Optimal. Leaf size=137 \[ -\frac{1}{4} a^2 \text{sech}^{-1}(a x)^3-\frac{3}{8} a^2 \text{sech}^{-1}(a x)+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{8 x^2}-\frac{(1-a x) (a x+1) \text{sech}^{-1}(a x)^3}{2 x^2}+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{4 x^2}-\frac{3 (1-a x) (a x+1) \text{sech}^{-1}(a x)}{4 x^2} \]
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Rubi [A] time = 0.0862309, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6285, 5372, 3311, 30, 2635, 8} \[ -\frac{1}{4} a^2 \text{sech}^{-1}(a x)^3-\frac{3}{8} a^2 \text{sech}^{-1}(a x)+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{8 x^2}-\frac{(1-a x) (a x+1) \text{sech}^{-1}(a x)^3}{2 x^2}+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{4 x^2}-\frac{3 (1-a x) (a x+1) \text{sech}^{-1}(a x)}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 5372
Rule 3311
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a x)^3}{x^3} \, dx &=-\left (a^2 \operatorname{Subst}\left (\int x^3 \cosh (x) \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\right )\\ &=-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^3}{2 x^2}+\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \sinh ^2(x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{3 (1-a x) (1+a x) \text{sech}^{-1}(a x)}{4 x^2}+\frac{3 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 x^2}-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^3}{2 x^2}-\frac{1}{4} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\text{sech}^{-1}(a x)\right )+\frac{1}{4} \left (3 a^2\right ) \operatorname{Subst}\left (\int \sinh ^2(x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=\frac{3 \sqrt{\frac{1-a x}{1+a x}} (1+a x)}{8 x^2}-\frac{3 (1-a x) (1+a x) \text{sech}^{-1}(a x)}{4 x^2}+\frac{3 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 x^2}-\frac{1}{4} a^2 \text{sech}^{-1}(a x)^3-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^3}{2 x^2}-\frac{1}{8} \left (3 a^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=\frac{3 \sqrt{\frac{1-a x}{1+a x}} (1+a x)}{8 x^2}-\frac{3}{8} a^2 \text{sech}^{-1}(a x)-\frac{3 (1-a x) (1+a x) \text{sech}^{-1}(a x)}{4 x^2}+\frac{3 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 x^2}-\frac{1}{4} a^2 \text{sech}^{-1}(a x)^3-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^3}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.133262, size = 147, normalized size = 1.07 \[ \frac{-3 a^2 x^2 \log (x)+3 a^2 x^2 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )+2 \left (a^2 x^2-2\right ) \text{sech}^{-1}(a x)^3+3 \sqrt{\frac{1-a x}{a x+1}} (a x+1)+6 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-6 \text{sech}^{-1}(a x)}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.217, size = 126, normalized size = 0.9 \begin{align*}{a}^{2} \left ( -{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{3}}{2\,{a}^{2}{x}^{2}}}+{\frac{3\, \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{4\,ax}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{3}}{4}}-{\frac{3\,{\rm arcsech} \left (ax\right )}{4\,{a}^{2}{x}^{2}}}+{\frac{3}{8\,ax}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{3\,{\rm arcsech} \left (ax\right )}{8}} \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*} \int \frac{\operatorname{arsech}\left (a x\right )^{3}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03459, size = 382, normalized size = 2.79 \begin{align*} \frac{6 \, a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} + 2 \,{\left (a^{2} x^{2} - 2\right )} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{3} + 3 \, a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 3 \,{\left (a^{2} x^{2} - 2\right )} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )}{8 \, x^{2}} \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{\operatorname{asech}^{3}{\left (a x \right )}}{x^{3}}\, 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{\operatorname{arsech}\left (a x\right )^{3}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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