Optimal. Leaf size=46 \[ \frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5713, 5676} \[ \frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 5676
Rule 5713
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.00 \[ \frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {\operatorname {arcosh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 41, normalized size = 0.89 \[ \frac {2 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {a x -1}\, \sqrt {a x +1}}{a \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {\operatorname {arcosh}\left (a x\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\mathrm {acosh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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