Optimal. Leaf size=137 \[ -\frac {3}{4} b^2 \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 x}+\frac {1}{2} \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^3-\frac {1}{4} c^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}+\frac {3}{8} b^3 c^2 \sec ^{-1}(c x) \]
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Rubi [A] time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5222, 4404, 3311, 32, 2635, 8} \[ -\frac {3}{4} b^2 \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 x}+\frac {1}{2} \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^3-\frac {1}{4} c^2 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}+\frac {3}{8} b^3 c^2 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 3311
Rule 4404
Rule 5222
Rubi steps
\begin {align*} \int \frac {\left (a+b \sec ^{-1}(c x)\right )^3}{x^3} \, dx &=c^2 \operatorname {Subst}\left (\int (a+b x)^3 \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 )^3-\frac {1}{2} \left (3 b c^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {3}{4} b^2 \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 x}+\frac {1}{2} \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^3-\frac {1}{4} \left (3 b c^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\sec ^{-1}(c x)\right )+\frac {1}{4} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \sin ^2(x) \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{4} b^2 \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 x}-\frac {1}{4} c^2 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {1}{2} \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^3+\frac {1}{8} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}+\frac {3}{8} b^3 c^2 \sec ^{-1}(c x)-\frac {3}{4} b^2 \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )+\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 x}-\frac {1}{4} c^2 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {1}{2} \left (c^2-\frac {1}{x^2}\right ) \left (a+b \sec ^{-1}(c x)\right )^3\\ \end {align*}
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Mathematica [A] time = 0.22, size = 185, normalized size = 1.35 \[ \frac {-4 a^3+3 b c^2 x^2 \left (b^2-2 a^2\right ) \sin ^{-1}\left (\frac {1}{c x}\right )+6 b \sec ^{-1}(c x) \left (-2 a^2+2 a b c x \sqrt {1-\frac {1}{c^2 x^2}}+b^2\right )+6 a^2 b c x \sqrt {1-\frac {1}{c^2 x^2}}+6 b^2 \sec ^{-1}(c x)^2 \left (a \left (c^2 x^2-2\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}}\right )+6 a b^2-3 b^3 c x \sqrt {1-\frac {1}{c^2 x^2}}+2 b^3 \left (c^2 x^2-2\right ) \sec ^{-1}(c x)^3}{8 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 150, normalized size = 1.09 \[ \frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right )^{3} - 4 \, a^{3} + 6 \, a b^{2} + 6 \, {\left (a b^{2} c^{2} x^{2} - 2 \, a b^{2}\right )} \operatorname {arcsec}\left (c x\right )^{2} + 3 \, {\left ({\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 4 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arcsec}\left (c x\right ) + 3 \, {\left (2 \, b^{3} \operatorname {arcsec}\left (c x\right )^{2} + 4 \, a b^{2} \operatorname {arcsec}\left (c x\right ) + 2 \, a^{2} b - b^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 278, normalized size = 2.03 \[ \frac {1}{8} \, {\left (2 \, b^{3} c \arccos \left (\frac {1}{c x}\right )^{3} + 6 \, a b^{2} c \arccos \left (\frac {1}{c x}\right )^{2} + 6 \, a^{2} b c \arccos \left (\frac {1}{c x}\right ) - 3 \, b^{3} c \arccos \left (\frac {1}{c x}\right ) + \frac {6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )^{2}}{x} - 3 \, a b^{2} c + \frac {12 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x} - \frac {4 \, b^{3} \arccos \left (\frac {1}{c x}\right )^{3}}{c x^{2}} + \frac {6 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {3 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {12 \, a b^{2} \arccos \left (\frac {1}{c x}\right )^{2}}{c x^{2}} - \frac {12 \, a^{2} b \arccos \left (\frac {1}{c x}\right )}{c x^{2}} + \frac {6 \, b^{3} \arccos \left (\frac {1}{c x}\right )}{c x^{2}} - \frac {4 \, a^{3}}{c x^{2}} + \frac {6 \, a b^{2}}{c x^{2}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 324, normalized size = 2.36 \[ -\frac {a^{3}}{2 x^{2}}-\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{3}}{2 x^{2}}+\frac {c^{2} b^{3} \mathrm {arcsec}\left (c x \right )^{3}}{4}+\frac {3 c \,b^{3} \mathrm {arcsec}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4 x}+\frac {3 b^{3} \mathrm {arcsec}\left (c x \right )}{4 x^{2}}-\frac {3 c \,b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 x}-\frac {3 b^{3} c^{2} \mathrm {arcsec}\left (c x \right )}{8}-\frac {3 a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2}}{2 x^{2}}+\frac {3 c^{2} a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2}}{4}+\frac {3 c a \,b^{2} \mathrm {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{2 x}-\frac {3 c^{2} a \,b^{2}}{4}+\frac {3 a \,b^{2}}{4 x^{2}}-\frac {3 a^{2} b \,\mathrm {arcsec}\left (c x \right )}{2 x^{2}}-\frac {3 c \,a^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 c \,a^{2} b}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {3 a^{2} b}{4 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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