Optimal. Leaf size=78 \[ \frac{1}{3} a^2 c^2 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{1}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.136212, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6175, 6178, 807, 266, 47, 63, 208} \[ \frac{1}{3} a^2 c^2 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{1}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 807
Rule 266
Rule 47
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 ) \sqrt{1-\frac{x^2}{a^2}}}{x^4} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{3} a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3+\left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3+\frac{1}{2} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a}\\ &=-\frac{1}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3+\frac{1}{2} \left (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{1}{2} a c^2 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{3} a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0961057, size = 64, normalized size = 0.82 \[ \frac{c^2 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-3 a x-2\right )+3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.141, size = 121, normalized size = 1.6 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{2}}{6\,a} \left ( 3\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-3\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.117, size = 244, normalized size = 3.13 \begin{align*} \frac{1}{6} \, a{\left (\frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (3 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 8 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 3 \, 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 = 1.49306, size = 231, normalized size = 2.96 \begin{align*} \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 5 \, a c^{2} x - 2 \, 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 - \frac{2 a x}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{a^{2} x^{2}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{1}{\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 [B] time = 1.22836, size = 224, normalized size = 2.87 \begin{align*} \frac{1}{6} \, a{\left (\frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, c^{2} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{8 \,{\left (a x - 1\right )} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 3 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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