Optimal. Leaf size=494 \[ -\frac{6 i a^2 \sec ^{-1}(a+b x) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 a^2 \text{PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 a^2 \text{PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{3 i a \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{\text{PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\text{PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b^3}+\frac{6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3 \]
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Rubi [A] time = 0.396809, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.167, Rules used = {5258, 4426, 4190, 4181, 2531, 2282, 6589, 4184, 3719, 2190, 2279, 2391, 4186, 3770} \[ -\frac{6 i a^2 \sec ^{-1}(a+b x) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 a^2 \text{PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 a^2 \text{PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{3 i a \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{\text{PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\text{PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b^3}+\frac{6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
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Rule 5258
Rule 4426
Rule 4190
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4184
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 4186
Rule 3770
Rubi steps
\begin{align*} \int x^2 \sec ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \sec (x) (-a+\sec (x))^2 \tan (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3-\frac{\operatorname{Subst}\left (\int x^2 (-a+\sec (x))^3 \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3-\frac{\operatorname{Subst}\left (\int \left (-a^3 x^2+3 a^2 x^2 \sec (x)-3 a x^2 \sec ^2(x)+x^2 \sec ^3(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3-\frac{\operatorname{Subst}\left (\int x^2 \sec ^3(x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}+\frac{(3 a) \operatorname{Subst}\left (\int x^2 \sec ^2(x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{2 b^3}-\frac{\operatorname{Subst}\left (\int \sec (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{(6 a) \operatorname{Subst}\left (\int x \tan (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}+\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b^3}-\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{\operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{\operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}+\frac{(12 i a) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}+\frac{\left (6 i a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{\left (6 i a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}\\ &=\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b^3}+\frac{6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{i \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}-\frac{(6 a) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^3}+\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sec ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b^3}+\frac{6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 a^2 \text{Li}_3\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 a^2 \text{Li}_3\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{(3 i a) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b^3}+\frac{6 a \sec ^{-1}(a+b x) \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{3 i a \text{Li}_2\left (-e^{2 i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{\text{Li}_3\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}+\frac{6 a^2 \text{Li}_3\left (-i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{\text{Li}_3\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}-\frac{6 a^2 \text{Li}_3\left (i e^{i \sec ^{-1}(a+b x)}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.490697, size = 442, normalized size = 0.89 \[ \frac{6 a^2 \left (\text{PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )-i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )\right )+6 i a^2 \left (\sec ^{-1}(a+b x) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )+i \text{PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )\right )-i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(a+b x)}\right )+i \sec ^{-1}(a+b x) \text{PolyLog}\left (2,i e^{i \sec ^{-1}(a+b x)}\right )+\text{PolyLog}\left (3,-i e^{i \sec ^{-1}(a+b x)}\right )-\text{PolyLog}\left (3,i e^{i \sec ^{-1}(a+b x)}\right )-3 i a \left (\text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(a+b x)}\right )+\sec ^{-1}(a+b x) \left (\sec ^{-1}(a+b x)+2 i \log \left (1+e^{2 i \sec ^{-1}(a+b x)}\right )\right )\right )+\frac{1}{3} a^3 \sec ^{-1}(a+b x)^3+6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )+\frac{1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3-\frac{1}{2} (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+(a+b x) \sec ^{-1}(a+b x)-\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )+i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left (e^{i \sec ^{-1}(a+b x)}\right )}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.59, size = 770, normalized size = 1.6 \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} \arctan \left (\sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )^{3} - \frac{1}{4} \, x^{3} \arctan \left (\sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} - \int \frac{{\left (4 \, b x^{3} \arctan \left (\sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )^{2} - b x^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2}\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} + 4 \,{\left (3 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} - 1\right )} b 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} - 1\right )} b x^{3} + 3 \,{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} - 1\right )} b x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} \arctan \left (\sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )}{4 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} - 1\right )} b 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{arcsec}\left (b x + a\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{asec}^{3}{\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{arcsec}\left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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