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 = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 8
Rule 30
Rule 2635
Rule 3311
Rule 5372
Rule 6285
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*}
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Mathematica [A] time = 0.15, 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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fricas [A] time = 0.63, size = 174, normalized size = 1.27 \[ \frac {6 \, a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} + 2 \, {\left (a^{2} x^{2} - 2\right )} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{3} + 3 \, a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 3 \, {\left (a^{2} x^{2} - 2\right )} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 126, normalized size = 0.92 \[ a^{2} \left (-\frac {\mathrm {arcsech}\left (a x \right )^{3}}{2 a^{2} x^{2}}+\frac {3 \mathrm {arcsech}\left (a x \right )^{2} \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{4 a x}+\frac {\mathrm {arcsech}\left (a x \right )^{3}}{4}-\frac {3 \,\mathrm {arcsech}\left (a x \right )}{4 a^{2} x^{2}}+\frac {3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{8 a x}+\frac {3 \,\mathrm {arcsech}\left (a x \right )}{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asech}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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