Optimal. Leaf size=72 \[ \frac{x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{3 c^2}+\frac{x^2}{2 c^3}-\frac{2 x \sqrt{\frac{1}{c^2 x^2}+1}}{3 c^4}-\frac{\log \left (c^2 x^2+1\right )}{2 c^5} \]
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Rubi [A] time = 0.0975695, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6342, 271, 191, 266, 43} \[ \frac{x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{3 c^2}+\frac{x^2}{2 c^3}-\frac{2 x \sqrt{\frac{1}{c^2 x^2}+1}}{3 c^4}-\frac{\log \left (c^2 x^2+1\right )}{2 c^5} \]
Antiderivative was successfully verified.
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Rule 6342
Rule 271
Rule 191
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)} x^4}{1+c^2 x^2} \, dx &=\frac{\int \frac{x^2}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{c^2}+\frac{\int \frac{x^3}{1+c^2 x^2} \, dx}{c}\\ &=\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^3}{3 c^2}-\frac{2 \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{3 c^4}+\frac{\operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{2 \sqrt{1+\frac{1}{c^2 x^2}} x}{3 c^4}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^3}{3 c^2}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{2 c}\\ &=-\frac{2 \sqrt{1+\frac{1}{c^2 x^2}} x}{3 c^4}+\frac{x^2}{2 c^3}+\frac{\sqrt{1+\frac{1}{c^2 x^2}} x^3}{3 c^2}-\frac{\log \left (1+c^2 x^2\right )}{2 c^5}\\ \end{align*}
Mathematica [A] time = 0.127156, size = 64, normalized size = 0.89 \[ \frac{c x \left (2 c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}-4 \sqrt{\frac{1}{c^2 x^2}+1}+3 c x\right )-3 \log \left (c^2 x^2+1\right )}{6 c^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.143, size = 120, normalized size = 1.7 \begin{align*}{\frac{x}{3\,{c}^{4}}\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}-3\,\sqrt{-{\frac{ \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }{{c}^{4}}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{{x}^{2}}{2\,{c}^{3}}}-{\frac{\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02718, size = 66, normalized size = 0.92 \begin{align*} \frac{x^{2}}{2 \, c^{3}} + \frac{\sqrt{c^{2} x^{2} + 1}{\left (c^{2} x^{2} - 2\right )}}{3 \, c^{5}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{2 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.98138, size = 127, normalized size = 1.76 \begin{align*} \frac{3 \, c^{2} x^{2} + 2 \,{\left (c^{3} x^{3} - 2 \, c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 3 \, \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{5}} \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^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac{c x^{4} \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 = 1.14811, size = 101, normalized size = 1.4 \begin{align*} \frac{2 \,{\left | c \right |} \mathrm{sgn}\left (x\right )}{3 \, c^{6}} + \frac{2 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left | c \right |} \mathrm{sgn}\left (x\right ) - 6 \, \sqrt{c^{2} x^{2} + 1}{\left | c \right |} \mathrm{sgn}\left (x\right ) + 3 \,{\left (c^{2} x^{2} + 1\right )} c - 3 \, c \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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