Optimal. Leaf size=75 \[ -\frac{x \sqrt{1-c x}}{2 c^3 \sqrt{\frac{1}{c x+1}}}-\frac{x}{c^3}+\frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{2 c^4}+\frac{\tanh ^{-1}(c x)}{c^4} \]
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Rubi [A] time = 0.165123, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6341, 1956, 90, 41, 216, 321, 206} \[ -\frac{x \sqrt{1-c x}}{2 c^3 \sqrt{\frac{1}{c x+1}}}-\frac{x}{c^3}+\frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{2 c^4}+\frac{\tanh ^{-1}(c x)}{c^4} \]
Antiderivative was successfully verified.
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Rule 6341
Rule 1956
Rule 90
Rule 41
Rule 216
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)} x^3}{1-c^2 x^2} \, dx &=\frac{\int \frac{x^2 \sqrt{\frac{1}{1+c x}}}{\sqrt{1-c x}} \, dx}{c}+\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{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{x}{c^3}-\frac{x \sqrt{1-c x}}{2 c^3 \sqrt{\frac{1}{1+c x}}}+\frac{\tanh ^{-1}(c x)}{c^4}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{2 c^3}\\ &=-\frac{x}{c^3}-\frac{x \sqrt{1-c x}}{2 c^3 \sqrt{\frac{1}{1+c x}}}+\frac{\tanh ^{-1}(c x)}{c^4}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 c^3}\\ &=-\frac{x}{c^3}-\frac{x \sqrt{1-c x}}{2 c^3 \sqrt{\frac{1}{1+c x}}}+\frac{\sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{2 c^4}+\frac{\tanh ^{-1}(c x)}{c^4}\\ \end{align*}
Mathematica [C] time = 0.153733, size = 110, normalized size = 1.47 \[ -\frac{c^2 x^2 \sqrt{\frac{1-c x}{c x+1}}+2 c x+c x \sqrt{\frac{1-c x}{c x+1}}+\log (1-c x)-\log (c x+1)-i \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{2 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.213, size = 117, normalized size = 1.6 \begin{align*} -{\frac{x{\it csgn} \left ( c \right ) }{2\,{c}^{3}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( x\sqrt{-{c}^{2}{x}^{2}+1}{\it csgn} \left ( c \right ) c-\arctan \left ({x{\it csgn} \left ( c \right ) c{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{x}{{c}^{3}}}-{\frac{\ln \left ( cx-1 \right ) }{2\,{c}^{4}}}+{\frac{\ln \left ( cx+1 \right ) }{2\,{c}^{4}}} \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{x}{c^{3}} + \frac{\log \left (c x + 1\right )}{2 \, c^{4}} - \frac{\log \left (c x - 1\right )}{2 \, c^{4}} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} x^{2}}{c^{3} x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0929, size = 213, normalized size = 2.84 \begin{align*} -\frac{c^{2} x^{2} \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 2 \, c x + \arctan \left (\sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}}\right ) - \log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \, c^{4}} \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^{3}{\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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