Optimal. Leaf size=39 \[ \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.18, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5656, 5781, 3301} \[ \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5656
Rule 5781
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{-1}(a x)^2} \, dx &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+a \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)} \, dx\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 60, normalized size = 1.54 \[ \frac {\sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )-a x+1}{a \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\operatorname {arcosh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 0.85 \[ \frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\mathrm {arccosh}\left (a x \right )}+\Chi \left (\mathrm {arccosh}\left (a x \right )\right )}{a} \]
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 \[ -\frac {a^{3} x^{3} + {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a x}{{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )} + \int \frac {a^{4} x^{4} - 2 \, a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (2 \, a^{3} x^{3} - a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 1}{{\left (a^{4} x^{4} + {\left (a x + 1\right )} {\left (a x - 1\right )} a^{2} x^{2} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 1\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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