715, 267} \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {24 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2}}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}} \]
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Rule 267
Rule 4715
Rule 4767
Rule 5351
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {x \arcsin (x)^4}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{x} \\ & = \frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}-\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \arcsin (x)^3 \, dx,x,\frac {1}{x}\right )}{x} \\ & = -\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}+\frac {\left (12 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x \arcsin (x)^2}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{x} \\ & = -\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}+\frac {\left (24 \sqrt {x^2}\right ) \text {Subst}\left (\int \arcsin (x) \, dx,x,\frac {1}{x}\right )}{x} \\ & = \frac {24 \csc ^{-1}(x)}{\sqrt {x^2}}-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}-\frac {\left (24 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{x} \\ & = \frac {24 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2}}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}}-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {24 \left (-1+x^2\right )+24 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)-12 \left (-1+x^2\right ) \csc ^{-1}(x)^2-4 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)^3+\left (-1+x^2\right ) \csc ^{-1}(x)^4}{x \sqrt {-1+x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, \left (\operatorname {arccsc}\left (x \right )^{4} x^{2}-\operatorname {arccsc}\left (x \right )^{4}-12 \operatorname {arccsc}\left (x \right )^{2} x^{2}+12 \operatorname {arccsc}\left (x \right )^{2}-4 \sqrt {\frac {x^{2}-1}{x^{2}}}\, \operatorname {arccsc}\left (x \right )^{3} x +24 x^{2}-24+24 \,\operatorname {arccsc}\left (x \right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right )}{x^{2}-1}\) | \(102\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.50 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=-\frac {4 \, \operatorname {arccsc}\left (x\right )^{3} - {\left (\operatorname {arccsc}\left (x\right )^{4} - 12 \, \operatorname {arccsc}\left (x\right )^{2} + 24\right )} \sqrt {x^{2} - 1} - 24 \, \operatorname {arccsc}\left (x\right )}{x} \]
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\[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {\operatorname {acsc}^{4}{\left (x \right )}}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {\sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right )^{4}}{x} - 12 \, \sqrt {-\frac {1}{x^{2}} + 1} \operatorname {arccsc}\left (x\right )^{2} - \frac {4 \, \operatorname {arccsc}\left (x\right )^{3}}{x} + 24 \, \sqrt {-\frac {1}{x^{2}} + 1} + \frac {24 \, \operatorname {arccsc}\left (x\right )}{x} \]
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\[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\int { \frac {\operatorname {arccsc}\left (x\right )^{4}}{\sqrt {x^{2} - 1} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {{\mathrm {asin}\left (\frac {1}{x}\right )}^4}{x^2\,\sqrt {x^2-1}} \,d x \]
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