Optimal. Leaf size=60 \[ -\frac{\sqrt{\frac{1}{c^2 x^2}+1}}{2 x}+\frac{1}{2} c \log \left (c^2 x^2+1\right )-\frac{1}{2 c x^2}-c \log (x)+\frac{1}{2} c \text{csch}^{-1}(c x) \]
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Rubi [A] time = 0.0970317, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6342, 335, 321, 215, 266, 44} \[ -\frac{\sqrt{\frac{1}{c^2 x^2}+1}}{2 x}+\frac{1}{2} c \log \left (c^2 x^2+1\right )-\frac{1}{2 c x^2}-c \log (x)+\frac{1}{2} c \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6342
Rule 335
Rule 321
Rule 215
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx &=\frac{\int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^4} \, dx}{c^2}+\frac{\int \frac{1}{x^3 \left (1+c^2 x^2\right )} \, dx}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{\sqrt{1+\frac{1}{c^2 x^2}}}{2 x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )}{2 c}\\ &=-\frac{1}{2 c x^2}-\frac{\sqrt{1+\frac{1}{c^2 x^2}}}{2 x}+\frac{1}{2} c \text{csch}^{-1}(c x)-c \log (x)+\frac{1}{2} c \log \left (1+c^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.108683, size = 58, normalized size = 0.97 \[ \frac{1}{2} \left (-\frac{\sqrt{\frac{1}{c^2 x^2}+1}}{x}+c \log \left (c^2 x^2+1\right )-\frac{1}{c x^2}-2 c \log (x)+c \sinh ^{-1}\left (\frac{1}{c x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.195, size = 210, normalized size = 3.5 \begin{align*} -{\frac{1}{2\,x}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ({c}^{2} \left ({\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{{c}^{-2}}+\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}\sqrt{{c}^{-2}}{x}^{2}{c}^{2}-2\,\sqrt{{c}^{-2}}\sqrt{-{\frac{ \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }{{c}^{4}}}}{x}^{2}{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 ){x}^{2} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}{\frac{1}{\sqrt{{c}^{-2}}}}}-{\frac{1}{2\,c{x}^{2}}}-c\ln \left ( x \right ) +{\frac{c\ln \left ({c}^{2}{x}^{2}+1 \right ) }{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{1}{2} \, c \log \left (c^{2} x^{2} + 1\right ) - c \log \left (x\right ) - \frac{1}{2 \, c x^{2}} + \int \frac{\sqrt{c^{2} x^{2} + 1}}{c^{3} x^{5} + c x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.61346, size = 293, normalized size = 4.88 \begin{align*} \frac{c^{2} x^{2} \log \left (c^{2} x^{2} + 1\right ) + c^{2} x^{2} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - c^{2} x^{2} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 2 \, c^{2} x^{2} \log \left (x\right ) - c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 1}{2 \, c x^{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^{5} + x^{3}}\, dx + \int \frac{1}{c^{2} x^{5} + x^{3}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14435, size = 154, normalized size = 2.57 \begin{align*} \frac{1}{2} \, c \log \left (c^{2} x^{2} + 1\right ) + \frac{1}{4} \,{\left ({\left | c \right |} \mathrm{sgn}\left (x\right ) - 2 \, c\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) - \frac{1}{4} \,{\left ({\left | c \right |} \mathrm{sgn}\left (x\right ) + 2 \, c\right )} \log \left (\sqrt{c^{2} x^{2} + 1} - 1\right ) - \frac{\sqrt{c^{2} x^{2} + 1}{\left | c \right |} \mathrm{sgn}\left (x\right ) + c}{2 \,{\left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{\left (\sqrt{c^{2} x^{2} + 1} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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