Optimal. Leaf size=80 \[ \frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {5223, 3296, 2638} \[ \frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5223
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx &=-\left (c \operatorname {Subst}\left (\int (a+b x)^3 \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}+\left (6 b^2 c\right ) \operatorname {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}-\left (6 b^3 c\right ) \operatorname {Subst}\left (\int \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}+\frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 135, normalized size = 1.69 \[ -\frac {a^3+3 b \csc ^{-1}(c x) \left (a^2+2 a b c x \sqrt {1-\frac {1}{c^2 x^2}}-2 b^2\right )+3 a^2 b c x \sqrt {1-\frac {1}{c^2 x^2}}+3 b^2 \csc ^{-1}(c x)^2 \left (a+b c x \sqrt {1-\frac {1}{c^2 x^2}}\right )-6 a b^2-6 b^3 c x \sqrt {1-\frac {1}{c^2 x^2}}+b^3 \csc ^{-1}(c x)^3}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 98, normalized size = 1.22 \[ -\frac {b^{3} \operatorname {arccsc}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2} + a^{3} - 6 \, a b^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) + 3 \, {\left (b^{3} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b^{2} \operatorname {arccsc}\left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 195, normalized size = 2.44 \[ -{\left (3 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2} + 6 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) + \frac {b^{3} \arcsin \left (\frac {1}{c x}\right )^{3}}{c x} + 3 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {3 \, a b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c x} + \frac {3 \, a^{2} b \arcsin \left (\frac {1}{c x}\right )}{c x} - \frac {6 \, b^{3} \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a^{3}}{c x} - \frac {6 \, a b^{2}}{c x}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 199, normalized size = 2.49 \[ c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{3}}{c x}-3 \mathrm {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+\frac {6 \,\mathrm {arccsc}\left (c x \right )}{c x}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{2}}{c x}+\frac {2}{c x}-2 \,\mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+3 a^{2} b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 147, normalized size = 1.84 \[ -\frac {b^{3} \operatorname {arccsc}\left (c x\right )^{3}}{x} - 3 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} a^{2} b - 6 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} - 3 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right )^{2} - 2 \, c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {2 \, \operatorname {arccsc}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 155, normalized size = 1.94 \[ \frac {b^3\,\left (6\,\mathrm {asin}\left (\frac {1}{c\,x}\right )-{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^3\right )}{x}-\frac {a^3}{x}-3\,a^2\,b\,c\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {\mathrm {asin}\left (\frac {1}{c\,x}\right )}{c\,x}\right )-b^3\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}\,\left (3\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-6\right )-3\,a\,b^2\,c\,\left (2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-2}{c\,x}\right ) \]
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 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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