Optimal. Leaf size=27 \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c^2}+\frac{\tan ^{-1}(c x)}{c^2} \]
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Rubi [A] time = 0.0592035, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6342, 266, 63, 208, 203} \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c^2}+\frac{\tan ^{-1}(c x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 6342
Rule 266
Rule 63
Rule 208
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)} x}{1+c^2 x^2} \, dx &=\frac{\int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x} \, dx}{c^2}+\frac{\int \frac{1}{1+c^2 x^2} \, dx}{c}\\ &=\frac{\tan ^{-1}(c x)}{c^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c^2}\\ &=\frac{\tan ^{-1}(c x)}{c^2}-\operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )\\ &=\frac{\tan ^{-1}(c x)}{c^2}+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c^2}\\ \end{align*}
Mathematica [A] time = 0.0486192, size = 38, normalized size = 1.41 \[ \frac{\log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{c^2}+\frac{\tan ^{-1}(c x)}{c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.14, size = 85, normalized size = 3.2 \begin{align*}{\frac{x}{{c}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}\ln \left ( x+\sqrt{-{\frac{1}{{c}^{4}} \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{\arctan \left ( cx \right ) }{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55592, size = 80, normalized size = 2.96 \begin{align*} \frac{\log \left (\sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1\right ) - \log \left (\sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 1\right )}{2 \, c^{2}} + \frac{\arctan \left (c x\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55209, size = 89, normalized size = 3.3 \begin{align*} \frac{\arctan \left (c x\right ) - \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right )}{c^{2}} \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^{2} + 1}\, dx + \int \frac{1}{c^{2} x^{2} + 1}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15771, size = 46, normalized size = 1.7 \begin{align*} -\frac{\log \left (-x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{c^{2}} + \frac{\arctan \left (c x\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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