Optimal. Leaf size=16 \[ -\frac{e^{-\tanh ^{-1}(a x)}}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0311933, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6137} \[ -\frac{e^{-\tanh ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6137
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac{e^{-\tanh ^{-1}(a x)}}{a c}\\ \end{align*}
Mathematica [A] time = 0.0101139, size = 27, normalized size = 1.69 \[ -\frac{\sqrt{1-a x}}{a c \sqrt{a x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 28, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( ax+1 \right ) ac}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}{\left (a x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.0768, size = 66, normalized size = 4.12 \begin{align*} -\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 1}{a^{2} c x + a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1947, size = 50, normalized size = 3.12 \begin{align*} \frac{2}{c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]