Optimal. Leaf size=88 \[ -\frac {3}{2} \text {sech}^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {3}{2} \text {sech}^{-1}(a x) \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{4} \text {Li}_4\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6285, 3718, 2190, 2531, 6609, 2282, 6589} \[ -\frac {3}{2} \text {sech}^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {3}{2} \text {sech}^{-1}(a x) \text {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{4} \text {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 6285
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a x)^3}{x} \, dx &=-\operatorname {Subst}\left (\int x^3 \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{4} \text {sech}^{-1}(a x)^4-2 \operatorname {Subst}\left (\int \frac {e^{2 x} x^3}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )+3 \operatorname {Subst}\left (\int x^2 \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{2} \text {sech}^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+3 \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{2} \text {sech}^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {3}{2} \text {sech}^{-1}(a x) \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{2} \text {sech}^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {3}{2} \text {sech}^{-1}(a x) \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a x)}\right )\\ &=\frac {1}{4} \text {sech}^{-1}(a x)^4-\text {sech}^{-1}(a x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{2} \text {sech}^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {3}{2} \text {sech}^{-1}(a x) \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a x)}\right )-\frac {3}{4} \text {Li}_4\left (-e^{2 \text {sech}^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 84, normalized size = 0.95 \[ \frac {1}{4} \left (6 \text {sech}^{-1}(a x)^2 \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(a x)}\right )+6 \text {sech}^{-1}(a x) \text {Li}_3\left (-e^{-2 \text {sech}^{-1}(a x)}\right )+3 \text {Li}_4\left (-e^{-2 \text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x)^4-4 \text {sech}^{-1}(a x)^3 \log \left (e^{-2 \text {sech}^{-1}(a x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (a x\right )^{3}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 181, normalized size = 2.06 \[ \frac {\mathrm {arcsech}\left (a x \right )^{4}}{4}-\mathrm {arcsech}\left (a x \right )^{3} \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )-\frac {3 \mathrm {arcsech}\left (a x \right )^{2} \polylog \left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2}+\frac {3 \,\mathrm {arcsech}\left (a x \right ) \polylog \left (3, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2}-\frac {3 \polylog \left (4, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{4} \]
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 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3}{x} \,d x \]
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 \[ \int \frac {\operatorname {asech}^{3}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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