Optimal. Leaf size=71 \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c}+\frac{\log (x)}{c}-\frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )}{c} \]
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Rubi [A] time = 0.119409, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6339, 1956, 92, 208, 266, 36, 29, 31} \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c}+\frac{\log (x)}{c}-\frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 6339
Rule 1956
Rule 92
Rule 208
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)}}{1-c^2 x^2} \, dx &=\frac{\int \frac{\sqrt{\frac{1}{1+c x}}}{x \sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{1}{x \left (1-c^2 x^2\right )} \, dx}{c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 c}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 c}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )-\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{c-c x^2} \, dx,x,\sqrt{1-c x} \sqrt{1+c x}\right )\\ &=-\frac{\sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{1+c x}\right )}{c}+\frac{\log (x)}{c}-\frac{\log \left (1-c^2 x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0398584, size = 73, normalized size = 1.03 \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c}+\frac{2 \log (x)}{c}-\frac{\log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )}{c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.204, size = 87, normalized size = 1.2 \begin{align*} -{x\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{\ln \left ( cx-1 \right ) }{2\,c}}+{\frac{\ln \left ( x \right ) }{c}}-{\frac{\ln \left ( cx+1 \right ) }{2\,c}} \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*} \frac{\log \left (x\right )}{c} - \frac{\log \left (c x + 1\right )}{2 \, c} - \frac{\log \left (c x - 1\right )}{2 \, c} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{3} - c x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07632, size = 213, normalized size = 3. \begin{align*} -\frac{\log \left (c^{2} x^{2} - 1\right ) + \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 1\right ) - \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - 1\right ) - 2 \, \log \left (x\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.4875, size = 48, normalized size = 0.68 \begin{align*} - \frac{\log{\left (-1 + \frac{1}{c x} \right )}}{2 c} - \frac{\log{\left (\sqrt{1 + \frac{1}{c x}} \right )}}{c} - \frac{2 \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{1 + \frac{1}{c x}}}{2} \right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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