Optimal. Leaf size=116 \[ \frac {1}{3} i a^3 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\frac {2}{3} a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\sin ^{-1}(a x)^2}{3 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4627, 4701, 4709, 4183, 2279, 2391, 30} \[ \frac {1}{3} i a^3 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {2}{3} a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2279
Rule 2391
Rule 4183
Rule 4627
Rule 4701
Rule 4709
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\sin ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\sin ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {\sin ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx+\frac {1}{3} a^3 \int \frac {\sin ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {\sin ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {\sin ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {1}{3} a^3 \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\frac {1}{3} a^3 \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {\sin ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+\frac {1}{3} \left (i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )-\frac {1}{3} \left (i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac {a^2}{3 x}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 x^2}-\frac {\sin ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+\frac {1}{3} i a^3 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {1}{3} i a^3 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.60, size = 139, normalized size = 1.20 \[ -\frac {-i a^3 x^3 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+i a^3 x^3 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-a^3 x^3 \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )+a^3 x^3 \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )+a^2 x^2+a x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+\sin ^{-1}(a x)^2}{3 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{2}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 157, normalized size = 1.35 \[ -\frac {a \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{2}}-\frac {a^{2}}{3 x}-\frac {\arcsin \left (a x \right )^{2}}{3 x^{3}}-\frac {a^{3} \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+\frac {i a^{3} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}+\frac {a^{3} \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {i a^{3} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3} \]
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 \[ -\frac {2 \, a x^{3} \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{a^{2} x^{5} - x^{3}}\,{d x} + \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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