Optimal. Leaf size=62 \[ -\frac {1}{10} i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (a x^5\right )}\right )-\frac {1}{10} i \cos ^{-1}\left (a x^5\right )^2+\frac {1}{5} \cos ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right ) \]
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Rubi [A] time = 0.06, 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 = {4831, 3719, 2190, 2279, 2391} \[ -\frac {1}{10} i \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (a x^5\right )}\right )-\frac {1}{10} i \cos ^{-1}\left (a x^5\right )^2+\frac {1}{5} \cos ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4831
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}\left (a x^5\right )}{x} \, dx &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (a x^5\right )\right )\right )\\ &=-\frac {1}{10} i \cos ^{-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,\cos ^{-1}\left (a x^5\right )\right )\\ &=-\frac {1}{10} i \cos ^{-1}\left (a x^5\right )^2+\frac {1}{5} \cos ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right )-\frac {1}{5} \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (a x^5\right )\right )\\ &=-\frac {1}{10} i \cos ^{-1}\left (a x^5\right )^2+\frac {1}{5} \cos ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \cos ^{-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 \cos ^{-1}\left (a x^5\right )}\right )\\ &=-\frac {1}{10} i \cos ^{-1}\left (a x^5\right )^2+\frac {1}{5} \cos ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right )-\frac {1}{10} i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (a x^5\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.90 \[ -\frac {1}{10} i \left (\text {Li}_2\left (-e^{2 i \cos ^{-1}\left (a x^5\right )}\right )+\cos ^{-1}\left (a x^5\right ) \left (\cos ^{-1}\left (a x^5\right )+2 i \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x^{5}\right )}{x}, x\right ) \]
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 \[ \int \frac {\arccos \left (a x^{5}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\arccos \left (a \,x^{5}\right )}{x}\, dx \]
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 \[ \int \frac {\arccos \left (a x^{5}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acos}\left (a\,x^5\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acos}{\left (a x^{5} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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