Optimal. Leaf size=64 \[ \frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{6 c}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{6 c^3} \]
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Rubi [A] time = 0.0348788, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5220, 266, 51, 63, 208} \[ \frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{6 c}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{6 c^3} \]
Antiderivative was successfully verified.
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Rule 5220
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \int \frac{x}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{3 c}\\ &=\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{12 c^3}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{c^2-c^2 x^2} \, dx,x,\sqrt{1-\frac{1}{c^2 x^2}}\right )}{6 c}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0503631, size = 85, normalized size = 1.33 \[ \frac{a x^3}{3}-\frac{b x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{6 c}-\frac{b \log \left (x \left (\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{6 c^3}+\frac{1}{3} b x^3 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.162, size = 123, normalized size = 1.9 \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{{x}^{3}b{\rm arcsec} \left (cx\right )}{3}}-{\frac{b{x}^{2}}{6\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b}{6\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b}{6\,{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 [A] time = 1.02803, size = 132, normalized size = 2.06 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arcsec}\left (c x\right ) - \frac{\frac{2 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.67018, size = 219, normalized size = 3.42 \begin{align*} \frac{2 \, a c^{3} x^{3} + 4 \, b c^{3} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) - \sqrt{c^{2} x^{2} - 1} b c x + 2 \,{\left (b c^{3} x^{3} - b c^{3}\right )} \operatorname{arcsec}\left (c x\right ) + b \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{6 \, c^{3}} \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 )\, 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 )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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