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 = 0.0608282, antiderivative size = 50, normalized size of antiderivative = 1., 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 5223
Rule 3296
Rule 2637
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*}
Mathematica [A] time = 0.146587, 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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Maple [B] time = 0.219, size = 118, normalized size = 2.4 \begin{align*} c \left ( -{\frac{{a}^{2}}{cx}}+{b}^{2} \left ( -{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{cx}}+2\,{\frac{1}{cx}}-2\,\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}{\rm arccsc} \left (cx\right ) \right ) +2\,ab \left ( -{\frac{{\rm arccsc} \left (cx\right )}{cx}}-{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00159, size = 107, normalized size = 2.14 \begin{align*} -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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8743, size = 140, normalized size = 2.8 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsc}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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