Optimal. Leaf size=103 \[ -\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^2 (3 a x+2) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}+\frac{c^2 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.169505, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6157, 6149, 811, 813, 844, 216, 266, 63, 208} \[ -\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^2 (3 a x+2) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}+\frac{c^2 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6149
Rule 811
Rule 813
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac{c^2 \int \frac{(1-a x) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{a^4}\\ &=-\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}-\frac{c^2 \int \frac{\left (4 a^2-6 a^3 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{4 a^4}\\ &=\frac{c^2 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^2 \int \frac{12 a^3+8 a^4 x}{x \sqrt{1-a^2 x^2}} \, dx}{8 a^4}\\ &=\frac{c^2 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+c^2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\frac{\left (3 c^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{c^2 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^2 \sin ^{-1}(a x)}{a}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac{c^2 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^2 \sin ^{-1}(a x)}{a}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{2 a^3}\\ &=\frac{c^2 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^2 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^2 \sin ^{-1}(a x)}{a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}\\ \end{align*}
Mathematica [C] time = 0.0245761, size = 70, normalized size = 0.68 \[ \frac{c^2 \left (3 a^3 \left (1-a^2 x^2\right )^{5/2} \text{Hypergeometric2F1}\left (2,\frac{5}{2},\frac{7}{2},1-a^2 x^2\right )-\frac{5 \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},a^2 x^2\right )}{x^3}\right )}{15 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 157, normalized size = 1.5 \begin{align*}{\frac{{c}^{2}}{{a}^{2}x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{c}^{2}x\sqrt{-{a}^{2}{x}^{2}+1}+{{c}^{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{3\,{c}^{2}}{2\,a}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{c}^{2}}{2\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{{c}^{2}}{2\,{x}^{2}{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{{c}^{2}}{3\,{a}^{4}{x}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \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*} c^{2}{\left (\frac{\arcsin \left (a x\right )}{a} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a}\right )} - \int \frac{{\left (2 \, a^{2} c^{2} x^{2} - c^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{a^{5} x^{5} + a^{4} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82589, size = 282, normalized size = 2.74 \begin{align*} -\frac{12 \, a^{3} c^{2} x^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 9 \, a^{3} c^{2} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{2} x^{3} -{\left (6 \, a^{3} c^{2} x^{3} + 8 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.0891, size = 381, normalized size = 3.7 \begin{align*} \frac{c^{2} \left (\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left (a x \right )} + \frac{\log{\left (a^{2} x^{2} \right )}}{2} + i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left (a^{2} x^{2} \right )}}{2} - \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )} & \text{otherwise} \end{cases}\right )}{a} - \frac{c^{2} \left (\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left (a x \right )} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left (a x \right )} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right )}{a^{2}} - \frac{c^{2} \left (\begin{cases} \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right )}{a^{3}} + \frac{c^{2} \left (\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2255, size = 354, normalized size = 3.44 \begin{align*} \frac{{\left (c^{2} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{2}}{a^{2} x} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{2}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} + \frac{c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{3 \, c^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{a} + \frac{\frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{2}}{x} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{2}}{a^{4} x^{3}}}{24 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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