Optimal. Leaf size=94 \[ \frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{2 x}+\frac{1}{2} \left (c^2-\frac{1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-\frac{1}{2} a b c^2 \sec ^{-1}(c x)-\frac{1}{4} b^2 c^2 \sec ^{-1}(c x)^2+\frac{b^2}{4 x^2} \]
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Rubi [A] time = 0.0793479, antiderivative size = 94, 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, 4404, 3310} \[ \frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{2 x}+\frac{1}{2} \left (c^2-\frac{1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-\frac{1}{2} a b c^2 \sec ^{-1}(c x)-\frac{1}{4} b^2 c^2 \sec ^{-1}(c x)^2+\frac{b^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4404
Rule 3310
Rubi steps
\begin{align*} \int \frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x^3} \, dx &=c^2 \operatorname{Subst}\left (\int (a+b x)^2 \cos (x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{1}{2} \left (c^2-\frac{1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-\left (b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \sin ^2(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{b^2}{4 x^2}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{2 x}+\frac{1}{2} \left (c^2-\frac{1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^2-\frac{1}{2} \left (b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{b^2}{4 x^2}-\frac{1}{2} a b c^2 \sec ^{-1}(c x)-\frac{1}{4} b^2 c^2 \sec ^{-1}(c x)^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{2 x}+\frac{1}{2} \left (c^2-\frac{1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.113702, size = 102, normalized size = 1.09 \[ \frac{-2 a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 a b c^2 x^2 \sin ^{-1}\left (\frac{1}{c x}\right )+2 b \sec ^{-1}(c x) \left (b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 a\right )+b^2 \left (c^2 x^2-2\right ) \sec ^{-1}(c x)^2+b^2}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.244, size = 199, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{{b}^{2} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{2\,{x}^{2}}}+{\frac{{b}^{2}{c}^{2} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{4}}+{\frac{c{b}^{2}{\rm arcsec} \left (cx\right )}{2\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{{b}^{2}{c}^{2}}{4}}+{\frac{{b}^{2}}{4\,{x}^{2}}}-{\frac{ab{\rm arcsec} \left (cx\right )}{{x}^{2}}}-{\frac{acb}{2\,x}\sqrt{{c}^{2}{x}^{2}-1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{acb}{2\,x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{ab}{2\,c{x}^{3}}{\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 [A] time = 2.38939, size = 196, normalized size = 2.09 \begin{align*} \frac{{\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \operatorname{arcsec}\left (c x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \,{\left (a b c^{2} x^{2} - 2 \, a b\right )} \operatorname{arcsec}\left (c x\right ) + 2 \, \sqrt{c^{2} x^{2} - 1}{\left (b^{2} \operatorname{arcsec}\left (c x\right ) + a b\right )}}{4 \, x^{2}} \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^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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