Optimal. Leaf size=62 \[ \frac{1}{10} i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} i \sec ^{-1}\left (a x^5\right )^2-\frac{1}{5} \sec ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right ) \]
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Rubi [A] time = 0.0870773, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5218, 4626, 3719, 2190, 2279, 2391} \[ \frac{1}{10} i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} i \sec ^{-1}\left (a x^5\right )^2-\frac{1}{5} \sec ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 5218
Rule 4626
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\sec ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int \frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x} \, dx,x,\frac{1}{x^5}\right )\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int x \tan (x) \, dx,x,\sec ^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} i \sec ^{-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,\sec ^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} i \sec ^{-1}\left (a x^5\right )^2-\frac{1}{5} \sec ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} i \sec ^{-1}\left (a x^5\right )^2-\frac{1}{5} \sec ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \sec ^{-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 \sec ^{-1}\left (a x^5\right )}\right )\\ &=\frac{1}{10} i \sec ^{-1}\left (a x^5\right )^2-\frac{1}{5} \sec ^{-1}\left (a x^5\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} i \text{Li}_2\left (-e^{2 i \sec ^{-1}\left (a x^5\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0336328, size = 56, normalized size = 0.9 \[ \frac{1}{10} i \left (\text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^5\right )}\right )+\sec ^{-1}\left (a x^5\right ) \left (\sec ^{-1}\left (a x^5\right )+2 i \log \left (1+e^{2 i \sec ^{-1}\left (a x^5\right )}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.26, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arcsec} \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*} -5 \, a^{2} \int \frac{\sqrt{a x^{5} + 1} \sqrt{a x^{5} - 1} \log \left (x\right )}{a^{4} x^{11} - a^{2} x}\,{d x} - 5 i \, a^{2} \int \frac{\log \left (x\right )}{a^{4} x^{11} - a^{2} x}\,{d x} + \arctan \left (\sqrt{a x^{5} + 1} \sqrt{a x^{5} - 1}\right ) \log \left (x\right ) - \frac{1}{2} i \, \log \left (a^{2} x^{10}\right ) \log \left (x\right ) + \frac{1}{2} i \, \log \left (a x^{5} + 1\right ) \log \left (x\right ) + \frac{1}{2} i \, \log \left (-a x^{5} + 1\right ) \log \left (x\right ) + i \, \log \left (a\right ) \log \left (x\right ) + \frac{5}{2} i \, \log \left (x\right )^{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 ) \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{\operatorname{arcsec}\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{asec}{\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{\operatorname{arcsec}\left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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