Optimal. Leaf size=78 \[ \frac{\sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{6 a}+\frac{x}{6 \sin ^{-1}(a x)^2} \]
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Rubi [A] time = 0.152712, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4621, 4719, 4723, 3299} \[ \frac{\sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{6 a}+\frac{x}{6 \sin ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 4621
Rule 4719
Rule 4723
Rule 3299
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{-1}(a x)^4} \, dx &=-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{1}{3} a \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{x}{6 \sin ^{-1}(a x)^2}-\frac{1}{6} \int \frac{1}{\sin ^{-1}(a x)^2} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{x}{6 \sin ^{-1}(a x)^2}+\frac{\sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}+\frac{1}{6} a \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{x}{6 \sin ^{-1}(a x)^2}+\frac{\sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{6 a}\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{x}{6 \sin ^{-1}(a x)^2}+\frac{\sqrt{1-a^2 x^2}}{6 a \sin ^{-1}(a x)}+\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{6 a}\\ \end{align*}
Mathematica [A] time = 0.0621412, size = 70, normalized size = 0.9 \[ \frac{-2 \sqrt{1-a^2 x^2}+\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+\sin ^{-1}(a x)^3 \text{Si}\left (\sin ^{-1}(a x)\right )+a x \sin ^{-1}(a x)}{6 a \sin ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 63, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{3\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{ax}{6\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}+{\frac{1}{6\,\arcsin \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{6}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x}{{\left (a^{2} x^{2} - 1\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}\,{d x} - a x \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right ) - \sqrt{a x + 1} \sqrt{-a x + 1}{\left (\arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2} - 2\right )}}{6 \, a \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\arcsin \left (a x\right )^{4}}, x\right ) \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{1}{\operatorname{asin}^{4}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2452, size = 89, normalized size = 1.14 \begin{align*} \frac{\operatorname{Si}\left (\arcsin \left (a x\right )\right )}{6 \, a} + \frac{x}{6 \, \arcsin \left (a x\right )^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{6 \, a \arcsin \left (a x\right )} - \frac{\sqrt{-a^{2} x^{2} + 1}}{3 \, a \arcsin \left (a x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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