Optimal. Leaf size=111 \[ -\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}+\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{5 c^2 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.295462, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6131, 6128, 1805, 1807, 1809, 844, 216, 266, 63, 208} \[ -\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}+\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{5 c^2 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6128
Rule 1805
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 x}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{-3 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \frac{(1-a x)^5}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \int \frac{-1+5 a x+5 a^2 x^2-a^3 x^3}{x^2 \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}+\frac{c^2 \int \frac{-5 a-5 a^2 x+a^3 x^2}{x \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c^2 \int \frac{5 a^3+5 a^4 x}{x \sqrt{1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\left (5 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{\left (5 c^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{5 c^2 \sin ^{-1}(a x)}{a}-\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{5 c^2 \sin ^{-1}(a x)}{a}+\frac{\left (5 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 )}{a^3}\\ &=-\frac{16 c^2 (1-a x)}{a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{1-a^2 x^2}}{a}-\frac{c^2 \sqrt{1-a^2 x^2}}{a^2 x}-\frac{5 c^2 \sin ^{-1}(a x)}{a}+\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.216785, size = 81, normalized size = 0.73 \[ \frac{c^2 \left (-\frac{\sqrt{1-a^2 x^2} \left (a^2 x^2+18 a x+1\right )}{a x (a x+1)}+5 \log \left (\sqrt{1-a^2 x^2}+1\right )-5 \log (a x)-5 \sin ^{-1}(a x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.052, size = 329, normalized size = 3. \begin{align*} -4\,{\frac{{c}^{2} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{4} \left ( x+{a}^{-1} \right ) ^{3}}}-4\,{\frac{{c}^{2} \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}}}-{\frac{7\,{c}^{2}}{3\,a} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{7\,x{c}^{2}}{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{7\,{c}^{2}}{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}^{2}}{{a}^{2}x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{c}^{2}x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}-{\frac{3\,x{c}^{2}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{c}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{5\,{c}^{2}}{3\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-5\,{\frac{{c}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{a}}+5\,{\frac{{c}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }{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 x}\right )}^{2}}{{\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 = 2.47556, size = 313, normalized size = 2.82 \begin{align*} -\frac{17 \, a^{2} c^{2} x^{2} + 17 \, a c^{2} x - 10 \,{\left (a^{2} c^{2} x^{2} + a c^{2} x\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 5 \,{\left (a^{2} c^{2} x^{2} + a c^{2} x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (a^{2} c^{2} x^{2} + 18 \, a c^{2} x + c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{a^{3} x^{2} + 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^{2} \left (\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int - \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int - \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 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 = 1.20234, size = 266, normalized size = 2.4 \begin{align*} -\frac{5 \, c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{5 \, 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 )}{{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{a} + \frac{{\left (c^{2} + \frac{65 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{2}}{a^{2} x}\right )} a^{2} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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