Optimal. Leaf size=30 \[ -\sqrt{\frac{1}{c^2 x^2}+1}-\frac{1}{c x}-\tan ^{-1}(c x) \]
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Rubi [A] time = 0.071148, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6342, 261, 325, 203} \[ -\sqrt{\frac{1}{c^2 x^2}+1}-\frac{1}{c x}-\tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6342
Rule 261
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)}}{x \left (1+c^2 x^2\right )} \, dx &=\frac{\int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^3} \, dx}{c^2}+\frac{\int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx}{c}\\ &=-\sqrt{1+\frac{1}{c^2 x^2}}-\frac{1}{c x}-c \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\sqrt{1+\frac{1}{c^2 x^2}}-\frac{1}{c x}-\tan ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.101257, size = 30, normalized size = 1. \[ -\sqrt{\frac{1}{c^2 x^2}+1}-\frac{1}{c x}-\tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 154, normalized size = 5.1 \begin{align*} -{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ( \left ({\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}} \right ) ^{{\frac{3}{2}}}{c}^{2}-\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{x}^{2}{c}^{2}-\ln \left ( x+\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}} \right ) x+\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 ) x \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}-{\frac{1}{cx}}-\arctan \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52835, size = 46, normalized size = 1.53 \begin{align*} -\frac{\sqrt{c^{2} x^{2} + 1}}{c x} - \frac{1}{c x} - \arctan \left (c x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58744, size = 97, normalized size = 3.23 \begin{align*} -\frac{c x \arctan \left (c x\right ) + c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + c x + 1}{c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1543, size = 58, normalized size = 1.93 \begin{align*} \frac{2 \, \mathrm{sgn}\left (x\right )}{{\left (x{\left | c \right |} - \sqrt{c^{2} x^{2} + 1}\right )}^{2} - 1} - \frac{1}{c x} - \arctan \left (c x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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