Optimal. Leaf size=102 \[ \frac {3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {3 \text {sech}^{-1}(a x) \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right )}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3 \]
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Rubi [A] time = 0.12, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6285, 5418, 4184, 3718, 2190, 2279, 2391} \[ \frac {3 \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a x)}\right )}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {3 \text {sech}^{-1}(a x) \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right )}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 4184
Rule 5418
Rule 6285
Rubi steps
\begin {align*} \int x \text {sech}^{-1}(a x)^3 \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \text {sech}^2(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^2}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3-\frac {3 \operatorname {Subst}\left (\int x^2 \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3+\frac {3 \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3+\frac {6 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3+\frac {3 \text {sech}^{-1}(a x) \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )}{a^2}-\frac {3 \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3+\frac {3 \text {sech}^{-1}(a x) \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )}{a^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a x)}\right )}{2 a^2}\\ &=-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3+\frac {3 \text {sech}^{-1}(a x) \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )}{a^2}+\frac {3 \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 101, normalized size = 0.99 \[ \frac {\text {sech}^{-1}(a x) \left (a^2 x^2 \text {sech}^{-1}(a x)^2-3 \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}-1\right ) \text {sech}^{-1}(a x)+6 \log \left (e^{-2 \text {sech}^{-1}(a x)}+1\right )\right )-3 \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(a x)}\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arsech}\left (a x\right )^{3}, x\right ) \]
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 \[ \int x \operatorname {arsech}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 152, normalized size = 1.49 \[ \frac {x^{2} \mathrm {arcsech}\left (a x \right )^{3}}{2}-\frac {3 \mathrm {arcsech}\left (a x \right )^{2} \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, x}{2 a}-\frac {3 \mathrm {arcsech}\left (a x \right )^{2}}{2 a^{2}}+\frac {3 \,\mathrm {arcsech}\left (a x \right ) \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{a^{2}}+\frac {3 \polylog \left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2 a^{2}} \]
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 \[ \int x \operatorname {arsech}\left (a x\right )^{3}\,{d x} \]
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 \[ \int x\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3 \,d x \]
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 \[ \int x \operatorname {asech}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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