Optimal. Leaf size=69 \[ \frac {i \text {Li}_2\left (e^{2 i \csc ^{-1}\left (a x^n\right )}\right )}{2 n}+\frac {i \csc ^{-1}\left (a x^n\right )^2}{2 n}-\frac {\csc ^{-1}\left (a x^n\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^n\right )}\right )}{n} \]
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Rubi [A] time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5219, 4625, 3717, 2190, 2279, 2391} \[ \frac {i \text {PolyLog}\left (2,e^{2 i \csc ^{-1}\left (a x^n\right )}\right )}{2 n}+\frac {i \csc ^{-1}\left (a x^n\right )^2}{2 n}-\frac {\csc ^{-1}\left (a x^n\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^n\right )}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 5219
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}\left (a x^n\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\csc ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x} \, dx,x,x^{-n}\right )}{n}\\ &=-\frac {\operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\frac {x^{-n}}{a}\right )\right )}{n}\\ &=\frac {i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )^2}{2 n}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac {x^{-n}}{a}\right )\right )}{n}\\ &=\frac {i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )^2}{2 n}-\frac {\sin ^{-1}\left (\frac {x^{-n}}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )}\right )}{n}+\frac {\operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac {x^{-n}}{a}\right )\right )}{n}\\ &=\frac {i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )^2}{2 n}-\frac {\sin ^{-1}\left (\frac {x^{-n}}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )}\right )}{n}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )}\right )}{2 n}\\ &=\frac {i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )^2}{2 n}-\frac {\sin ^{-1}\left (\frac {x^{-n}}{a}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )}\right )}{n}+\frac {i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\frac {x^{-n}}{a}\right )}\right )}{2 n}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 63, normalized size = 0.91 \[ \log (x) \left (\csc ^{-1}\left (a x^n\right )-\sin ^{-1}\left (\frac {x^{-n}}{a}\right )\right )-\frac {x^{-n} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {x^{-2 n}}{a^2}\right )}{a n} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccsc}\left (a x^{n}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 166, normalized size = 2.41 \[ \frac {i \mathrm {arccsc}\left (a \,x^{n}\right )^{2}}{2 n}-\frac {\mathrm {arccsc}\left (a \,x^{n}\right ) \ln \left (1+\frac {i x^{-n}}{a}+\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )}{n}-\frac {\mathrm {arccsc}\left (a \,x^{n}\right ) \ln \left (1-\frac {i x^{-n}}{a}-\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )}{n}+\frac {i \polylog \left (2, -\frac {i x^{-n}}{a}-\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )}{n}+\frac {i \polylog \left (2, \frac {i x^{-n}}{a}+\sqrt {1-\frac {x^{-2 n}}{a^{2}}}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} n \int \frac {\sqrt {a x^{n} + 1} \sqrt {a x^{n} - 1} \log \relax (x)}{a^{4} x x^{2 \, n} - a^{2} x}\,{d x} + \arctan \left (1, \sqrt {a x^{n} + 1} \sqrt {a x^{n} - 1}\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asin}\left (\frac {1}{a\,x^n}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acsc}{\left (a x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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