Optimal. Leaf size=448 \[ -3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 i \text {Li}_4\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 i \text {Li}_4\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\frac {3}{2} i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {3}{2} \csc ^{-1}(a+b x) \text {Li}_3\left (e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {3}{4} i \text {Li}_4\left (e^{2 i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]
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Rubi [A] time = 0.54, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5259, 4552, 4529, 3717, 2190, 2531, 6609, 2282, 6589, 4519} \[ -3 i \csc ^{-1}(a+b x)^2 \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-3 i \csc ^{-1}(a+b x)^2 \text {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 \csc ^{-1}(a+b x) \text {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \csc ^{-1}(a+b x) \text {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 i \text {PolyLog}\left (4,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 i \text {PolyLog}\left (4,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\frac {3}{2} i \csc ^{-1}(a+b x)^2 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {3}{2} \csc ^{-1}(a+b x) \text {PolyLog}\left (3,e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {3}{4} i \text {PolyLog}\left (4,e^{2 i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4519
Rule 4529
Rule 4552
Rule 5259
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}(a+b x)^3}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {x^3 \cot (x) \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {x^3 \cot (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\left (a \operatorname {Subst}\left (\int \frac {x^3 \cos (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\right )-\operatorname {Subst}\left (\int x^3 \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x^3}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \operatorname {Subst}\left (\int \frac {e^{i x} x^3}{1-\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \operatorname {Subst}\left (\int \frac {e^{i x} x^3}{1+\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-3 \operatorname {Subst}\left (\int x^2 \log \left (1+\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-3 \operatorname {Subst}\left (\int x^2 \log \left (1+\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+3 \operatorname {Subst}\left (\int x^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )-3 i \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+6 i \operatorname {Subst}\left (\int x \text {Li}_2\left (-\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+6 i \operatorname {Subst}\left (\int x \text {Li}_2\left (-\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \csc ^{-1}(a+b x) \text {Li}_3\left (e^{2 i \csc ^{-1}(a+b x)}\right )+\frac {3}{2} \operatorname {Subst}\left (\int \text {Li}_3\left (e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-6 \operatorname {Subst}\left (\int \text {Li}_3\left (-\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-6 \operatorname {Subst}\left (\int \text {Li}_3\left (-\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \csc ^{-1}(a+b x) \text {Li}_3\left (e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {3}{4} i \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )+6 i \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i a x}{-1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )+6 i \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {i a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )\\ &=\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^3 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \csc ^{-1}(a+b x) \text {Li}_3\left (e^{2 i \csc ^{-1}(a+b x)}\right )+6 i \text {Li}_4\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 i \text {Li}_4\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{4} i \text {Li}_4\left (e^{2 i \csc ^{-1}(a+b x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.33, size = 554, normalized size = 1.24 \[ -3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}-1}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}-1}\right )+6 \csc ^{-1}(a+b x) \text {Li}_3\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+6 i \text {Li}_4\left (\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}-1}\right )+6 i \text {Li}_4\left (-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\csc ^{-1}(a+b x)^3 \log \left (1-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}-1}\right )+\csc ^{-1}(a+b x)^3 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (e^{-i \csc ^{-1}(a+b x)}\right )+3 i \csc ^{-1}(a+b x)^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )-6 \csc ^{-1}(a+b x) \text {Li}_3\left (e^{-i \csc ^{-1}(a+b x)}\right )-6 \csc ^{-1}(a+b x) \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )+6 i \text {Li}_4\left (e^{-i \csc ^{-1}(a+b x)}\right )-6 i \text {Li}_4\left (-e^{i \csc ^{-1}(a+b x)}\right )-\frac {1}{4} i \csc ^{-1}(a+b x)^4-\csc ^{-1}(a+b x)^3 \log \left (1-e^{-i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x)^3 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )+\frac {i \pi ^4}{8} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccsc}\left (b x + a\right )^{3}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arccsc}\left (b x + a\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.08, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccsc}\left (b x +a \right )^{3}}{x}\, dx \]
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 \[ \int \frac {\operatorname {arccsc}\left (b x + a\right )^{3}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^3}{x} \,d x \]
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 \[ \int \frac {\operatorname {acsc}^{3}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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