Optimal. Leaf size=65 \[ -\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}+2 c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A] time = 0.167051, antiderivative size = 65, 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, 1807, 807, 266, 63, 208} \[ -\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}+2 c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
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) \, dx &=-\left ((a c) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right ) x \, dx\right )\\ &=(a c) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{2} (a c) \operatorname{Subst}\left (\int \frac{\frac{4}{a}-\frac{3 x}{a^2}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=2 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=2 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a}\\ &=2 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{2} (3 a c) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=2 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{3 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0737628, size = 53, normalized size = 0.82 \[ -\frac{c \left (a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-4)+3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.124, size = 153, normalized size = 2.4 \begin{align*} -{\frac{c \left ( ax+1 \right ) }{2\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( \sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) a-4\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+4\,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 = 1.02672, size = 182, normalized size = 2.8 \begin{align*} \frac{1}{2} \, a{\left (\frac{2 \,{\left (5 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 3 \, c \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{2 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac{3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58394, size = 197, normalized size = 3.03 \begin{align*} -\frac{3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a^{2} c x^{2} - 3 \, a c x - 4 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, 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 \left (\int a x \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 = 1.17012, size = 92, normalized size = 1.42 \begin{align*} \frac{3 \, c \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}{2} \, \sqrt{a^{2} x^{2} - 1}{\left (c x \mathrm{sgn}\left (a x + 1\right ) - \frac{4 \, c \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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