Optimal. Leaf size=62 \[ \frac {1}{10} i \text {Li}_2\left (e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac {1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac {1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 62, 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} \[ \frac {1}{10} i \text {PolyLog}\left (2,e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac {1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac {1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right ) \]
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^5\right )}{x} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {\csc ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x} \, dx,x,\frac {1}{x^5}\right )\right )\\ &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}\left (a x^5\right )\right )\right )\\ &=\frac {1}{10} i \csc ^{-1}\left (a x^5\right )^2+\frac {2}{5} i \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}\left (a x^5\right )\right )\\ &=\frac {1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac {1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}\left (a x^5\right )\right )\\ &=\frac {1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac {1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )-\frac {1}{10} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}\left (a x^5\right )}\right )\\ &=\frac {1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac {1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac {1}{10} i \text {Li}_2\left (e^{2 i \csc ^{-1}\left (a x^5\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.90 \[ \frac {1}{10} i \left (\text {Li}_2\left (e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\csc ^{-1}\left (a x^5\right ) \left (\csc ^{-1}\left (a x^5\right )+2 i \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccsc}\left (a x^{5}\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arccsc}\left (a x^{5}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccsc}\left (a \,x^{5}\right )}{x}\, dx \]
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 \[ 5 \, a^{2} \int \frac {\sqrt {a x^{5} + 1} \sqrt {a x^{5} - 1} \log \relax (x)}{a^{4} x^{11} - a^{2} x}\,{d x} - 5 i \, a^{2} \int \frac {\log \relax (x)}{a^{4} x^{11} - a^{2} x}\,{d x} + {\left (\arctan \left (1, \sqrt {a x^{5} + 1} \sqrt {a x^{5} - 1}\right ) + i \, \log \relax (a)\right )} \log \relax (x) - \frac {1}{2} i \, \log \left (a^{2} x^{10}\right ) \log \relax (x) + \frac {1}{2} i \, \log \left (a x^{5} + 1\right ) \log \relax (x) + \frac {1}{2} i \, \log \left (-a x^{5} + 1\right ) \log \relax (x) + \frac {5}{2} i \, \log \relax (x)^{2} + \frac {1}{10} i \, {\rm Li}_2\left (a x^{5}\right ) + \frac {1}{10} i \, {\rm Li}_2\left (-a x^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 58, normalized size = 0.94 \[ -\frac {\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x^5}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{a\,x^5}\right )}{5}+\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x^5}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{10}+\frac {{\mathrm {asin}\left (\frac {1}{a\,x^5}\right )}^2\,1{}\mathrm {i}}{10} \]
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 \[ \int \frac {\operatorname {acsc}{\left (a x^{5} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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