Optimal. Leaf size=88 \[ -\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2+\frac {b^2}{4 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 88, 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, 4404, 3310} \[ -\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2+\frac {b^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 4404
Rule 5223
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx &=-\left (c^2 \operatorname {Subst}\left (\int (a+b x)^2 \cos (x) \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\left (b c^2\right ) \operatorname {Subst}\left (\int (a+b x) \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b^2}{4 x^2}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{2} \left (b c^2\right ) \operatorname {Subst}\left (\int (a+b x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b^2}{4 x^2}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 102, normalized size = 1.16 \[ \frac {-2 a^2-2 a b c x \sqrt {1-\frac {1}{c^2 x^2}}+2 a b c^2 x^2 \sin ^{-1}\left (\frac {1}{c x}\right )-2 b \csc ^{-1}(c x) \left (2 a+b c x \sqrt {1-\frac {1}{c^2 x^2}}\right )+b^2 \left (c^2 x^2-2\right ) \csc ^{-1}(c x)^2+b^2}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 82, normalized size = 0.93 \[ \frac {{\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left (a b c^{2} x^{2} - 2 \, a b\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arccsc}\left (c x\right ) + a b\right )}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 163, normalized size = 1.85 \[ -\frac {1}{8} \, {\left (4 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 8 \, a b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 2 \, b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2} + 4 \, a^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 2 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 4 \, a b c \arcsin \left (\frac {1}{c x}\right ) - b^{2} c + \frac {4 \, b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {4 \, a b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 191, normalized size = 2.17 \[ -\frac {a^{2}}{2 x^{2}}+\frac {b^{2} c^{2} \mathrm {arccsc}\left (c x \right )^{2}}{4}-\frac {b^{2} \mathrm {arccsc}\left (c x \right )^{2}}{2 x^{2}}-\frac {c \,b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{2 x}+\frac {b^{2}}{4 x^{2}}-\frac {a b \,\mathrm {arccsc}\left (c x \right )}{x^{2}}+\frac {c a b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {c a b}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {a b}{2 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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