Optimal. Leaf size=104 \[ -\frac {\log (x)}{3 a^4}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2}{12 a^2}-\frac {x^2 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2 \]
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Rubi [A] time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6285, 5418, 4185, 4184, 3475} \[ -\frac {x^2}{12 a^2}-\frac {x^2 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{6 a^2}-\frac {\log (x)}{3 a^4}-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 4185
Rule 5418
Rule 6285
Rubi steps
\begin {align*} \int x^3 \text {sech}^{-1}(a x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}^4(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^4}\\ &=\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\operatorname {Subst}\left (\int x \text {sech}^4(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{2 a^4}\\ &=-\frac {x^2}{12 a^2}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\operatorname {Subst}\left (\int x \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^2}{12 a^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2+\frac {\operatorname {Subst}\left (\int \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^2}{12 a^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{6 a^2}+\frac {1}{4} x^4 \text {sech}^{-1}(a x)^2-\frac {\log (x)}{3 a^4}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 77, normalized size = 0.74 \[ -\frac {-3 a^4 x^4 \text {sech}^{-1}(a x)^2+a^2 x^2+2 \sqrt {\frac {1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2+2 a x+2\right ) \text {sech}^{-1}(a x)+4 \log (x)}{12 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 125, normalized size = 1.20 \[ \frac {3 \, a^{4} x^{4} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - a^{2} x^{2} - 2 \, {\left (a^{3} x^{3} + 2 \, a x\right )} \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) - 4 \, \log \relax (x)}{12 \, a^{4}} \]
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 \[ \int x^{3} \operatorname {arsech}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 151, normalized size = 1.45 \[ -\frac {\mathrm {arcsech}\left (a x \right )}{3 a^{4}}+\frac {x^{4} \mathrm {arcsech}\left (a x \right )^{2}}{4}-\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \mathrm {arcsech}\left (a x \right ) x^{3}}{6 a}-\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \mathrm {arcsech}\left (a x \right ) x}{3 a^{3}}-\frac {x^{2}}{12 a^{2}}+\frac {\ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{3 a^{4}} \]
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 \[ \int x^{3} \operatorname {arsech}\left (a x\right )^{2}\,{d x} \]
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 \[ \int x^3\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \]
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 \[ \int x^{3} \operatorname {asech}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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