Optimal. Leaf size=279 \[ \frac{2 i a^2 \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}-\frac{2 i a^2 \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{i \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{i \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}-\frac{4 a^2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{2 a \log (a+b x)}{b^3}+\frac{2 a \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)}{b^3}-\frac{(a+b x) \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)}{3 b^3}-\frac{2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{x}{3 b^2} \]
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Rubi [A] time = 0.241625, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6321, 5468, 4190, 4180, 2279, 2391, 4184, 3475, 4185} \[ \frac{2 i a^2 \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}-\frac{2 i a^2 \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{i \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{i \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}-\frac{4 a^2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{2 a \log (a+b x)}{b^3}+\frac{2 a \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)}{b^3}-\frac{(a+b x) \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)}{3 b^3}-\frac{2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{x}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 4190
Rule 4180
Rule 2279
Rule 2391
Rule 4184
Rule 3475
Rule 4185
Rubi steps
\begin{align*} \int x^2 \text{sech}^{-1}(a+b x)^2 \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \text{sech}(x) (-a+\text{sech}(x))^2 \tanh (x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^3}\\ &=\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{2 \operatorname{Subst}\left (\int x (-a+\text{sech}(x))^3 \, dx,x,\text{sech}^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{2 \operatorname{Subst}\left (\int \left (-a^3 x+3 a^2 x \text{sech}(x)-3 a x \text{sech}^2(x)+x \text{sech}^3(x)\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{2 \operatorname{Subst}\left (\int x \text{sech}^3(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{3 b^3}+\frac{(2 a) \operatorname{Subst}\left (\int x \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^3}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^3}\\ &=-\frac{x}{3 b^2}+\frac{2 a \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{b^3}-\frac{(a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{3 b^3}+\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{4 a^2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}-\frac{\operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{3 b^3}-\frac{(2 a) \operatorname{Subst}\left (\int \tanh (x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^3}+\frac{\left (2 i a^2\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^3}-\frac{\left (2 i a^2\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^3}\\ &=-\frac{x}{3 b^2}+\frac{2 a \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{b^3}-\frac{(a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{3 b^3}+\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{4 a^2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{2 a \log (a+b x)}{b^3}+\frac{i \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{3 b^3}-\frac{i \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{3 b^3}+\frac{\left (2 i a^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}-\frac{\left (2 i a^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}\\ &=-\frac{x}{3 b^2}+\frac{2 a \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{b^3}-\frac{(a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{3 b^3}+\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{4 a^2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{2 a \log (a+b x)}{b^3}+\frac{2 i a^2 \text{Li}_2\left (-i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}-\frac{2 i a^2 \text{Li}_2\left (i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}\\ &=-\frac{x}{3 b^2}+\frac{2 a \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{b^3}-\frac{(a+b x) \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{3 b^3}+\frac{a^3 \text{sech}^{-1}(a+b x)^2}{3 b^3}+\frac{1}{3} x^3 \text{sech}^{-1}(a+b x)^2-\frac{2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{4 a^2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}+\frac{2 a \log (a+b x)}{b^3}+\frac{i \text{Li}_2\left (-i e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}+\frac{2 i a^2 \text{Li}_2\left (-i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}-\frac{i \text{Li}_2\left (i e^{\text{sech}^{-1}(a+b x)}\right )}{3 b^3}-\frac{2 i a^2 \text{Li}_2\left (i e^{\text{sech}^{-1}(a+b x)}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 1.91494, size = 305, normalized size = 1.09 \[ -\frac{-\left (6 a^2+1\right ) \left (2 i \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a+b x)}\right )-2 i \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a+b x)}\right )-2 i \text{sech}^{-1}(a+b x) \log \left (1-i e^{\text{sech}^{-1}(a+b x)}\right )+\pi \log \left (1-i e^{\text{sech}^{-1}(a+b x)}\right )+2 i \text{sech}^{-1}(a+b x) \log \left (1+i e^{\text{sech}^{-1}(a+b x)}\right )-\pi \log \left (1+i e^{\text{sech}^{-1}(a+b x)}\right )-\pi \log \left (\cot \left (\frac{1}{4} \left (\pi +2 i \text{sech}^{-1}(a+b x)\right )\right )\right )\right )+2 \left (-3 a^2 (a+b x) \text{sech}^{-1}(a+b x)^2-6 a \sqrt{-\frac{a+b x-1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)+a+b x\right )+12 a \log \left (\frac{1}{a+b x}\right )-2 (a+b x)^3 \text{sech}^{-1}(a+b x)^2+6 a (a+b x)^2 \text{sech}^{-1}(a+b x)^2+2 \sqrt{-\frac{a+b x-1}{a+b x+1}} (a+b x+1) (a+b x) \text{sech}^{-1}(a+b x)}{6 b^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.483, size = 655, normalized size = 2.4 \begin{align*} \text{result too large to display} \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*} \frac{1}{3} \, x^{3} \log \left (\sqrt{b x + a + 1} \sqrt{-b x - a + 1} b x + \sqrt{b x + a + 1} \sqrt{-b x - a + 1} a + b x + a\right )^{2} - \int -\frac{2 \,{\left (6 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} \log \left (b x + a\right )^{2} + 6 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )^{2} -{\left (b^{3} x^{5} + 2 \, a b^{2} x^{4} +{\left (a^{2} b - b\right )} x^{3} + 6 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right ) +{\left (3 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \sqrt{b x + a + 1} \log \left (b x + a\right ) +{\left (2 \, b^{3} x^{5} + 4 \, a b^{2} x^{4} +{\left (2 \, a^{2} b - b\right )} x^{3} + 3 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )\right )} \sqrt{b x + a + 1}\right )} \sqrt{-b x - a + 1}\right )} \log \left (\sqrt{b x + a + 1} \sqrt{-b x - a + 1} b x + \sqrt{b x + a + 1} \sqrt{-b x - a + 1} a + b x + a\right )\right )}}{3 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} +{\left (3 \, a^{2} b - b\right )} x - a\right )}}\,{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 (b x + a\right )^{2}, 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}^{2}{\left (a + b 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 (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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