Optimal. Leaf size=37 \[ \frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c^2}+\frac{\tanh ^{-1}(c x)}{c^2} \]
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Rubi [A] time = 0.113641, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6341, 6677, 41, 216, 206} \[ \frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c^2}+\frac{\tanh ^{-1}(c x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 6341
Rule 6677
Rule 41
Rule 216
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)} x}{1-c^2 x^2} \, dx &=\frac{\int \frac{\sqrt{\frac{1}{1+c x}}}{\sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{1}{1-c^2 x^2} \, dx}{c}\\ &=\frac{\tanh ^{-1}(c x)}{c^2}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=\frac{\tanh ^{-1}(c x)}{c^2}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{c}\\ &=\frac{\sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{c^2}+\frac{\tanh ^{-1}(c x)}{c^2}\\ \end{align*}
Mathematica [C] time = 0.044183, size = 68, normalized size = 1.84 \[ -\frac{\log (1-c x)}{2 c^2}+\frac{\log (c x+1)}{2 c^2}+\frac{i \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{c^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.227, size = 92, normalized size = 2.5 \begin{align*}{\frac{x{\it csgn} \left ( c \right ) }{c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}\arctan \left ({{\it csgn} \left ( c \right ) cx{\frac{1}{\sqrt{- \left ( cx+1 \right ) \left ( cx-1 \right ) }}}} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{\ln \left ( cx-1 \right ) }{2\,{c}^{2}}}+{\frac{\ln \left ( cx+1 \right ) }{2\,{c}^{2}}} \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 x + 1\right )}{2 \, c^{2}} - \frac{\log \left (c x - 1\right )}{2 \, c^{2}} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1015, size = 131, normalized size = 3.54 \begin{align*} -\frac{2 \, \arctan \left (\sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}}\right ) - \log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \, 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 x}} \sqrt{1 + \frac{1}{c x}}}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x{\left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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