Optimal. Leaf size=134 \[ \frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac{3}{16} a b c^4 \sec ^{-1}(c x)-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac{b^2}{32 x^4} \]
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Rubi [A] time = 0.111133, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5222, 4405, 3310} \[ \frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac{3}{16} a b c^4 \sec ^{-1}(c x)-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac{b^2}{32 x^4} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4405
Rule 3310
Rubi steps
\begin{align*} \int \frac{\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx &=c^4 \operatorname{Subst}\left (\int (a+b x)^2 \cos ^3(x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac{1}{2} \left (b c^4\right ) \operatorname{Subst}\left (\int (a+b x) \cos ^4(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{b^2}{32 x^4}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac{1}{8} \left (3 b c^4\right ) \operatorname{Subst}\left (\int (a+b x) \cos ^2(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{b^2}{32 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac{1}{16} \left (3 b c^4\right ) \operatorname{Subst}\left (\int (a+b x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{b^2}{32 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{3}{16} a b c^4 \sec ^{-1}(c x)+\frac{3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac{\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.173987, size = 148, normalized size = 1.1 \[ \frac{-8 a^2+6 a b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+4 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-6 a b c^4 x^4 \sin ^{-1}\left (\frac{1}{c x}\right )+2 b \sec ^{-1}(c x) \left (b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 c^2 x^2+2\right )-8 a\right )+3 b^2 c^2 x^2+b^2 \left (3 c^4 x^4-8\right ) \sec ^{-1}(c x)^2+b^2}{32 x^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.253, size = 265, normalized size = 2. \begin{align*} -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{{b}^{2} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{3\,{b}^{2}{c}^{4} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{32}}+{\frac{3\,{c}^{3}{b}^{2}{\rm arcsec} \left (cx\right )}{16\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{c{b}^{2}{\rm arcsec} \left (cx\right )}{8\,{x}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{{b}^{2}}{32\,{x}^{4}}}+{\frac{3\,{b}^{2}{c}^{2}}{32\,{x}^{2}}}-{\frac{ab{\rm arcsec} \left (cx\right )}{2\,{x}^{4}}}-{\frac{3\,a{c}^{3}b}{16\,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{3\,a{c}^{3}b}{16\,x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{acb}{16\,{x}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{ab}{8\,c{x}^{5}}{\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 = 2.21055, size = 275, normalized size = 2.05 \begin{align*} \frac{3 \, b^{2} c^{2} x^{2} +{\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \operatorname{arcsec}\left (c x\right )^{2} - 8 \, a^{2} + b^{2} + 2 \,{\left (3 \, a b c^{4} x^{4} - 8 \, a b\right )} \operatorname{arcsec}\left (c x\right ) + 2 \,{\left (3 \, a b c^{2} x^{2} + 2 \, a b +{\left (3 \, b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \operatorname{arcsec}\left (c x\right )\right )} \sqrt{c^{2} x^{2} - 1}}{32 \, x^{4}} \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^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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