Optimal. Leaf size=63 \[ \frac {2 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {2 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}+x \text {sech}^{-1}(a x)^2-\frac {4 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6279, 5418, 4180, 2279, 2391} \[ \frac {2 i \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {2 i \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{a}+x \text {sech}^{-1}(a x)^2-\frac {4 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 5418
Rule 6279
Rubi steps
\begin {align*} \int \text {sech}^{-1}(a x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^2-\frac {2 \operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^2-\frac {4 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {(2 i) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}-\frac {(2 i) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^2-\frac {4 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a}\\ &=x \text {sech}^{-1}(a x)^2-\frac {4 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {2 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {2 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 90, normalized size = 1.43 \[ \frac {i \left (2 \text {Li}_2\left (-i e^{-\text {sech}^{-1}(a x)}\right )-2 \text {Li}_2\left (i e^{-\text {sech}^{-1}(a x)}\right )+\text {sech}^{-1}(a x) \left (-i a x \text {sech}^{-1}(a x)+2 \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arsech}\left (a x\right )^{2}, x\right ) \]
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 \[ \int \operatorname {arsech}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 190, normalized size = 3.02 \[ x \mathrm {arcsech}\left (a x \right )^{2}+\frac {2 i \mathrm {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{a}-\frac {2 i \mathrm {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{a}+\frac {2 i \dilog \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{a}-\frac {2 i \dilog \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{a} \]
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 \[ x \log \left (\sqrt {a x + 1} \sqrt {-a x + 1} + 1\right )^{2} - \int -\frac {a^{2} x^{2} \log \relax (a)^{2} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x)^{2} + {\left (a^{2} x^{2} \log \relax (a)^{2} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x)^{2} - \log \relax (a)^{2} + 2 \, {\left (a^{2} x^{2} \log \relax (a) - \log \relax (a)\right )} \log \relax (x)\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - 2 \, {\left (a^{2} x^{2} \log \relax (a) + {\left (a^{2} x^{2} {\left (\log \relax (a) + 1\right )} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x) - \log \relax (a)\right )} \sqrt {a x + 1} \sqrt {-a x + 1} + {\left (a^{2} x^{2} - 1\right )} \log \relax (x) - \log \relax (a)\right )} \log \left (\sqrt {a x + 1} \sqrt {-a x + 1} + 1\right ) - \log \relax (a)^{2} + 2 \, {\left (a^{2} x^{2} \log \relax (a) - \log \relax (a)\right )} \log \relax (x)}{a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - 1}\,{d x} \]
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 \[ \int {\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \]
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 \[ \int \operatorname {asech}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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