Optimal. Leaf size=50 \[ -2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x}+\frac {2 b^2}{x} \]
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Rubi [A] time = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = {5223, 3296, 2637} \[ -2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x}+\frac {2 b^2}{x} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5223
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^2} \, dx &=-\left (c \operatorname {Subst}\left (\int (a+b x)^2 \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x}+(2 b c) \operatorname {Subst}\left (\int (a+b x) \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x}+\left (2 b^2 c\right ) \operatorname {Subst}\left (\int \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {2 b^2}{x}-2 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 71, normalized size = 1.42 \[ -\frac {a^2+2 a b c x \sqrt {1-\frac {1}{c^2 x^2}}+2 b \csc ^{-1}(c x) \left (a+b c x \sqrt {1-\frac {1}{c^2 x^2}}\right )+b^2 \csc ^{-1}(c x)^2-2 b^2}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.27, size = 57, normalized size = 1.14 \[ -\frac {b^{2} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b \operatorname {arccsc}\left (c x\right ) + a^{2} - 2 \, b^{2} + 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arccsc}\left (c x\right ) + a b\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 104, normalized size = 2.08 \[ -{\left (2 \, b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) + 2 \, a b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c x} + \frac {2 \, a b \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a^{2}}{c x} - \frac {2 \, b^{2}}{c x}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 118, normalized size = 2.36 \[ c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{2}}{c x}+\frac {2}{c x}-2 \,\mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+2 a b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 79, normalized size = 1.58 \[ -2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} a b - 2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right ) - \frac {1}{x}\right )} b^{2} - \frac {b^{2} \operatorname {arccsc}\left (c x\right )^{2}}{x} - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 88, normalized size = 1.76 \[ -\frac {a^2}{x}-\frac {b^2\,\left ({\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-2\right )}{x}-2\,b^2\,c\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\sqrt {1-\frac {1}{c^2\,x^2}}-2\,a\,b\,c\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {\mathrm {asin}\left (\frac {1}{c\,x}\right )}{c\,x}\right ) \]
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 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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