Optimal. Leaf size=102 \[ -\frac{6 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}+\frac{6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x} \]
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Rubi [A] time = 0.102711, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6285, 3296, 2637} \[ -\frac{6 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}+\frac{6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x^2} \, dx &=-\left (c \operatorname{Subst}\left (\int (a+b x)^3 \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname{Subst}\left (\int (a+b x)^2 \cosh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}-\left (6 b^2 c\right ) \operatorname{Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{6 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x}+\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}+\left (6 b^3 c\right ) \operatorname{Subst}\left (\int \cosh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=\frac{6 b^3 \sqrt{\frac{1-c x}{1+c x}} (1+c x)}{x}-\frac{6 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x}+\frac{3 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}\\ \end{align*}
Mathematica [A] time = 0.321964, size = 165, normalized size = 1.62 \[ -\frac{3 b \text{sech}^{-1}(c x) \left (a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)+2 b^2\right )-3 a^2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)+a^3-3 b^2 \text{sech}^{-1}(c x)^2 \left (b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a\right )+6 a b^2-6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)+b^3 \text{sech}^{-1}(c x)^3}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.252, size = 227, normalized size = 2.2 \begin{align*} c \left ( -{\frac{{a}^{3}}{cx}}+{b}^{3} \left ( -{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{3}}{cx}}+3\, \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}-6\,{\frac{{\rm arcsech} \left (cx\right )}{cx}}+6\,\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \right ) +3\,a{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 ) +3\,{a}^{2}b \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 = 1.03547, size = 194, normalized size = 1.9 \begin{align*} -\frac{b^{3} \operatorname{arsech}\left (c x\right )^{3}}{x} + 3 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{\operatorname{arsech}\left (c x\right )}{x}\right )} a^{2} b + 6 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right ) - \frac{1}{x}\right )} a b^{2} + 3 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right )^{2} + 2 \, c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{2 \, \operatorname{arsech}\left (c x\right )}{x}\right )} b^{3} - \frac{3 \, a b^{2} \operatorname{arsech}\left (c x\right )^{2}}{x} - \frac{a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68919, size = 485, normalized size = 4.75 \begin{align*} -\frac{b^{3} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \,{\left (b^{3} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \,{\left (2 \, a b^{2} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\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 )^{3}}{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 )}^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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