Optimal. Leaf size=47 \[ \frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}-\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}} \]
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Rubi [A] time = 0.0706478, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6175, 6178, 266, 47, 63, 208} \[ \frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}-\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 266
Rule 47
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{\sqrt{1-\frac{x^2}{a^2}}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} (a c) \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 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{c \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 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{2} (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 )\\ &=-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0502585, size = 51, normalized size = 1.09 \[ \frac{c \left (\log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )-a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.135, size = 93, normalized size = 2. \begin{align*} -{\frac{c \left ( ax-1 \right ) }{2} \left ( x\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) \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 = 1.0445, size = 178, normalized size = 3.79 \begin{align*} \frac{1}{2} \, a{\left (\frac{2 \,{\left (c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 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{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{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 = 1.57671, size = 180, normalized size = 3.83 \begin{align*} \frac{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a^{2} c x^{2} + a c x\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 \frac{a x}{\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 = 1.26605, size = 211, normalized size = 4.49 \begin{align*} \frac{1}{4} \, a{\left (\frac{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}} + 2\right )}{a^{2}} - \frac{c \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}} - 2 \right |}\right )}{a^{2}} - \frac{4 \, c{\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}}{{\left ({\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{2} - 4\right )} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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