Optimal. Leaf size=120 \[ -\frac{8 x^3}{225 a^2}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125} \]
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Rubi [A] time = 0.195771, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 4708, 4678, 8, 30} \[ -\frac{8 x^3}{225 a^2}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4708
Rule 4678
Rule 8
Rule 30
Rubi steps
\begin{align*} \int x^4 \cos ^{-1}(a x)^2 \, dx &=\frac{1}{5} x^5 \cos ^{-1}(a x)^2+\frac{1}{5} (2 a) \int \frac{x^5 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 \int x^4 \, dx}{25}+\frac{8 \int \frac{x^3 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac{2 x^5}{125}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2+\frac{16 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{75 a^3}-\frac{8 \int x^2 \, dx}{75 a^2}\\ &=-\frac{8 x^3}{225 a^2}-\frac{2 x^5}{125}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{16 \int 1 \, dx}{75 a^4}\\ &=-\frac{16 x}{75 a^4}-\frac{8 x^3}{225 a^2}-\frac{2 x^5}{125}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2\\ \end{align*}
Mathematica [A] time = 0.0561761, size = 82, normalized size = 0.68 \[ -\frac{8 x^3}{225 a^2}-\frac{2 \sqrt{1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cos ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 76, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{2}{a}^{5}{x}^{5}}{5}}-{\frac{2\,\arccos \left ( ax \right ) \left ( 3\,{a}^{4}{x}^{4}+4\,{a}^{2}{x}^{2}+8 \right ) }{75}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{5}{x}^{5}}{125}}-{\frac{8\,{a}^{3}{x}^{3}}{225}}-{\frac{16\,ax}{75}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49582, size = 138, normalized size = 1.15 \begin{align*} \frac{1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac{2}{75} \,{\left (\frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right ) - \frac{2 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00245, size = 189, normalized size = 1.58 \begin{align*} \frac{225 \, a^{5} x^{5} \arccos \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 30 \,{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.14903, size = 121, normalized size = 1.01 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acos}^{2}{\left (a x \right )}}{5} - \frac{2 x^{5}}{125} - \frac{2 x^{4} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{25 a} - \frac{8 x^{3}}{225 a^{2}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{75 a^{3}} - \frac{16 x}{75 a^{4}} - \frac{16 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{75 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x^{5}}{20} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17294, size = 135, normalized size = 1.12 \begin{align*} \frac{1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac{2}{125} \, x^{5} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{25 \, a} - \frac{8 \, x^{3}}{225 \, a^{2}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{75 \, a^{3}} - \frac{16 \, x}{75 \, a^{4}} - \frac{16 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{75 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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