Optimal. Leaf size=26 \[ -\frac{\text{Chi}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a} \]
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Rubi [A] time = 0.038764, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {6681, 3301} \[ -\frac{\text{Chi}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6681
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh \left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ &=-\frac{\text{Chi}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0325367, size = 26, normalized size = 1. \[ -\frac{\text{Chi}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1}\cosh \left ({\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\cosh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cosh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )}{a^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\cosh{\left (\frac{\sqrt{- a x + 1}}{\sqrt{a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\cosh \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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