Optimal. Leaf size=122 \[ \frac{4 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x}+\frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{3 x^3}-\frac{4 b^2 c^2}{9 x}-\frac{2 b^2}{27 x^3} \]
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Rubi [A] time = 0.101873, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6285, 5447, 3310, 3296, 2638} \[ \frac{4 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x}+\frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{3 x^3}-\frac{4 b^2 c^2}{9 x}-\frac{2 b^2}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 5447
Rule 3310
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{x^4} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int (a+b x)^2 \cosh ^2(x) \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \cosh ^3(x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{2 b^2}{27 x^3}+\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{9} \left (4 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{2 b^2}{27 x^3}+\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x^3}+\frac{4 b c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{9} \left (4 b^2 c^3\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{sech}^{-1}(c x)\right )\\ &=-\frac{2 b^2}{27 x^3}-\frac{4 b^2 c^2}{9 x}+\frac{2 b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x^3}+\frac{4 b c^2 \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{9 x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.249804, size = 134, normalized size = 1.1 \[ \frac{-9 a^2+6 a b \sqrt{\frac{1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )+6 b \text{sech}^{-1}(c x) \left (b \sqrt{\frac{1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )-3 a\right )-2 b^2 \left (6 c^2 x^2+1\right )-9 b^2 \text{sech}^{-1}(c x)^2}{27 x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.225, size = 226, normalized size = 1.9 \begin{align*}{c}^{3} \left ( -{\frac{{a}^{2}}{3\,{c}^{3}{x}^{3}}}+{b}^{2} \left ({\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2} \left ( cx-1 \right ) \left ( cx+1 \right ) }{3\,{c}^{3}{x}^{3}}}-{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{3\,cx}}+{\frac{2\,{\rm arcsech} \left (cx\right )}{9\,{c}^{2}{x}^{2}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{4\,{\rm arcsech} \left (cx\right )}{9}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{ \left ( 2\,cx-2 \right ) \left ( cx+1 \right ) }{27\,{c}^{3}{x}^{3}}}-{\frac{14}{27\,cx}} \right ) +2\,ab \left ( -1/3\,{\frac{{\rm arcsech} \left (cx\right )}{{c}^{3}{x}^{3}}}+1/9\,{\frac{2\,{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{9} \, a b{\left (\frac{c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 3 \, c^{4} \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c} - \frac{3 \, \operatorname{arsech}\left (c x\right )}{x^{3}}\right )} + b^{2} \int \frac{\log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )^{2}}{x^{4}}\,{d x} - \frac{a^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65368, size = 392, normalized size = 3.21 \begin{align*} -\frac{12 \, b^{2} c^{2} x^{2} + 9 \, b^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 9 \, a^{2} + 2 \, b^{2} + 6 \,{\left (3 \, a b -{\left (2 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 6 \,{\left (2 \, a b c^{3} x^{3} + a b c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{27 \, 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*} \int \frac{\left (a + b \operatorname{asech}{\left (c x \right )}\right )^{2}}{x^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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