Optimal. Leaf size=88 \[ -\frac{x^2 \sqrt{1-c x}}{3 c^3 \sqrt{\frac{1}{c x+1}}}-\frac{x^2}{2 c^3}-\frac{\log \left (1-c^2 x^2\right )}{2 c^5}-\frac{2 \sqrt{1-c x}}{3 c^5 \sqrt{\frac{1}{c x+1}}} \]
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Rubi [A] time = 0.180897, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6341, 1956, 100, 12, 74, 266, 43} \[ -\frac{x^2 \sqrt{1-c x}}{3 c^3 \sqrt{\frac{1}{c x+1}}}-\frac{x^2}{2 c^3}-\frac{\log \left (1-c^2 x^2\right )}{2 c^5}-\frac{2 \sqrt{1-c x}}{3 c^5 \sqrt{\frac{1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 6341
Rule 1956
Rule 100
Rule 12
Rule 74
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)} x^4}{1-c^2 x^2} \, dx &=\frac{\int \frac{x^3 \sqrt{\frac{1}{1+c x}}}{\sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{x^3}{1-c^2 x^2} \, dx}{c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )}{2 c}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{x^2 \sqrt{1-c x}}{3 c^3 \sqrt{\frac{1}{1+c x}}}+\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{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{2 x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{3 c^3}\\ &=-\frac{x^2}{2 c^3}-\frac{x^2 \sqrt{1-c x}}{3 c^3 \sqrt{\frac{1}{1+c x}}}-\frac{\log \left (1-c^2 x^2\right )}{2 c^5}+\frac{\left (2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{3 c^3}\\ &=-\frac{x^2}{2 c^3}-\frac{2 \sqrt{1-c x}}{3 c^5 \sqrt{\frac{1}{1+c x}}}-\frac{x^2 \sqrt{1-c x}}{3 c^3 \sqrt{\frac{1}{1+c x}}}-\frac{\log \left (1-c^2 x^2\right )}{2 c^5}\\ \end{align*}
Mathematica [A] time = 0.18891, size = 69, normalized size = 0.78 \[ -\frac{3 c^2 x^2+2 \sqrt{\frac{1-c x}{c x+1}} \left (c^3 x^3+c^2 x^2+2 c x+2\right )+3 \log \left (1-c^2 x^2\right )}{6 c^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.211, size = 69, normalized size = 0.8 \begin{align*} -{\frac{x \left ({c}^{2}{x}^{2}+2 \right ) }{3\,{c}^{4}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}-{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{1}{2} \, x^{2}}{c^{3}} - \frac{\log \left (c x + 1\right )}{2 \, c^{5}} - \frac{\log \left (c x - 1\right )}{2 \, c^{5}} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} x^{3}}{c^{3} x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06668, size = 149, normalized size = 1.69 \begin{align*} -\frac{3 \, c^{2} x^{2} + 2 \,{\left (c^{3} x^{3} + 2 \, c x\right )} \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{4}{\left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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