Optimal. Leaf size=198 \[ \frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {Li}_3\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {i \text {Li}_3\left (i e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )}{a^3}-\frac {\text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {x \text {sech}^{-1}(a x)}{a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3 \]
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Rubi [A] time = 0.14, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6285, 5418, 4186, 3770, 4180, 2531, 2282, 6589} \[ \frac {i \text {sech}^{-1}(a x) \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {sech}^{-1}(a x) \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {i \text {PolyLog}\left (3,i e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )}{a^3}-\frac {x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {x \text {sech}^{-1}(a x)}{a^2}-\frac {\text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3 \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3770
Rule 4180
Rule 4186
Rule 5418
Rule 6285
Rule 6589
Rubi steps
\begin {align*} \int x^2 \text {sech}^{-1}(a x)^3 \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \text {sech}^3(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}\\ &=\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}^3(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}\\ &=-\frac {x \text {sech}^{-1}(a x)}{a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3-\frac {\operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{2 a^3}+\frac {\operatorname {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}\\ &=-\frac {x \text {sech}^{-1}(a x)}{a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3-\frac {\text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x}\right )}{a^3}+\frac {i \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}-\frac {i \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}\\ &=-\frac {x \text {sech}^{-1}(a x)}{a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3-\frac {\text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x}\right )}{a^3}+\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}+\frac {i \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^3}\\ &=-\frac {x \text {sech}^{-1}(a x)}{a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3-\frac {\text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x}\right )}{a^3}+\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a^3}\\ &=-\frac {x \text {sech}^{-1}(a x)}{a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^3-\frac {\text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x}\right )}{a^3}+\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a^3}-\frac {i \text {Li}_3\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a^3}+\frac {i \text {Li}_3\left (i e^{\text {sech}^{-1}(a x)}\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 199, normalized size = 1.01 \[ \frac {2 a^3 x^3 \text {sech}^{-1}(a x)^3+3 i \left (2 \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{-\text {sech}^{-1}(a x)}\right )-2 \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{-\text {sech}^{-1}(a x)}\right )+2 \text {Li}_3\left (-i e^{-\text {sech}^{-1}(a x)}\right )-2 \text {Li}_3\left (i e^{-\text {sech}^{-1}(a x)}\right )+\text {sech}^{-1}(a x)^2 \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x)^2 \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(a x)\right )\right )\right )-3 a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2-6 a x \text {sech}^{-1}(a x)}{6 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arsech}\left (a x\right )^{3}, x\right ) \]
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 \[ \int x^{2} \operatorname {arsech}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int x^{2} \mathrm {arcsech}\left (a x \right )^{3}\, dx \]
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 \[ \int x^{2} \operatorname {arsech}\left (a x\right )^{3}\,{d x} \]
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 \[ \int x^2\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3 \,d x \]
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 \[ \int x^{2} \operatorname {asech}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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