Optimal. Leaf size=169 \[ -\frac {1}{2} i a^4 \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{2} i a^4 \sin ^{-1}(a x)^2+a^4 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4} \]
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Rubi [A] time = 0.29, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {4627, 4701, 4681, 4625, 3717, 2190, 2279, 2391, 264} \[ -\frac {1}{2} i a^4 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {1}{2} i a^4 \sin ^{-1}(a x)^2+a^4 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^3}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 264
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4627
Rule 4681
Rule 4701
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\sin ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\sin ^{-1}(a x)^2}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {\sin ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\sin ^{-1}(a x)}{x^3} \, dx+\frac {1}{2} a^3 \int \frac {\sin ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+a^4 \int \frac {\sin ^{-1}(a x)}{x} \, dx\\ &=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4}+a^4 \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \sin ^{-1}(a x)^2-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4}-\left (2 i a^4\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \sin ^{-1}(a x)^2-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4}+a^4 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-a^4 \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \sin ^{-1}(a x)^2-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4}+a^4 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \left (i a^4\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \sin ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \sin ^{-1}(a x)^2-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{4 x^4}+a^4 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{2} i a^4 \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.63, size = 116, normalized size = 0.69 \[ \frac {1}{4} \left (-\frac {\sin ^{-1}(a x)^3}{x^4}+a^4 \left (-\frac {\sqrt {1-a^2 x^2} \left (\left (\frac {1}{a^2 x^2}+2\right ) \sin ^{-1}(a x)^2+1\right )}{a x}-\sin ^{-1}(a x) \left (\frac {1}{a^2 x^2}+2 i \sin ^{-1}(a x)-4 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )\right )-2 i \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{3}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 225, normalized size = 1.33 \[ -\frac {i a^{4} \arcsin \left (a x \right )^{2}}{2}-\frac {a^{3} \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{2 x}+\frac {i a^{4}}{4}-\frac {a^{3} \sqrt {-a^{2} x^{2}+1}}{4 x}-\frac {a \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{4 x^{3}}-\frac {a^{2} \arcsin \left (a x \right )}{4 x^{2}}-\frac {\arcsin \left (a x \right )^{3}}{4 x^{4}}+a^{4} \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+a^{4} \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-i a^{4} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-i a^{4} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\frac {1}{4} \, {\left ({\left (2 \, a^{2} x^{2} + 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} + 12 \, x^{3} \int \frac {9 \, \sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} - 2 \, {\left (2 \, a^{5} x^{5} - a^{3} x^{3} - a x\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{12 \, {\left (a^{2} x^{6} - x^{4}\right )}}\,{d x}\right )} a x + \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{4 \, x^{4}} \]
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 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^5} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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