Optimal. Leaf size=117 \[ \frac {i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {i \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {2 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2 \]
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Rubi [A] time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6285, 5418, 4185, 4180, 2279, 2391} \[ \frac {i \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {i \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^2}-\frac {2 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 4185
Rule 5418
Rule 6285
Rubi steps
\begin {align*} \int x^2 \text {sech}^{-1}(a x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}^3(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}\\ &=\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2-\frac {2 \operatorname {Subst}\left (\int x \text {sech}^3(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^3}\\ &=-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2-\frac {\operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^3}\\ &=-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2-\frac {2 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}+\frac {i \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^3}-\frac {i \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^3}\\ &=-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2-\frac {2 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}\\ &=-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2-\frac {2 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}+\frac {i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {i \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 138, normalized size = 1.18 \[ \frac {a^3 x^3 \text {sech}^{-1}(a x)^2+i \text {Li}_2\left (-i e^{-\text {sech}^{-1}(a x)}\right )-i \text {Li}_2\left (i e^{-\text {sech}^{-1}(a x)}\right )-a x-a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)+i \text {sech}^{-1}(a x) \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-i \text {sech}^{-1}(a x) \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )}{3 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arsech}\left (a x\right )^{2}, x\right ) \]
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 \[ \int x^{2} \operatorname {arsech}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 240, normalized size = 2.05 \[ \frac {x^{3} \mathrm {arcsech}\left (a x \right )^{2}}{3}-\frac {\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, x^{2}}{3 a}-\frac {x}{3 a^{2}}+\frac {i \mathrm {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^{3}}-\frac {i \mathrm {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^{3}}+\frac {i \dilog \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^{3}}-\frac {i \dilog \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^{3}} \]
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 \[ \int x^{2} \operatorname {arsech}\left (a x\right )^{2}\,{d x} \]
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 \[ \int x^2\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \]
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 \[ \int x^{2} \operatorname {asech}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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