Optimal. Leaf size=62 \[ -\frac{1}{10} i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (a x^5\right )}\right )-\frac{1}{10} i \cos ^{-1}\left (a x^5\right )^2+\frac{1}{5} \cos ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right ) \]
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Rubi [A] time = 0.0574571, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 4831
Rule 3719
Rule 2190
Rule 2279
Rule 2391
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*}
Mathematica [A] time = 0.0313788, size = 56, normalized size = 0.9 \[ -\frac{1}{10} i \left (\text{PolyLog}\left (2,-e^{2 i \cos ^{-1}\left (a x^5\right )}\right )+\cos ^{-1}\left (a x^5\right ) \left (\cos ^{-1}\left (a x^5\right )+2 i \log \left (1+e^{2 i \cos ^{-1}\left (a x^5\right )}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{\arccos \left ( a{x}^{5} \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arccos \left (a x^{5}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}{\left (a x^{5} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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