Optimal. Leaf size=33 \[ -\frac{\log \left (c^2 x^2+1\right )}{2 c}+\frac{\log (x)}{c}-\frac{\text{csch}^{-1}(c x)}{c} \]
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Rubi [A] time = 0.0428856, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {6340, 335, 215, 266, 36, 29, 31} \[ -\frac{\log \left (c^2 x^2+1\right )}{2 c}+\frac{\log (x)}{c}-\frac{\text{csch}^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 6340
Rule 335
Rule 215
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)}}{1+c^2 x^2} \, dx &=\frac{\int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c^2}+\frac{\int \frac{1}{x \left (1+c^2 x^2\right )} \, dx}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{\text{csch}^{-1}(c x)}{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 )\\ &=-\frac{\text{csch}^{-1}(c x)}{c}+\frac{\log (x)}{c}-\frac{\log \left (1+c^2 x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0331613, size = 37, normalized size = 1.12 \[ -\frac{\log \left (c^2 x^2+1\right )}{2 c}+\frac{\log (x)}{c}-\frac{\sinh ^{-1}\left (\frac{1}{c x}\right )}{c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.15, size = 172, normalized size = 5.2 \begin{align*}{\frac{x}{{c}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ( \sqrt{{c}^{-2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{c}^{2}-\sqrt{-{\frac{1}{{c}^{4}} \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }}{c}^{2}\sqrt{{c}^{-2}}-\ln \left ( 2\,{\frac{1}{x{c}^{2}} \left ( \sqrt{{c}^{-2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{c}^{2}+1 \right ) } \right ) \right ){\frac{1}{\sqrt{{c}^{-2}}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{\ln \left ( x \right ) }{c}}-{\frac{\ln \left ({c}^{2}{x}^{2}+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 (c^{2} x^{2} + 1\right )}{2 \, c} + \frac{\log \left (x\right )}{c} + \int \frac{\sqrt{c^{2} x^{2} + 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 = 2.67739, size = 194, normalized size = 5.88 \begin{align*} -\frac{\log \left (c^{2} x^{2} + 1\right ) + 2 \, \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{c x \sqrt{1 + \frac{1}{c^{2} x^{2}}}}{c^{2} x^{3} + x}\, dx + \int \frac{1}{c^{2} x^{3} + x}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15889, size = 95, normalized size = 2.88 \begin{align*} -\frac{\log \left (c^{2} x^{2} + 1\right )}{2 \, c} - \frac{{\left ({\left | c \right |} \mathrm{sgn}\left (x\right ) - c\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{2 \, c^{2}} + \frac{{\left ({\left | c \right |} \mathrm{sgn}\left (x\right ) + c\right )} \log \left (\sqrt{c^{2} x^{2} + 1} - 1\right )}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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