Optimal. Leaf size=37 \[ b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {a+b \cosh ^{-1}(c x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5662, 92, 205} \[ b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {a+b \cosh ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 5662
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{x}+(b c) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {a+b \cosh ^{-1}(c x)}{x}+\left (b c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=-\frac {a+b \cosh ^{-1}(c x)}{x}+b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 65, normalized size = 1.76 \[ -\frac {a}{x}+\frac {b c \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b \cosh ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 74, normalized size = 2.00 \[ \frac {2 \, b c x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + b x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b x - b\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - a}{x} \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 1.59 \[ -\frac {a}{x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{x}-\frac {c b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 30, normalized size = 0.81 \[ -{\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b - \frac {a}{x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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