Integrand size = 14, antiderivative size = 72 \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=-\frac {a+b \arcsin \left (c x^n\right )}{2 x^2}-\frac {b c n x^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right ),\frac {1}{2} \left (3-\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (2-n)} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4926, 12, 371} \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=-\frac {a+b \arcsin \left (c x^n\right )}{2 x^2}-\frac {b c n x^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right ),\frac {1}{2} \left (3-\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (2-n)} \]
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Rule 12
Rule 371
Rule 4926
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arcsin \left (c x^n\right )}{2 x^2}+\frac {1}{2} b \int \frac {c n x^{-3+n}}{\sqrt {1-c^2 x^{2 n}}} \, dx \\ & = -\frac {a+b \arcsin \left (c x^n\right )}{2 x^2}+\frac {1}{2} (b c n) \int \frac {x^{-3+n}}{\sqrt {1-c^2 x^{2 n}}} \, dx \\ & = -\frac {a+b \arcsin \left (c x^n\right )}{2 x^2}-\frac {b c n x^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right ),\frac {1}{2} \left (3-\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (2-n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {b \arcsin \left (c x^n\right )}{2 x^2}+\frac {b c n x^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {-2+n}{2 n},1+\frac {-2+n}{2 n},c^2 x^{2 n}\right )}{2 (-2+n)} \]
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\[\int \frac {a +b \arcsin \left (c \,x^{n}\right )}{x^{3}}d x\]
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Exception generated. \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Result contains complex when optimal does not.
Time = 4.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=- \frac {a}{2 x^{2}} - \frac {i b c c^{- \frac {2}{n}} c^{-1 + \frac {2}{n}} \Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {x^{- 2 n}}{c^{2}}} \right )}}{4 x^{2} \Gamma \left (1 - \frac {1}{n}\right )} - \frac {b \operatorname {asin}{\left (c x^{n} \right )}}{2 x^{2}} \]
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\[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=\int { \frac {b \arcsin \left (c x^{n}\right ) + a}{x^{3}} \,d x } \]
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\[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=\int { \frac {b \arcsin \left (c x^{n}\right ) + a}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x^n\right )}{x^3} \,d x \]
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