Optimal. Leaf size=137 \[ 6 i b^2 c \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 i b^2 c \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 b^3 c \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )+6 b^3 c \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \]
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Rubi [A] time = 0.212895, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4627, 4709, 4183, 2531, 2282, 6589} \[ 6 i b^2 c \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 i b^2 c \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 b^3 c \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )+6 b^3 c \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x^2} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}+(3 b c) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (6 b^2 c\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (6 b^2 c\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\left (6 i b^3 c\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (6 i b^3 c\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\left (6 b^3 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\left (6 b^3 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-6 b^3 c \text{Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )+6 b^3 c \text{Li}_3\left (e^{i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [B] time = 0.314443, size = 283, normalized size = 2.07 \[ 3 a b^2 c \left (2 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\sin ^{-1}(c x) \left (\frac{\sin ^{-1}(c x)}{c x}-2 \log \left (1-e^{i \sin ^{-1}(c x)}\right )+2 \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )+b^3 c \left (6 i \sin ^{-1}(c x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-6 i \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-6 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )+6 \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )-\frac{\sin ^{-1}(c x)^3}{c x}+3 \sin ^{-1}(c x)^2 \log \left (1-e^{i \sin ^{-1}(c x)}\right )-3 \sin ^{-1}(c x)^2 \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )-3 a^2 b c \log \left (\sqrt{1-c^2 x^2}+1\right )+3 a^2 b c \log (x)-\frac{3 a^2 b \sin ^{-1}(c x)}{x}-\frac{a^3}{x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.042, size = 378, normalized size = 2.8 \begin{align*} -{\frac{{a}^{3}}{x}}-{\frac{{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{3}}{x}}-3\,c{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,ic{b}^{3}\arcsin \left ( cx \right ){\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -6\,{b}^{3}c{\it polylog} \left ( 3,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,c{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -6\,ic{b}^{3}\arcsin \left ( cx \right ){\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,{b}^{3}c{\it polylog} \left ( 3,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -3\,{\frac{a{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{x}}+6\,ca{b}^{2}\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -6\,ca{b}^{2}\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -6\,ica{b}^{2}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,ica{b}^{2}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -3\,{\frac{{a}^{2}b\arcsin \left ( cx \right ) }{x}}-3\,c{a}^{2}b{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -3 \,{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} a^{2} b - \frac{a^{3}}{x} - \frac{b^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{3} + \frac{3}{2} \,{\left (a b^{2} c^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} \arctan \left (\frac{c x}{\sqrt{c x + 1} \sqrt{-c x + 1}}\right )^{2} -{\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac{2}{x}\right )} a b^{2} \arctan \left (\frac{c x}{\sqrt{c x + 1} \sqrt{-c x + 1}}\right )^{2} + 8 \, b^{3} c \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2}}{4 \,{\left (c^{2} x^{3} - x\right )}}\,{d x}\right )} x}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}}{x^{2}}, x\right ) \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{asin}{\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 \arcsin \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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