Optimal. Leaf size=198 \[ \frac{i \text{sech}^{-1}(a x) \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a x)}\right )}{a^3}-\frac{i \text{sech}^{-1}(a x) \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a x)}\right )}{a^3}-\frac{i \text{PolyLog}\left (3,-i e^{\text{sech}^{-1}(a x)}\right )}{a^3}+\frac{i \text{PolyLog}\left (3,i e^{\text{sech}^{-1}(a x)}\right )}{a^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a x}\right )}{a^3}-\frac{x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{2 a^2}-\frac{x \text{sech}^{-1}(a x)}{a^2}-\frac{\text{sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text{sech}^{-1}(a x)}\right )}{a^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a x)^3 \]
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Rubi [A] time = 0.140893, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, 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 6285
Rule 5418
Rule 4186
Rule 3770
Rule 4180
Rule 2531
Rule 2282
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*}
Mathematica [A] time = 0.460283, size = 199, normalized size = 1.01 \[ \frac{3 i \left (2 \text{sech}^{-1}(a x) \text{PolyLog}\left (2,-i e^{-\text{sech}^{-1}(a x)}\right )-2 \text{sech}^{-1}(a x) \text{PolyLog}\left (2,i e^{-\text{sech}^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\text{sech}^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{-\text{sech}^{-1}(a x)}\right )+\text{sech}^{-1}(a x)^2 \log \left (1-i e^{-\text{sech}^{-1}(a x)}\right )-\text{sech}^{-1}(a x)^2 \log \left (1+i e^{-\text{sech}^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(a x)\right )\right )\right )+2 a^3 x^3 \text{sech}^{-1}(a x)^3-3 a x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-6 a x \text{sech}^{-1}(a x)}{6 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\rm arcsech} \left (ax\right ) \right ) ^{3}\, dx \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 x^{2} \operatorname{arsech}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{arsech}\left (a x\right )^{3}, x\right ) \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 x^{2} \operatorname{asech}^{3}{\left (a x \right )}\, 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 x^{2} \operatorname{arsech}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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