Optimal. Leaf size=220 \[ -\frac {i b^2 \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {i b^2 \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}+\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b^3 \text {Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3} \]
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Rubi [A] time = 0.19, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5223, 4410, 4186, 3770, 4183, 2531, 2282, 6589} \[ -\frac {i b^2 \text {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {i b^2 \text {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {b^3 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3770
Rule 4183
Rule 4186
Rule 4410
Rule 5223
Rule 6589
Rubi steps
\begin {align*} \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^3 \cot (x) \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^3}-\frac {b^3 \operatorname {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^2 \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac {b^2 \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}\\ \end {align*}
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Mathematica [B] time = 7.82, size = 580, normalized size = 2.64 \[ \frac {a^3 x^3}{3}+\frac {a^2 b x^2 \sqrt {\frac {c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac {a^2 b \log \left (x \left (\sqrt {\frac {c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{2 c^3}+a^2 b x^3 \csc ^{-1}(c x)+\frac {a b^2 \left (2 c^3 x^3 \left (\frac {4 i \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3 x^3}+4 \csc ^{-1}(c x)^2-2 \cos \left (2 \csc ^{-1}(c x)\right )-\frac {3 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )}{c x}+\frac {3 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c x}+2 \csc ^{-1}(c x) \sin \left (2 \csc ^{-1}(c x)\right )+\csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )-\csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )+2\right )-8 i \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )\right )}{8 c^3}+\frac {b^3 \left (16 c^3 x^3 \csc ^{-1}(c x)^3 \sin ^4\left (\frac {1}{2} \csc ^{-1}(c x)\right )-48 i \csc ^{-1}(c x) \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )+48 i \csc ^{-1}(c x) \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )+48 \text {Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )-48 \text {Li}_3\left (e^{i \csc ^{-1}(c x)}\right )+\frac {\csc ^{-1}(c x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(c x)\right )}{c x}+6 \csc ^{-1}(c x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )+24 \csc ^{-1}(c x) \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )-24 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+24 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )+4 \csc ^{-1}(c x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )+24 \csc ^{-1}(c x) \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )-6 \csc ^{-1}(c x)^2 \sec ^2\left (\frac {1}{2} \csc ^{-1}(c x)\right )-48 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )\right )}{48 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \operatorname {arccsc}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname {arccsc}\left (c x\right ) + a^{3} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.55, size = 648, normalized size = 2.95 \[ \frac {a^{3} x^{3}}{3}+\frac {x^{3} b^{3} \mathrm {arccsc}\left (c x \right )^{3}}{3}+\frac {b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arccsc}\left (c x \right )^{2} x^{2}}{2 c}+\frac {b^{3} \mathrm {arccsc}\left (c x \right ) x}{c^{2}}-\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2 c^{3}}+\frac {i b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}-\frac {b^{3} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+\frac {b^{3} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2 c^{3}}-\frac {i a \,b^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+\frac {b^{3} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+\frac {2 b^{3} \arctanh \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+b^{2} x^{3} a \mathrm {arccsc}\left (c x \right )^{2}+\frac {a \,b^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arccsc}\left (c x \right ) x^{2}}{c}-\frac {i b^{3} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+\frac {i a \,b^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}-\frac {a \,b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+\frac {a \,b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{3}}+\frac {a \,b^{2} x}{c^{2}}+x^{3} a^{2} b \,\mathrm {arccsc}\left (c x \right )+\frac {a^{2} b \,x^{2}}{2 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {a^{2} b}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {a^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
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 = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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