Optimal. Leaf size=36 \[ \frac{x \sqrt{\frac{1}{c^2 x^2}+1}}{c^2}+\frac{\log \left (c^2 x^2+1\right )}{2 c^3} \]
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Rubi [A] time = 0.0698466, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6342, 191, 260} \[ \frac{x \sqrt{\frac{1}{c^2 x^2}+1}}{c^2}+\frac{\log \left (c^2 x^2+1\right )}{2 c^3} \]
Antiderivative was successfully verified.
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Rule 6342
Rule 191
Rule 260
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)} x^2}{1+c^2 x^2} \, dx &=\frac{\int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{c^2}+\frac{\int \frac{x}{1+c^2 x^2} \, dx}{c}\\ &=\frac{\sqrt{1+\frac{1}{c^2 x^2}} x}{c^2}+\frac{\log \left (1+c^2 x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0671598, size = 35, normalized size = 0.97 \[ \frac{2 c x \sqrt{\frac{1}{c^2 x^2}+1}+\log \left (c^2 x^2+1\right )}{2 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.151, size = 89, normalized size = 2.5 \begin{align*}{\frac{x}{{c}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}\sqrt{-{\frac{1}{{c}^{4}} \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03118, size = 42, normalized size = 1.17 \begin{align*} \frac{\log \left (c^{3} x^{2} + c\right )}{2 \, c^{3}} + \frac{\sqrt{c^{2} x^{2} + 1}}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.60734, size = 89, normalized size = 2.47 \begin{align*} \frac{2 \, c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + \log \left (c^{2} x^{2} + 1\right )}{2 \, c^{3}} \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{x}{c^{2} x^{2} + 1}\, dx + \int \frac{c x^{2} \sqrt{1 + \frac{1}{c^{2} x^{2}}}}{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 = 1.13129, size = 61, normalized size = 1.69 \begin{align*} -\frac{{\left | c \right |} \mathrm{sgn}\left (x\right )}{c^{4}} + \frac{2 \, \sqrt{c^{2} x^{2} + 1}{\left | c \right |} \mathrm{sgn}\left (x\right ) + c \log \left (c^{2} x^{2} + 1\right )}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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