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 = 0.0857982, antiderivative size = 80, normalized size of antiderivative = 1., 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 5223
Rule 3296
Rule 2638
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*}
Mathematica [A] time = 0.200749, size = 135, normalized size = 1.69 \[ -\frac{3 b \csc ^{-1}(c x) \left (a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 b^2\right )+3 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+a^3+3 b^2 \csc ^{-1}(c x)^2 \left (a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right )-6 a b^2-6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+b^3 \csc ^{-1}(c x)^3}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.25, size = 199, normalized size = 2.5 \begin{align*} c \left ( -{\frac{{a}^{3}}{cx}}+{b}^{3} \left ( -{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}}{cx}}-3\, \left ({\rm arccsc} \left (cx\right ) \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}}}}+6\,{\frac{{\rm arccsc} \left (cx\right )}{cx}} \right ) +3\,a{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 ) +3\,{a}^{2}b \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 = 1.02653, size = 198, normalized size = 2.48 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37818, size = 238, normalized size = 2.98 \begin{align*} -\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} \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 )^{3}}{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 )}^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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