Optimal. Leaf size=87 \[ -\frac{a^2}{12 x^2}-\frac{a^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{6 x^3}+\frac{1}{3} a^4 \log (x)-\frac{\sin ^{-1}(a x)^2}{4 x^4} \]
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Rubi [A] time = 0.140384, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4627, 4701, 4681, 29, 30} \[ -\frac{a^2}{12 x^2}-\frac{a^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{6 x^3}+\frac{1}{3} a^4 \log (x)-\frac{\sin ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4701
Rule 4681
Rule 29
Rule 30
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{x^5} \, dx &=-\frac{\sin ^{-1}(a x)^2}{4 x^4}+\frac{1}{2} a \int \frac{\sin ^{-1}(a x)}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{6 x^3}-\frac{\sin ^{-1}(a x)^2}{4 x^4}+\frac{1}{6} a^2 \int \frac{1}{x^3} \, dx+\frac{1}{3} a^3 \int \frac{\sin ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2}{12 x^2}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{6 x^3}-\frac{a^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 x}-\frac{\sin ^{-1}(a x)^2}{4 x^4}+\frac{1}{3} a^4 \int \frac{1}{x} \, dx\\ &=-\frac{a^2}{12 x^2}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{6 x^3}-\frac{a^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 x}-\frac{\sin ^{-1}(a x)^2}{4 x^4}+\frac{1}{3} a^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0380599, size = 69, normalized size = 0.79 \[ -\frac{a^2}{12 x^2}-\frac{a \sqrt{1-a^2 x^2} \left (2 a^2 x^2+1\right ) \sin ^{-1}(a x)}{6 x^3}+\frac{1}{3} a^4 \log (x)-\frac{\sin ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 76, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{4\,{x}^{4}}}-{\frac{a\arcsin \left ( ax \right ) }{6\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{12\,{x}^{2}}}-{\frac{{a}^{3}\arcsin \left ( ax \right ) }{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{a}^{4}\ln \left ( ax \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61304, size = 100, normalized size = 1.15 \begin{align*} \frac{1}{12} \,{\left (4 \, a^{2} \log \left (x\right ) - \frac{1}{x^{2}}\right )} a^{2} - \frac{1}{6} \,{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x} + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{3}}\right )} a \arcsin \left (a x\right ) - \frac{\arcsin \left (a x\right )^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30886, size = 149, normalized size = 1.71 \begin{align*} \frac{4 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} - 2 \,{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right ) - 3 \, \arcsin \left (a x\right )^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{2}{\left (a x \right )}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.4574, size = 223, normalized size = 2.56 \begin{align*} \frac{1}{48} \,{\left ({\left (\frac{{\left (a^{4} + \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} - \frac{\frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2}{\left | a \right |}}\right )} \arcsin \left (a x\right ) + \frac{4 \,{\left (2 \, a^{4} \log \left (a^{2} x^{2}\right ) - \frac{a^{2}}{x^{2}}\right )}}{a}\right )} a - \frac{\arcsin \left (a x\right )^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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