Integrand size = 10, antiderivative size = 97 \[ \int \frac {\arcsin (a x)^3}{x} \, dx=-\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {3}{2} \arcsin (a x) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4721, 3798, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\arcsin (a x)^3}{x} \, dx=-\frac {3}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {3}{2} \arcsin (a x) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right )-\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4721
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^3 \cot (x) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{4} i \arcsin (a x)^4-2 i \text {Subst}\left (\int \frac {e^{2 i x} x^3}{1-e^{2 i x}} \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+3 i \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {3}{2} \arcsin (a x) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )-\frac {3}{2} \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {3}{2} \arcsin (a x) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )+\frac {3}{4} i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i \arcsin (a x)}\right ) \\ & = -\frac {1}{4} i \arcsin (a x)^4+\arcsin (a x)^3 \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {3}{2} \arcsin (a x) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,e^{2 i \arcsin (a x)}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)^3}{x} \, dx=-\frac {1}{64} i \left (\pi ^4-16 \arcsin (a x)^4+64 i \arcsin (a x)^3 \log \left (1-e^{-2 i \arcsin (a x)}\right )-96 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (a x)}\right )+96 i \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arcsin (a x)}\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.36
method | result | size |
derivativedivides | \(-\frac {i \arcsin \left (a x \right )^{4}}{4}+\arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(229\) |
default | \(-\frac {i \arcsin \left (a x \right )^{4}}{4}+\arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(229\) |
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\[ \int \frac {\arcsin (a x)^3}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\arcsin (a x)^3}{x} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\arcsin (a x)^3}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\arcsin (a x)^3}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\arcsin (a x)^3}{x} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x} \,d x \]
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