Optimal. Leaf size=166 \[ -\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{160 x}{27 a^2}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{9} x^3 \cos ^{-1}(a x)^2+\frac{8 x^3}{81} \]
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Rubi [A] time = 0.365734, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4628, 4708, 4678, 4620, 8, 30} \[ -\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{160 x}{27 a^2}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{9} x^3 \cos ^{-1}(a x)^2+\frac{8 x^3}{81} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4708
Rule 4678
Rule 4620
Rule 8
Rule 30
Rubi steps
\begin{align*} \int x^2 \cos ^{-1}(a x)^4 \, dx &=\frac{1}{3} x^3 \cos ^{-1}(a x)^4+\frac{1}{3} (4 a) \int \frac{x^3 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{3} \int x^2 \cos ^{-1}(a x)^2 \, dx+\frac{8 \int \frac{x \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{8 \int \cos ^{-1}(a x)^2 \, dx}{3 a^2}-\frac{1}{9} (8 a) \int \frac{x^3 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4+\frac{8 \int x^2 \, dx}{27}-\frac{16 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{27 a}-\frac{16 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=\frac{8 x^3}{81}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4+\frac{16 \int 1 \, dx}{27 a^2}+\frac{16 \int 1 \, dx}{3 a^2}\\ &=\frac{160 x}{27 a^2}+\frac{8 x^3}{81}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0722178, size = 114, normalized size = 0.69 \[ \frac{8 a x \left (a^2 x^2+60\right )+27 a^3 x^3 \cos ^{-1}(a x)^4-36 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \cos ^{-1}(a x)^3-36 a x \left (a^2 x^2+6\right ) \cos ^{-1}(a x)^2+24 \sqrt{1-a^2 x^2} \left (a^2 x^2+20\right ) \cos ^{-1}(a x)}{81 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 130, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{4}}{3}}-{\frac{4\, \left ( \arccos \left ( ax \right ) \right ) ^{3} \left ({a}^{2}{x}^{2}+2 \right ) }{9}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{8\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}}{3}}+{\frac{160\,ax}{27}}+{\frac{16\,\arccos \left ( ax \right ) }{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4\,{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{9}}+{\frac{8\,\arccos \left ( ax \right ) \left ({a}^{2}{x}^{2}+2 \right ) }{27}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{8\,{a}^{3}{x}^{3}}{81}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52923, size = 197, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac{4}{9} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{3} + \frac{4}{81} \,{\left (2 \, a{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{3}} + \frac{a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac{9 \,{\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )^{2}}{a^{3}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42694, size = 247, normalized size = 1.49 \begin{align*} \frac{27 \, a^{3} x^{3} \arccos \left (a x\right )^{4} + 8 \, a^{3} x^{3} - 36 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right )^{2} + 480 \, a x - 12 \, \sqrt{-a^{2} x^{2} + 1}{\left (3 \,{\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{3} - 2 \,{\left (a^{2} x^{2} + 20\right )} \arccos \left (a x\right )\right )}}{81 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.37318, size = 165, normalized size = 0.99 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acos}^{4}{\left (a x \right )}}{3} - \frac{4 x^{3} \operatorname{acos}^{2}{\left (a x \right )}}{9} + \frac{8 x^{3}}{81} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{9 a} + \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{27 a} - \frac{8 x \operatorname{acos}^{2}{\left (a x \right )}}{3 a^{2}} + \frac{160 x}{27 a^{2}} - \frac{8 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{9 a^{3}} + \frac{160 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{27 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{3}}{48} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17803, size = 189, normalized size = 1.14 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac{4}{9} \, x^{3} \arccos \left (a x\right )^{2} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{9 \, a} + \frac{8}{81} \, x^{3} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{27 \, a} - \frac{8 \, x \arccos \left (a x\right )^{2}}{3 \, a^{2}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{9 \, a^{3}} + \frac{160 \, x}{27 \, a^{2}} + \frac{160 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{27 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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