Integrand size = 26, antiderivative size = 126 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {b x \sqrt {1-c^4 x^4}}{2 c^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}+\frac {b x \arctan \left (\frac {\sqrt {1-c^4 x^4}}{\sqrt {-1+c^2 x^2}}\right )}{2 c^3 \sqrt {c^2 x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {267, 5355, 12, 1586, 1266, 862, 52, 65, 214} \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}+\frac {b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{2 c^5 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{2 c^5 x \sqrt {1-\frac {1}{c^2 x^2}}} \]
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Rule 12
Rule 52
Rule 65
Rule 214
Rule 267
Rule 862
Rule 1266
Rule 1586
Rule 5355
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}+\frac {b \int -\frac {\sqrt {1-c^4 x^4}}{2 c^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac {b \int \frac {\sqrt {1-c^4 x^4}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{2 c^5} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {1-c^4 x^4}}{x \sqrt {1-c^2 x^2}} \, dx}{2 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-c^4 x^2}}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+c^2 x}}{x} \, dx,x,x^2\right )}{4 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{4 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{2 c^7 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{2 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^4}+\frac {b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{2 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {\left (a-b c \sqrt {1-\frac {1}{c^2 x^2}} x-a c^2 x^2\right ) \sqrt {1-c^4 x^4}-b \left (-1+c^2 x^2\right ) \sqrt {1-c^4 x^4} \csc ^{-1}(c x)+\left (b-b c^2 x^2\right ) \arctan \left (\frac {c \sqrt {1-\frac {1}{c^2 x^2}} x}{\sqrt {1-c^4 x^4}}\right )}{2 c^4 \left (-1+c^2 x^2\right )} \]
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\[\int \frac {x^{3} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {\sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} b - {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1}}{\sqrt {c^{2} x^{2} - 1}}\right ) + \sqrt {-c^{4} x^{4} + 1} {\left (a c^{2} x^{2} + {\left (b c^{2} x^{2} - b\right )} \operatorname {arccsc}\left (c x\right ) - a\right )}}{2 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
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\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
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\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]
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