Optimal. Leaf size=59 \[ \frac{x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{2 c^2}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{2 c^4}+\frac{x}{c^3}-\frac{\tan ^{-1}(c x)}{c^4} \]
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Rubi [A] time = 0.0897483, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6342, 266, 51, 63, 208, 321, 203} \[ \frac{x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{2 c^2}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{2 c^4}+\frac{x}{c^3}-\frac{\tan ^{-1}(c x)}{c^4} \]
Antiderivative was successfully verified.
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Rule 6342
Rule 266
Rule 51
Rule 63
Rule 208
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)} x^3}{1+c^2 x^2} \, dx &=\frac{\int \frac{x}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{c^2}+\frac{\int \frac{x^2}{1+c^2 x^2} \, dx}{c}\\ &=\frac{x}{c^3}-\frac{\int \frac{1}{1+c^2 x^2} \, dx}{c^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c^2}\\ &=\frac{x}{c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^2}{2 c^2}-\frac{\tan ^{-1}(c x)}{c^4}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 c^4}\\ &=\frac{x}{c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^2}{2 c^2}-\frac{\tan ^{-1}(c x)}{c^4}+\frac{\operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{2 c^2}\\ &=\frac{x}{c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^2}{2 c^2}-\frac{\tan ^{-1}(c x)}{c^4}-\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.131322, size = 54, normalized size = 0.92 \[ -\frac{-c x \left (c x \sqrt{\frac{1}{c^2 x^2}+1}+2\right )+\log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+2 \tan ^{-1}(c x)}{2 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.15, size = 133, normalized size = 2.3 \begin{align*}{\frac{x}{2\,{c}^{4}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ( x\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{c}^{2}+\ln \left ( x+\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}} \right ) -2\,\ln \left ( x+\sqrt{-{\frac{ \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }{{c}^{4}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{x}{{c}^{3}}}-{\frac{\arctan \left ( cx \right ) }{{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.49774, size = 142, normalized size = 2.41 \begin{align*} \frac{x}{c^{3}} + \frac{\frac{2 \, \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}} - 1} - \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 )}{4 \, c^{4}} - \frac{\arctan \left (c x\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61218, size = 162, normalized size = 2.75 \begin{align*} \frac{c^{2} x^{2} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, c x - 2 \, \arctan \left (c x\right ) + \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right )}{2 \, c^{4}} \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^{2}}{c^{2} x^{2} + 1}\, dx + \int \frac{c x^{3} \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 = 1.13793, size = 82, normalized size = 1.39 \begin{align*} \frac{\sqrt{c^{2} x^{2} + 1} x{\left | c \right |} \mathrm{sgn}\left (x\right )}{2 \, c^{4}} + \frac{x}{c^{3}} + \frac{\log \left (-x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{2 \, c^{4}} - \frac{\arctan \left (c x\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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