Optimal. Leaf size=144 \[ -\frac{6 i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c}+x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c} \]
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Rubi [A] time = 0.112179, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5217, 4410, 4183, 2531, 2282, 6589} \[ -\frac{6 i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c}+x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 5217
Rule 4410
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 b^3 \text{Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.293682, size = 265, normalized size = 1.84 \[ \frac{-6 i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+6 i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+6 b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-6 b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )+3 a^2 b \log \left (c x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )+3 a^2 b c x \csc ^{-1}(c x)+a^3 c x+3 a b^2 c x \csc ^{-1}(c x)^2-6 a b^2 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )+6 a b^2 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )+b^3 c x \csc ^{-1}(c x)^3-3 b^3 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+3 b^3 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.319, size = 437, normalized size = 3. \begin{align*} x{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}+3\,xa{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}-3\,{\frac{{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{c}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+3\,{\frac{{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{c}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{6\,i{b}^{3}{\rm arccsc} \left (cx\right )}{c}{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{6\,i{b}^{3}{\rm arccsc} \left (cx\right )}{c}{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+3\,x{a}^{2}b{\rm arccsc} \left (cx\right )-6\,{\frac{a{b}^{2}{\rm arccsc} \left (cx\right )}{c}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+6\,{\frac{a{b}^{2}{\rm arccsc} \left (cx\right )}{c}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{6\,ia{b}^{2}}{c}{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{6\,ia{b}^{2}}{c}{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+x{a}^{3}-6\,{\frac{{b}^{3}}{c}{\it polylog} \left ( 3,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+6\,{\frac{{b}^{3}}{c}{\it polylog} \left ( 3,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+3\,{\frac{{a}^{2}b}{c}\ln \left ( cx+cx\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \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*} \text{result too large to display} \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 (b^{3} \operatorname{arccsc}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arccsc}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arccsc}\left (c x\right ) + a^{3}, 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 \left (a + b \operatorname{acsc}{\left (c x \right )}\right )^{3}\, 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{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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