Optimal. Leaf size=69 \[ \frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{24 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]
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Rubi [A] time = 0.117854, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4619, 4677, 8} \[ \frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{24 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
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Rule 4619
Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \sin ^{-1}(a x)^4 \, dx &=x \sin ^{-1}(a x)^4-(4 a) \int \frac{x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4-12 \int \sin ^{-1}(a x)^2 \, dx\\ &=-12 x \sin ^{-1}(a x)^2+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4+(24 a) \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{24 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-12 x \sin ^{-1}(a x)^2+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x-\frac{24 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-12 x \sin ^{-1}(a x)^2+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+x \sin ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0155383, size = 69, normalized size = 1. \[ \frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{24 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^4-12 x \sin ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 67, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arcsin \left ( ax \right ) \right ) ^{4}+4\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sqrt{-{a}^{2}{x}^{2}+1}-12\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{2}+24\,ax-24\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76751, size = 101, normalized size = 1.46 \begin{align*} x \arcsin \left (a x\right )^{4} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} - 12 \,{\left (\frac{x \arcsin \left (a x\right )^{2}}{a} - \frac{2 \,{\left (x - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92647, size = 149, normalized size = 2.16 \begin{align*} \frac{a x \arcsin \left (a x\right )^{4} - 12 \, a x \arcsin \left (a x\right )^{2} + 24 \, a x + 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.98801, size = 65, normalized size = 0.94 \begin{align*} \begin{cases} x \operatorname{asin}^{4}{\left (a x \right )} - 12 x \operatorname{asin}^{2}{\left (a x \right )} + 24 x + \frac{4 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{a} - \frac{24 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38978, size = 88, normalized size = 1.28 \begin{align*} x \arcsin \left (a x\right )^{4} - 12 \, x \arcsin \left (a x\right )^{2} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a} + 24 \, x - \frac{24 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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