Optimal. Leaf size=170 \[ \frac{4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac{2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\frac{2}{27} b^3 c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{14}{9} b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}} \]
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Rubi [A] time = 0.146838, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5222, 4405, 3311, 3296, 2637, 2633} \[ \frac{4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac{2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\frac{2}{27} b^3 c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{14}{9} b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4405
Rule 3311
Rule 3296
Rule 2637
Rule 2633
Rubi steps
\begin{align*} \int \frac{\left (a+b \sec ^{-1}(c x)\right )^3}{x^4} \, dx &=c^3 \operatorname{Subst}\left (\int (a+b x)^3 \cos ^2(x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \cos ^3(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}+\frac{1}{3} \left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \cos (x) \, dx,x,\sec ^{-1}(c x)\right )-\frac{1}{9} \left (2 b^3 c^3\right ) \operatorname{Subst}\left (\int \cos ^3(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}-\frac{1}{3} \left (4 b^2 c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right )+\frac{1}{9} \left (2 b^3 c^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sqrt{1-\frac{1}{c^2 x^2}}\right )\\ &=-\frac{2}{9} b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}+\frac{2}{27} b^3 c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}+\frac{2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac{4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}-\frac{1}{3} \left (4 b^3 c^3\right ) \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{14}{9} b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}+\frac{2}{27} b^3 c^3 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}+\frac{2 b^2 \left (a+b \sec ^{-1}(c x)\right )}{9 x^3}+\frac{4 b^2 c^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x}+\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{3 x^2}-\frac{\left (a+b \sec ^{-1}(c x)\right )^3}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.280422, size = 204, normalized size = 1.2 \[ \frac{3 b \sec ^{-1}(c x) \left (-9 a^2+6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )+2 b^2 \left (6 c^2 x^2+1\right )\right )+9 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )-9 a^3+6 a b^2 \left (6 c^2 x^2+1\right )+9 b^2 \sec ^{-1}(c x)^2 \left (b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )-3 a\right )-2 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}} \left (20 c^2 x^2+1\right )-9 b^3 \sec ^{-1}(c x)^3}{27 x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.319, size = 299, normalized size = 1.8 \begin{align*}{c}^{3} \left ( -{\frac{{a}^{3}}{3\,{c}^{3}{x}^{3}}}+{b}^{3} \left ( -{\frac{ \left ({\rm arcsec} \left (cx\right ) \right ) ^{3}}{3\,{c}^{3}{x}^{3}}}+{\frac{ \left ({\rm arcsec} \left (cx\right ) \right ) ^{2} \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{3\,{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{4}{3}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{4\,{\rm arcsec} \left (cx\right )}{3\,cx}}+{\frac{2\,{\rm arcsec} \left (cx\right )}{9\,{c}^{3}{x}^{3}}}-{\frac{4\,{c}^{2}{x}^{2}+2}{27\,{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}} \right ) +3\,a{b}^{2} \left ( -1/3\,{\frac{ \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{{c}^{3}{x}^{3}}}+2/9\,{\frac{{\rm arcsec} \left (cx\right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{2}{27\,{c}^{3}{x}^{3}}}+4/9\,{\frac{1}{cx}} \right ) +3\,{a}^{2}b \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 [B] time = 3.22333, size = 787, normalized size = 4.63 \begin{align*} -\frac{1}{216} \,{\left (\frac{72 \,{\left (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}\right )} \operatorname{arcsec}\left (c x\right )^{2}}{c} + \frac{72 \, c^{4}{\left (\frac{c^{2} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{2 \, \sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\sqrt{c^{2} x^{2} - 1}}{x^{2}}}{c} - \frac{c^{2} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{2 \, \sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\sqrt{c^{2} x^{2} - 1}}{x^{2}}}{c} - \frac{4 \, \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )}{x}\right )} + c^{2}{\left (\frac{9 \, c^{4} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{16 \, \sqrt{c^{2} x^{2} - 1} c^{3}}{x} - \frac{9 \, \sqrt{c^{2} x^{2} - 1} c^{2}}{x^{2}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} c}{x^{3}} - \frac{6 \, \sqrt{c^{2} x^{2} - 1}}{x^{4}}}{c} - \frac{9 \, c^{4} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{16 \, \sqrt{c^{2} x^{2} - 1} c^{3}}{x} - \frac{9 \, \sqrt{c^{2} x^{2} - 1} c^{2}}{x^{2}} - \frac{8 \, \sqrt{c^{2} x^{2} - 1} c}{x^{3}} - \frac{6 \, \sqrt{c^{2} x^{2} - 1}}{x^{4}}}{c} - \frac{48 \, \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )}{x^{3}}\right )}}{c^{2}}\right )} b^{3} - \frac{1}{3} \, a^{2} 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^{3} \operatorname{arcsec}\left (c x\right )^{3}}{3 \, x^{3}} - \frac{a b^{2} \operatorname{arcsec}\left (c x\right )^{2}}{x^{3}} - \frac{a^{3}}{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 )} a b^{2}}{9 \, \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 = 2.24396, size = 401, normalized size = 2.36 \begin{align*} \frac{36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \operatorname{arcsec}\left (c x\right )^{3} - 27 \, a b^{2} \operatorname{arcsec}\left (c x\right )^{2} - 9 \, a^{3} + 6 \, a b^{2} + 3 \,{\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b + 2 \, b^{3}\right )} \operatorname{arcsec}\left (c x\right ) +{\left (2 \,{\left (9 \, a^{2} b - 20 \, b^{3}\right )} c^{2} x^{2} + 9 \, a^{2} b - 2 \, b^{3} + 9 \,{\left (2 \, b^{3} c^{2} x^{2} + b^{3}\right )} \operatorname{arcsec}\left (c x\right )^{2} + 18 \,{\left (2 \, a b^{2} c^{2} x^{2} + a 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 )^{3}}{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 )}^{3}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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