Optimal. Leaf size=62 \[ -\frac {1}{10} i \text {ArcSin}\left (a x^5\right )^2+\frac {1}{5} \text {ArcSin}\left (a x^5\right ) \log \left (1-e^{2 i \text {ArcSin}\left (a x^5\right )}\right )-\frac {1}{10} i \text {PolyLog}\left (2,e^{2 i \text {ArcSin}\left (a x^5\right )}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4914, 3798,
2221, 2317, 2438} \begin {gather*} -\frac {1}{10} i \text {Li}_2\left (e^{2 i \text {ArcSin}\left (a x^5\right )}\right )-\frac {1}{10} i \text {ArcSin}\left (a x^5\right )^2+\frac {1}{5} \text {ArcSin}\left (a x^5\right ) \log \left (1-e^{2 i \text {ArcSin}\left (a x^5\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4914
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}\left (a x^5\right )}{x} \, dx &=\frac {1}{5} \text {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (a x^5\right )\right )\\ &=-\frac {1}{10} i \sin ^{-1}\left (a x^5\right )^2-\frac {2}{5} i \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (a x^5\right )\right )\\ &=-\frac {1}{10} i \sin ^{-1}\left (a x^5\right )^2+\frac {1}{5} \sin ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \sin ^{-1}\left (a x^5\right )}\right )-\frac {1}{5} \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (a x^5\right )\right )\\ &=-\frac {1}{10} i \sin ^{-1}\left (a x^5\right )^2+\frac {1}{5} \sin ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \sin ^{-1}\left (a x^5\right )}\right )+\frac {1}{10} i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (a x^5\right )}\right )\\ &=-\frac {1}{10} i \sin ^{-1}\left (a x^5\right )^2+\frac {1}{5} \sin ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \sin ^{-1}\left (a x^5\right )}\right )-\frac {1}{10} i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (a x^5\right )}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 58, normalized size = 0.94 \begin {gather*} \frac {1}{5} \left (\text {ArcSin}\left (a x^5\right ) \log \left (1-e^{2 i \text {ArcSin}\left (a x^5\right )}\right )-\frac {1}{2} i \left (\text {ArcSin}\left (a x^5\right )^2+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}\left (a x^5\right )}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (a \,x^{5}\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asin}{\left (a x^{5} \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 50, normalized size = 0.81 \begin {gather*} -\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (a\,x^5\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{10}+\frac {\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (a\,x^5\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (a\,x^5\right )}{5}-\frac {{\mathrm {asin}\left (a\,x^5\right )}^2\,1{}\mathrm {i}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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