Optimal. Leaf size=156 \[ 12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-24 i a \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+24 i a \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4627, 4709, 4183, 2531, 6609, 2282, 6589} \[ 12 i a \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-24 i a \text {PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )+24 i a \text {PolyLog}\left (4,e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4183
Rule 4627
Rule 4709
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^4}{x^2} \, dx &=-\frac {\sin ^{-1}(a x)^4}{x}+(4 a) \int \frac {\sin ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sin ^{-1}(a x)^4}{x}+(4 a) \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-(12 a) \operatorname {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+(12 a) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-(24 i a) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+(24 i a) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+(24 a) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-(24 a) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-(24 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+(24 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-24 i a \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+24 i a \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.26, size = 198, normalized size = 1.27 \[ a \left (12 i \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{-i \sin ^{-1}(a x)}\right )+12 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+24 \sin ^{-1}(a x) \text {Li}_3\left (e^{-i \sin ^{-1}(a x)}\right )-24 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )-24 i \text {Li}_4\left (e^{-i \sin ^{-1}(a x)}\right )-24 i \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{a x}+i \sin ^{-1}(a x)^4+4 \sin ^{-1}(a x)^3 \log \left (1-e^{-i \sin ^{-1}(a x)}\right )-4 \sin ^{-1}(a x)^3 \log \left (1+e^{i \sin ^{-1}(a x)}\right )-\frac {i \pi ^4}{2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{4}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{4}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 241, normalized size = 1.54 \[ -\frac {\arcsin \left (a x \right )^{4}}{x}-4 a \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 a \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-24 a \arcsin \left (a x \right ) \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 a \arcsin \left (a x \right ) \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+12 i a \arcsin \left (a x \right )^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-12 i a \arcsin \left (a x \right )^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 i a \polylog \left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 i a \polylog \left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{4} + 4 \, a x \int \frac {\sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{\sqrt {a x + 1} {\left (a x - 1\right )} x}\,{d x}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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