Optimal. Leaf size=45 \[ \frac{2 (a x+1)}{c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.167767, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 12, 266, 63, 208} \[ \frac{2 (a x+1)}{c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6128
Rule 852
Rule 1805
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x (c-a c x)} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x (c-a c x)^2} \, dx\\ &=\frac{\int \frac{(c+a c x)^2}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{2 (1+a x)}{c \sqrt{1-a^2 x^2}}+\frac{\int \frac{c^2}{x \sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=\frac{2 (1+a x)}{c \sqrt{1-a^2 x^2}}+\frac{\int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{2 (1+a x)}{c \sqrt{1-a^2 x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{2 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^2 c}\\ &=\frac{2 (1+a x)}{c \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0245177, size = 55, normalized size = 1.22 \[ \frac{-\sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+2 a x+2}{c \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 61, normalized size = 1.4 \begin{align*} -{\frac{1}{c} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +2\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) } \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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53129, size = 124, normalized size = 2.76 \begin{align*} \frac{2 \, a x +{\left (a x - 1\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1} - 2}{a c x - 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{a x}{a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \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 = 1.27417, size = 108, normalized size = 2.4 \begin{align*} -\frac{a \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{c{\left | a \right |}} + \frac{4 \, a}{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]