Optimal. Leaf size=61 \[ \frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{2 b^2}{x} \]
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Rubi [A] time = 0.0699774, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6285, 3296, 2638} \[ \frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{2 b^2}{x} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x^2} \, dx &=-\left (c \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}+(2 b c) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{2 b^2}{x}+\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x}\\ \end{align*}
Mathematica [A] time = 0.228817, size = 87, normalized size = 1.43 \[ -\frac{a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 b \text{sech}^{-1}(c x) \left (b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a\right )+b^2 \text{sech}^{-1}(c x)^2+2 b^2}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.217, size = 124, normalized size = 2. \begin{align*} c \left ( -{\frac{{a}^{2}}{cx}}+{b}^{2} \left ( -{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{cx}}+2\,{\rm arcsech} \left (cx\right )\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}-2\,{\frac{1}{cx}} \right ) +2\,ab \left ( -{\frac{{\rm arcsech} \left (cx\right )}{cx}}+\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02864, size = 105, normalized size = 1.72 \begin{align*} 2 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{\operatorname{arsech}\left (c x\right )}{x}\right )} a b + 2 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right ) - \frac{1}{x}\right )} b^{2} - \frac{b^{2} \operatorname{arsech}\left (c x\right )^{2}}{x} - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65977, size = 301, normalized size = 4.93 \begin{align*} \frac{2 \, a b c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - b^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - a^{2} - 2 \, b^{2} + 2 \,{\left (b^{2} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \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*} \int \frac{\left (a + b \operatorname{asech}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \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{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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