Optimal. Leaf size=147 \[ -\frac{i b^2 \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac{i b^2 \text{PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac{2 i b \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b^2 x}{3 c^2} \]
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Rubi [A] time = 0.123132, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5222, 4409, 4185, 4181, 2279, 2391} \[ -\frac{i b^2 \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac{i b^2 \text{PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac{2 i b \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b^2 x}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4409
Rule 4185
Rule 4181
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^2 \sec ^3(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}\\ &=\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \sec ^3(x) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac{b \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{2 i b \left (a+b \sec ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{2 i b \left (a+b \sec ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{2 i b \left (a+b \sec ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac{i b^2 \text{Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac{i b^2 \text{Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}\\ \end{align*}
Mathematica [A] time = 1.1968, size = 225, normalized size = 1.53 \[ \frac{1}{3} \left (\frac{b^2 \left (-i \text{PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )+i \text{PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )+c^3 x^3 \sec ^{-1}(c x)^2-c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}} \sec ^{-1}(c x)+c x-\sec ^{-1}(c x) \log \left (1-i e^{i \sec ^{-1}(c x)}\right )+\sec ^{-1}(c x) \log \left (1+i e^{i \sec ^{-1}(c x)}\right )\right )}{c^3}+a^2 x^3+\frac{a b \left (2 x^4 \sec ^{-1}(c x)-\frac{c^3 x^3+\sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )-c x}{c^4 \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.392, size = 343, normalized size = 2.3 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{x}^{3}{b}^{2} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{3}}-{\frac{{b}^{2}{\rm arcsec} \left (cx\right ){x}^{2}}{3\,c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{{b}^{2}x}{3\,{c}^{2}}}+{\frac{{b}^{2}{\rm arcsec} \left (cx\right )}{3\,{c}^{3}}\ln \left ( 1+i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }-{\frac{{b}^{2}{\rm arcsec} \left (cx\right )}{3\,{c}^{3}}\ln \left ( 1-i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }-{\frac{{\frac{i}{3}}{b}^{2}}{{c}^{3}}{\it dilog} \left ( 1+i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }+{\frac{{\frac{i}{3}}{b}^{2}}{{c}^{3}}{\it dilog} \left ( 1-i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }+{\frac{2\,{x}^{3}ab{\rm arcsec} \left (cx\right )}{3}}-{\frac{ab{x}^{2}}{3\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{ab}{3\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{ab}{3\,{c}^{4}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \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*} \text{result too large to display} \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 (b^{2} x^{2} \operatorname{arcsec}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arcsec}\left (c x\right ) + a^{2} x^{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} \left (a + b \operatorname{asec}{\left (c x \right )}\right )^{2}\, 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{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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