Optimal. Leaf size=85 \[ c^2 \tanh ^{-1}(c x)-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{c x+1}}}-\frac{1}{3 c x^3}-\frac{2 c \sqrt{1-c x}}{3 x \sqrt{\frac{1}{c x+1}}}-\frac{c}{x} \]
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Rubi [A] time = 0.165316, antiderivative size = 85, 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, 103, 12, 95, 325, 206} \[ c^2 \tanh ^{-1}(c x)-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{c x+1}}}-\frac{1}{3 c x^3}-\frac{2 c \sqrt{1-c x}}{3 x \sqrt{\frac{1}{c x+1}}}-\frac{c}{x} \]
Antiderivative was successfully verified.
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Rule 6341
Rule 1956
Rule 103
Rule 12
Rule 95
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)}}{x^3 \left (1-c^2 x^2\right )} \, dx &=\frac{\int \frac{\sqrt{\frac{1}{1+c x}}}{x^4 \sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{1}{x^4 \left (1-c^2 x^2\right )} \, dx}{c}\\ &=-\frac{1}{3 c x^3}+c \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^4 \sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{1}{3 c x^3}-\frac{c}{x}-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{1+c x}}}+c^3 \int \frac{1}{1-c^2 x^2} \, dx-\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{2 c^2}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx}{3 c}\\ &=-\frac{1}{3 c x^3}-\frac{c}{x}-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{1+c x}}}+c^2 \tanh ^{-1}(c x)+\frac{1}{3} \left (2 c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{1}{3 c x^3}-\frac{c}{x}-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{1+c x}}}-\frac{2 c \sqrt{1-c x}}{3 x \sqrt{\frac{1}{1+c x}}}+c^2 \tanh ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.241577, size = 90, normalized size = 1.06 \[ -\frac{6 c^2 x^2+2 \sqrt{\frac{1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )+3 c^3 x^3 \log (1-c x)-3 c^3 x^3 \log (c x+1)+2}{6 c x^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.22, size = 86, normalized size = 1. \begin{align*} -{\frac{ \left ({\it csgn} \left ( c \right ) \right ) ^{2} \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{3\,{x}^{2}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}-{\frac{{c}^{2}\ln \left ( cx-1 \right ) }{2}}-{\frac{1}{3\,c{x}^{3}}}-{\frac{c}{x}}+{\frac{{c}^{2}\ln \left ( cx+1 \right ) }{2}} \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{1}{2} \, c^{2} \log \left (c x + 1\right ) - \frac{1}{2} \, c^{2} \log \left (c x - 1\right ) + c \int \frac{1}{x^{2}}\,{d x} + \frac{-\frac{1}{3 \, x^{3}}}{c} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{6} - c x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20589, size = 197, normalized size = 2.32 \begin{align*} \frac{3 \, c^{3} x^{3} \log \left (c x + 1\right ) - 3 \, c^{3} x^{3} \log \left (c x - 1\right ) - 6 \, c^{2} x^{2} - 2 \,{\left (2 \, c^{3} x^{3} + c x\right )} \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - 2}{6 \, c x^{3}} \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{c x \sqrt{-1 + \frac{1}{c x}} \sqrt{1 + \frac{1}{c x}}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac{1}{c^{2} x^{6} - x^{4}}\, dx}{c} \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{\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}}{{\left (c^{2} x^{2} - 1\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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