Optimal. Leaf size=276 \[ 2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {\sin ^{-1}(a x)^4}{3 x^3} \]
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Rubi [A] time = 0.41, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4627, 4701, 4709, 4183, 2531, 6609, 2282, 6589, 2279, 2391} \[ 2 i a^3 \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text {PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text {PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {PolyLog}\left (4,e^{i \sin ^{-1}(a x)}\right )-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4183
Rule 4627
Rule 4701
Rule 4709
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^4}{x^4} \, dx &=-\frac {\sin ^{-1}(a x)^4}{3 x^3}+\frac {1}{3} (4 a) \int \frac {\sin ^{-1}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac {\sin ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\sin ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 a^3\right ) \int \frac {\sin ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\left (2 a^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (2 a^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+\left (4 a^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac {2 a^2 \sin ^{-1}(a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac {\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 3.97, size = 399, normalized size = 1.45 \[ \frac {1}{24} a^3 \left (-\frac {8 \sin ^4\left (\frac {1}{2} \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)^4}{a^3 x^3}+48 i \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{-i \sin ^{-1}(a x)}\right )+96 \sin ^{-1}(a x) \text {Li}_3\left (e^{-i \sin ^{-1}(a x)}\right )-96 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+48 i \left (\sin ^{-1}(a x)^2+2\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-96 i \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-96 i \text {Li}_4\left (e^{-i \sin ^{-1}(a x)}\right )-96 i \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+4 i \sin ^{-1}(a x)^4+16 \sin ^{-1}(a x)^3 \log \left (1-e^{-i \sin ^{-1}(a x)}\right )-16 \sin ^{-1}(a x)^3 \log \left (1+e^{i \sin ^{-1}(a x)}\right )+96 \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )-96 \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x)^4 \tan \left (\frac {1}{2} \sin ^{-1}(a x)\right )-24 \sin ^{-1}(a x)^2 \tan \left (\frac {1}{2} \sin ^{-1}(a x)\right )-2 \sin ^{-1}(a x)^4 \cot \left (\frac {1}{2} \sin ^{-1}(a x)\right )-24 \sin ^{-1}(a x)^2 \cot \left (\frac {1}{2} \sin ^{-1}(a x)\right )-\frac {1}{2} a x \sin ^{-1}(a x)^4 \csc ^4\left (\frac {1}{2} \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x)^3 \csc ^2\left (\frac {1}{2} \sin ^{-1}(a x)\right )+4 \sin ^{-1}(a x)^3 \sec ^2\left (\frac {1}{2} \sin ^{-1}(a x)\right )-2 i \pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{4}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{4}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 409, normalized size = 1.48 \[ -\frac {2 a \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}}{3 x^{2}}-\frac {2 a^{2} \arcsin \left (a x \right )^{2}}{x}-\frac {\arcsin \left (a x \right )^{4}}{3 x^{3}}-\frac {2 a^{3} \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i a^{3} \arcsin \left (a x \right )^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 a^{3} \arcsin \left (a x \right ) \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i a^{3} \polylog \left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\frac {2 a^{3} \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i a^{3} \arcsin \left (a x \right )^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 a^{3} \arcsin \left (a x \right ) \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i a^{3} \polylog \left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 a^{3} \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i a^{3} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 a^{3} \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i a^{3} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {4 \, a x^{3} \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{a^{2} x^{5} - x^{3}}\,{d x} + \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{4}}{3 \, x^{3}} \]
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 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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