Optimal. Leaf size=90 \[ -\frac {1}{4} a^2 \text {sech}^{-1}(a x)^2-\frac {(1-a x) (a x+1)}{4 x^2}-\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^2}{2 x^2}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6285, 5372, 3310, 30} \[ -\frac {1}{4} a^2 \text {sech}^{-1}(a x)^2-\frac {(1-a x) (a x+1)}{4 x^2}-\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^2}{2 x^2}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3310
Rule 5372
Rule 6285
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a x)^2}{x^3} \, dx &=-\left (a^2 \operatorname {Subst}\left (\int x^2 \cosh (x) \sinh (x) \, dx,x,\text {sech}^{-1}(a x)\right )\right )\\ &=-\frac {(1-a x) (1+a x) \text {sech}^{-1}(a x)^2}{2 x^2}+a^2 \operatorname {Subst}\left (\int x \sinh ^2(x) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=-\frac {(1-a x) (1+a x)}{4 x^2}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{2 x^2}-\frac {(1-a x) (1+a x) \text {sech}^{-1}(a x)^2}{2 x^2}-\frac {1}{2} a^2 \operatorname {Subst}\left (\int x \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=-\frac {(1-a x) (1+a x)}{4 x^2}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{2 x^2}-\frac {1}{4} a^2 \text {sech}^{-1}(a x)^2-\frac {(1-a x) (1+a x) \text {sech}^{-1}(a x)^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.60 \[ \frac {\left (a^2 x^2-2\right ) \text {sech}^{-1}(a x)^2+2 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)-1}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 106, normalized size = 1.18 \[ \frac {2 \, a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) + {\left (a^{2} x^{2} - 2\right )} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 1}{4 \, x^{2}} \]
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 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 77, normalized size = 0.86 \[ a^{2} \left (-\frac {\mathrm {arcsech}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{2 a x}+\frac {\mathrm {arcsech}\left (a x \right )^{2}}{4}-\frac {1}{4 x^{2} a^{2}}\right ) \]
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 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{2}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asech}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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