Optimal. Leaf size=50 \[ 2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x}+\frac{2 b^2}{x} \]
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Rubi [A] time = 0.0586854, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5222, 3296, 2638} \[ 2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x}+\frac{2 b^2}{x} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x^2} \, dx &=c \operatorname{Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x}+(2 b c) \operatorname{Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x}-\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{2 b^2}{x}+2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x}\\ \end{align*}
Mathematica [A] time = 0.122778, size = 75, normalized size = 1.5 \[ \frac{-a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b \sec ^{-1}(c x) \left (b c x \sqrt{1-\frac{1}{c^2 x^2}}-a\right )-b^2 \sec ^{-1}(c x)^2+2 b^2}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 117, normalized size = 2.3 \begin{align*} c \left ( -{\frac{{a}^{2}}{cx}}+{b}^{2} \left ( -{\frac{ \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{cx}}+2\,{\frac{1}{cx}}+2\,{\rm arcsec} \left (cx\right )\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}} \right ) +2\,ab \left ( -{\frac{{\rm arcsec} \left (cx\right )}{cx}}+{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0221, size = 105, normalized size = 2.1 \begin{align*} 2 \,{\left (c \sqrt{-\frac{1}{c^{2} x^{2}} + 1} - \frac{\operatorname{arcsec}\left (c x\right )}{x}\right )} a b + 2 \,{\left (c \sqrt{-\frac{1}{c^{2} x^{2}} + 1} \operatorname{arcsec}\left (c x\right ) + \frac{1}{x}\right )} b^{2} - \frac{b^{2} \operatorname{arcsec}\left (c x\right )^{2}}{x} - \frac{a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.66604, size = 140, normalized size = 2.8 \begin{align*} -\frac{b^{2} \operatorname{arcsec}\left (c x\right )^{2} + 2 \, a b \operatorname{arcsec}\left (c x\right ) + a^{2} - 2 \, b^{2} - 2 \, \sqrt{c^{2} x^{2} - 1}{\left (b^{2} \operatorname{arcsec}\left (c x\right ) + a b\right )}}{x} \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 \frac{\left (a + b \operatorname{asec}{\left (c x \right )}\right )^{2}}{x^{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 \frac{{\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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