Optimal. Leaf size=100 \[ \frac{1}{3} a^2 c^2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{11}{3} c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.286164, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {6175, 6178, 1807, 807, 266, 63, 208} \[ \frac{1}{3} a^2 c^2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{11}{3} c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx &=\left (a^2 c^2\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^2 x^2 \, dx\\ &=-\left (\left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{1}{3} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\frac{9}{a}-\frac{11 x}{a^2}+\frac{3 x^2}{a^3}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{6} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\frac{22}{a^2}-\frac{15 x}{a^3}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{11}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a}\\ &=\frac{11}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{2} \left (5 a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{11}{3} c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.113812, size = 64, normalized size = 0.64 \[ \frac{c^2 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-9 a x+22\right )-15 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.132, size = 176, normalized size = 1.8 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{2}}{6\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-9\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+24\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02074, size = 244, normalized size = 2.44 \begin{align*} -\frac{1}{6} \, a{\left (\frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{2 \,{\left (33 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 40 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 15 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6813, size = 240, normalized size = 2.4 \begin{align*} -\frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (2 \, a^{3} c^{2} x^{3} - 7 \, a^{2} c^{2} x^{2} + 13 \, a c^{2} x + 22 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, a} \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*} c^{2} \left (\int - 2 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16438, size = 122, normalized size = 1.22 \begin{align*} \frac{5 \, c^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{2 \,{\left | a \right |}} + \frac{1}{6} \, \sqrt{a^{2} x^{2} - 1}{\left ({\left (2 \, a c^{2} x \mathrm{sgn}\left (a x + 1\right ) - 9 \, c^{2} \mathrm{sgn}\left (a x + 1\right )\right )} x + \frac{22 \, c^{2} \mathrm{sgn}\left (a x + 1\right )}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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