Optimal. Leaf size=92 \[ \frac{x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{4 c^2}+\frac{x^3}{3 c^3}-\frac{3 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{8 c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{8 c^6}-\frac{x}{c^5}+\frac{\tan ^{-1}(c x)}{c^6} \]
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Rubi [A] time = 0.108045, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 9, 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, 302, 203} \[ \frac{x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{4 c^2}+\frac{x^3}{3 c^3}-\frac{3 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{8 c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{8 c^6}-\frac{x}{c^5}+\frac{\tan ^{-1}(c x)}{c^6} \]
Antiderivative was successfully verified.
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Rule 6342
Rule 266
Rule 51
Rule 63
Rule 208
Rule 302
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)} x^5}{1+c^2 x^2} \, dx &=\frac{\int \frac{x^3}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{c^2}+\frac{\int \frac{x^4}{1+c^2 x^2} \, dx}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c^2}+\frac{\int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{c}\\ &=-\frac{x}{c^5}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\int \frac{1}{1+c^2 x^2} \, dx}{c^5}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{8 c^4}\\ &=-\frac{x}{c^5}-\frac{3 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{8 c^4}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\tan ^{-1}(c x)}{c^6}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 c^6}\\ &=-\frac{x}{c^5}-\frac{3 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{8 c^4}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\tan ^{-1}(c x)}{c^6}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{8 c^4}\\ &=-\frac{x}{c^5}-\frac{3 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{8 c^4}+\frac{x^3}{3 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^4}{4 c^2}+\frac{\tan ^{-1}(c x)}{c^6}+\frac{3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{8 c^6}\\ \end{align*}
Mathematica [A] time = 0.207719, size = 85, normalized size = 0.92 \[ \frac{c x \left (6 c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+8 c^2 x^2-9 c x \sqrt{\frac{1}{c^2 x^2}+1}-24\right )+9 \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+24 \tan ^{-1}(c x)}{24 c^6} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.201, size = 165, normalized size = 1.8 \begin{align*}{\frac{x}{8\,{c}^{6}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ( 2\,x \left ({\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}} \right ) ^{3/2}{c}^{4}-5\,x\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{c}^{2}-5\,\ln \left ( x+\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}} \right ) +8\,\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}^{3}}{3\,{c}^{3}}}-{\frac{x}{{c}^{5}}}+{\frac{\arctan \left ( cx \right ) }{{c}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53694, size = 216, normalized size = 2.35 \begin{align*} \frac{c^{2} x^{3} - 3 \, x}{3 \, c^{5}} + \frac{\frac{2 \,{\left (3 \, \left (\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}\right )^{\frac{3}{2}} - 5 \, \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )}}{\frac{2 \,{\left (c^{2} x^{2} + 1\right )}}{c^{2} x^{2}} - \frac{{\left (c^{2} x^{2} + 1\right )}^{2}}{c^{4} x^{4}} - 1} + 3 \, \log \left (\sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1\right ) - 3 \, \log \left (\sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 1\right )}{16 \, c^{6}} + \frac{\arctan \left (c x\right )}{c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.02309, size = 209, normalized size = 2.27 \begin{align*} \frac{8 \, c^{3} x^{3} - 24 \, c x + 3 \,{\left (2 \, c^{4} x^{4} - 3 \, c^{2} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 24 \, \arctan \left (c x\right ) - 9 \, \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right )}{24 \, c^{6}} \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^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac{c x^{5} \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 = 1.20418, size = 120, normalized size = 1.3 \begin{align*} \frac{1}{8} \, \sqrt{c^{2} x^{2} + 1} x{\left (\frac{2 \, x^{2}{\left | c \right |} \mathrm{sgn}\left (x\right )}{c^{4}} - \frac{3 \,{\left | c \right |} \mathrm{sgn}\left (x\right )}{c^{6}}\right )} - \frac{3 \, \log \left (-x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{8 \, c^{6}} + \frac{\arctan \left (c x\right )}{c^{6}} + \frac{c^{6} x^{3} - 3 \, c^{4} x}{3 \, c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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