Integrand size = 10, antiderivative size = 276 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4723, 4789, 4803, 4268, 2611, 6744, 2320, 6724, 2317, 2438} \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right )-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {\arcsin (a x)^4}{3 x^3} \]
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4268
Rule 4723
Rule 4789
Rule 4803
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin (a x)^4}{3 x^3}+\frac {1}{3} (4 a) \int \frac {\arcsin (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac {\arcsin (a x)^2}{x^2} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arcsin (a x)\right )+\left (4 a^3\right ) \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (4 a^3\right ) \text {Subst}(\int x \csc (x) \, dx,x,\arcsin (a x)) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )+\left (4 a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arcsin (a x)}\right ) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \\ \end{align*}
Time = 3.42 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.45 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\frac {1}{24} a^3 \left (-2 i \pi ^4+4 i \arcsin (a x)^4-24 \arcsin (a x)^2 \cot \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \cot \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {1}{2} a x \arcsin (a x)^4 \csc ^4\left (\frac {1}{2} \arcsin (a x)\right )+16 \arcsin (a x)^3 \log \left (1-e^{-i \arcsin (a x)}\right )+96 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )-96 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-16 \arcsin (a x)^3 \log \left (1+e^{i \arcsin (a x)}\right )+48 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (a x)}\right )+48 i \left (2+\arcsin (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )+96 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (a x)}\right )-96 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 \arcsin (a x)^3 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {8 \arcsin (a x)^4 \sin ^4\left (\frac {1}{2} \arcsin (a x)\right )}{a^3 x^3}-24 \arcsin (a x)^2 \tan \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \]
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Time = 0.10 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(377\) |
default | \(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(377\) |
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\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]
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\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]
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\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^4} \,d x \]
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