Optimal. Leaf size=53 \[ -\frac {\log (x)}{a^2}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^2 \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6285, 5418, 4184, 3475} \[ -\frac {\log (x)}{a^2}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 5418
Rule 6285
Rubi steps
\begin {align*} \int x \text {sech}^{-1}(a x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \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)^2-\frac {\operatorname {Subst}\left (\int x \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^2+\frac {\operatorname {Subst}\left (\int \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^2-\frac {\log (x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 53, normalized size = 1.00 \[ -\frac {\log (x)}{a^2}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 106, normalized size = 2.00 \[ \frac {a^{2} x^{2} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 2 \, a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) - 2 \, \log \relax (x)}{2 \, a^{2}} \]
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 \[ \int x \operatorname {arsech}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.61, size = 101, normalized size = 1.91 \[ -\frac {\mathrm {arcsech}\left (a x \right )}{a^{2}}+\frac {x^{2} \mathrm {arcsech}\left (a x \right )^{2}}{2}-\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \mathrm {arcsech}\left (a x \right ) x}{a}+\frac {\ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 40, normalized size = 0.75 \[ \frac {1}{2} \, x^{2} \operatorname {arsech}\left (a x\right )^{2} - \frac {x \sqrt {\frac {1}{a^{2} x^{2}} - 1} \operatorname {arsech}\left (a x\right )}{a} - \frac {\log \relax (x)}{a^{2}} \]
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 \[ \int x\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 42, normalized size = 0.79 \[ \begin {cases} \frac {x^{2} \operatorname {asech}^{2}{\left (a x \right )}}{2} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {asech}{\left (a x \right )}}{a^{2}} - \frac {\log {\relax (x )}}{a^{2}} & \text {for}\: a \neq 0 \\\infty x^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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