Optimal. Leaf size=74 \[ -\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-\frac{3 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{3 c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.207967, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6157, 6149, 1807, 1809, 844, 216, 266, 63, 208} \[ -\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-\frac{3 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{3 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6149
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right ) \, dx &=-\frac{c \int \frac{e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac{c \int \frac{(1-a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a^2 x}+\frac{c \int \frac{3 a-3 a^2 x+a^3 x^2}{x \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c \int \frac{-3 a^3+3 a^4 x}{x \sqrt{1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-(3 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\frac{(3 c) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-\frac{3 c \sin ^{-1}(a x)}{a}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-\frac{3 c \sin ^{-1}(a x)}{a}-\frac{(3 c) \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^3}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \sqrt{1-a^2 x^2}}{a^2 x}-\frac{3 c \sin ^{-1}(a x)}{a}-\frac{3 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0628591, size = 57, normalized size = 0.77 \[ -\frac{c \left (\sqrt{1-a^2 x^2} (a x-1)+3 a x \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+3 a x \sin ^{-1}(a x)\right )}{a^2 x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 266, normalized size = 3.6 \begin{align*}{\frac{c}{{a}^{2}x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+cx \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}+{\frac{3\,cx}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,c}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-2\,{\frac{c \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{3} \left ( x+{a}^{-1} \right ) ^{2}}}-3\,{\frac{c \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{3/2}}{a}}-{\frac{9\,cx}{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{9\,c}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{c}{a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-3\,{\frac{c{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }{a}}+3\,{\frac{c\sqrt{-{a}^{2}{x}^{2}+1}}{a}} \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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1854, size = 190, normalized size = 2.57 \begin{align*} \frac{6 \, a c x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a c x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - a c x - \sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )}}{a^{2} x} \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*} \frac{c \left (\int - \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int - \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25703, size = 176, normalized size = 2.38 \begin{align*} -\frac{a^{2} c x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{3 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{3 \, c \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1} c}{a} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c}{2 \, a^{2} x{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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