Optimal. Leaf size=42 \[ \frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5666, 3301} \[ \frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5666
Rubi steps
\begin {align*} \int \frac {x}{\cosh ^{-1}(a x)^2} \, dx &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 44, normalized size = 1.05 \[ \frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )-\frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\cosh ^{-1}(a x)}}{a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\operatorname {arcosh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {x}{\operatorname {arcosh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 0.67 \[ \frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2 \,\mathrm {arccosh}\left (a x \right )}+\Chi \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{2}} \]
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 \[ -\frac {a^{3} x^{4} - a x^{2} + {\left (a^{2} x^{3} - x\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )} + \int \frac {2 \, a^{5} x^{5} + 2 \, {\left (a x + 1\right )} {\left (a x - 1\right )} a^{3} x^{3} - 4 \, a^{3} x^{3} + {\left (4 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 2 \, a x}{{\left (a^{5} x^{4} + {\left (a x + 1\right )} {\left (a x - 1\right )} a^{3} x^{2} - 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} - a^{2} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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