Integrand size = 14, antiderivative size = 88 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\frac {b^2}{4 x^2}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2} \]
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Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5331, 4489, 3391} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{2 x}+\frac {1}{2} a b c^2 \csc ^{-1}(c x)-\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{4} b^2 c^2 \csc ^{-1}(c x)^2+\frac {b^2}{4 x^2} \]
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Rule 3391
Rule 4489
Rule 5331
Rubi steps \begin{align*} \text {integral}& = -\left (c^2 \text {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 ) \text {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 ) \text {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*}
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\frac {-2 a^2+b^2-2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x-2 b \left (2 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)+b^2 \left (-2+c^2 x^2\right ) \csc ^{-1}(c x)^2+2 a b c^2 x^2 \arcsin \left (\frac {1}{c x}\right )}{4 x^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs. \(2(76)=152\).
Time = 0.59 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.09
method | result | size |
parts | \(-\frac {a^{2}}{2 x^{2}}+b^{2} c^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\) | \(184\) |
derivativedivides | \(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\right )\) | \(186\) |
default | \(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\right )\) | \(186\) |
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none
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\frac {{\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left (a b c^{2} x^{2} - 2 \, a b\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arccsc}\left (c x\right ) + a b\right )}}{4 \, x^{2}} \]
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (76) = 152\).
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {1}{8} \, {\left (4 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 8 \, a b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 2 \, b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2} + 4 \, a^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 2 \, b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 4 \, a b c \arcsin \left (\frac {1}{c x}\right ) - b^{2} c + \frac {4 \, b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {4 \, a b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]
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Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \]
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