Optimal. Leaf size=54 \[ -\frac {1}{10} \text {Li}_2\left (-e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )+\frac {1}{10} \text {sech}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {sech}^{-1}\left (a x^5\right ) \log \left (e^{2 \text {sech}^{-1}\left (a x^5\right )}+1\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 54, 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 = {6281, 5660, 3718, 2190, 2279, 2391} \[ -\frac {1}{10} \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )+\frac {1}{10} \text {sech}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {sech}^{-1}\left (a x^5\right ) \log \left (e^{2 \text {sech}^{-1}\left (a x^5\right )}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 6281
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}\left (a x^5\right )}{x} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {\text {sech}^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )}{x} \, dx,x,\frac {1}{x^5}\right )\right )\\ &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\text {sech}^{-1}\left (a x^5\right )\right )\right )\\ &=\frac {1}{10} \text {sech}^{-1}\left (a x^5\right )^2-\frac {2}{5} \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}\left (a x^5\right )\right )\\ &=\frac {1}{10} \text {sech}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {sech}^{-1}\left (a x^5\right ) \log \left (1+e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}\left (a x^5\right )\right )\\ &=\frac {1}{10} \text {sech}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {sech}^{-1}\left (a x^5\right ) \log \left (1+e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )+\frac {1}{10} \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )\\ &=\frac {1}{10} \text {sech}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {sech}^{-1}\left (a x^5\right ) \log \left (1+e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )-\frac {1}{10} \text {Li}_2\left (-e^{2 \text {sech}^{-1}\left (a x^5\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.91 \[ \frac {1}{10} \left (\text {Li}_2\left (-e^{-2 \text {sech}^{-1}\left (a x^5\right )}\right )-\text {sech}^{-1}\left (a x^5\right ) \left (\text {sech}^{-1}\left (a x^5\right )+2 \log \left (e^{-2 \text {sech}^{-1}\left (a x^5\right )}+1\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (a x^{5}\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arsech}\left (a x^{5}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arcsech}\left (a \,x^{5}\right )}{x}\, dx \]
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 \[ \int \frac {\operatorname {arsech}\left (a x^{5}\right )}{x}\,{d x} \]
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 \[ \int \frac {\mathrm {acosh}\left (\frac {1}{a\,x^5}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {asech}{\left (a x^{5} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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