Optimal. Leaf size=102 \[ \frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )+\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{2 b^2}{27 x^3} \]
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Rubi [A] time = 0.0940474, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5222, 4405, 3310, 3296, 2638} \[ \frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )+\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{2 b^2}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4405
Rule 3310
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x^4} \, dx &=c^3 \operatorname{Subst}\left (\int (a+b x)^2 \cos ^2(x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \cos ^3(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{2 b^2}{27 x^3}+\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{9} \left (4 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{2 b^2}{27 x^3}+\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )+\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{9} \left (4 b^2 c^3\right ) \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{2 b^2}{27 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )+\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.183608, size = 108, normalized size = 1.06 \[ \frac{-9 a^2+6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )+6 b \sec ^{-1}(c x) \left (b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )-3 a\right )+2 b^2 \left (6 c^2 x^2+1\right )-9 b^2 \sec ^{-1}(c x)^2}{27 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 154, normalized size = 1.5 \begin{align*}{c}^{3} \left ( -{\frac{{a}^{2}}{3\,{c}^{3}{x}^{3}}}+{b}^{2} \left ( -{\frac{ \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{3\,{c}^{3}{x}^{3}}}+{\frac{2\,{\rm arcsec} \left (cx\right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{9\,{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{2}{27\,{c}^{3}{x}^{3}}}+{\frac{4}{9\,cx}} \right ) +2\,ab \left ( -1/3\,{\frac{{\rm arcsec} \left (cx\right )}{{c}^{3}{x}^{3}}}+1/9\,{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{{c}^{4}{x}^{4}}{\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 = 2.18582, size = 221, normalized size = 2.17 \begin{align*} -\frac{2}{9} \, a b{\left (\frac{c^{4}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, c^{4} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} + \frac{3 \, \operatorname{arcsec}\left (c x\right )}{x^{3}}\right )} - \frac{b^{2} \operatorname{arcsec}\left (c x\right )^{2}}{3 \, x^{3}} - \frac{a^{2}}{3 \, x^{3}} + \frac{2 \,{\left ({\left (6 \, c^{3} x^{2} + c\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 3 \,{\left (2 \, c^{5} x^{4} - c^{3} x^{2} - c\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} b^{2}}{27 \, \sqrt{c x + 1} \sqrt{c x - 1} c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22573, size = 224, normalized size = 2.2 \begin{align*} \frac{12 \, b^{2} c^{2} x^{2} - 9 \, b^{2} \operatorname{arcsec}\left (c x\right )^{2} - 18 \, a b \operatorname{arcsec}\left (c x\right ) - 9 \, a^{2} + 2 \, b^{2} + 6 \,{\left (2 \, a b c^{2} x^{2} + a b +{\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname{arcsec}\left (c x\right )\right )} \sqrt{c^{2} x^{2} - 1}}{27 \, x^{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 \frac{\left (a + b \operatorname{asec}{\left (c x \right )}\right )^{2}}{x^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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