Optimal. Leaf size=282 \[ \frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}+\frac{65 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}-\frac{5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}+\frac{245 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}-\frac{5 \cos ^{-1}(a x)^4}{96 a^6}+\frac{245 \cos ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4-\frac{1}{18} x^6 \cos ^{-1}(a x)^2+\frac{x^6}{324} \]
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Rubi [A] time = 0.874223, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4628, 4708, 4642, 30} \[ \frac{65 x^4}{3456 a^2}+\frac{245 x^2}{1152 a^4}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}+\frac{65 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}-\frac{5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}+\frac{245 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}-\frac{5 \cos ^{-1}(a x)^4}{96 a^6}+\frac{245 \cos ^{-1}(a x)^2}{1152 a^6}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4-\frac{1}{18} x^6 \cos ^{-1}(a x)^2+\frac{x^6}{324} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4708
Rule 4642
Rule 30
Rubi steps
\begin{align*} \int x^5 \cos ^{-1}(a x)^4 \, dx &=\frac{1}{6} x^6 \cos ^{-1}(a x)^4+\frac{1}{3} (2 a) \int \frac{x^6 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4-\frac{1}{3} \int x^5 \cos ^{-1}(a x)^2 \, dx+\frac{5 \int \frac{x^4 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{1}{18} x^6 \cos ^{-1}(a x)^2-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4+\frac{5 \int \frac{x^2 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{12 a^3}-\frac{5 \int x^3 \cos ^{-1}(a x)^2 \, dx}{12 a^2}-\frac{1}{9} a \int \frac{x^6 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \cos ^{-1}(a x)^2-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4+\frac{\int x^5 \, dx}{54}+\frac{5 \int \frac{\cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{24 a^5}-\frac{5 \int x \cos ^{-1}(a x)^2 \, dx}{8 a^4}-\frac{5 \int \frac{x^4 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{54 a}-\frac{5 \int \frac{x^4 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{24 a}\\ &=\frac{x^6}{324}+\frac{65 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac{5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \cos ^{-1}(a x)^2-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{5 \cos ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4-\frac{5 \int \frac{x^2 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{72 a^3}-\frac{5 \int \frac{x^2 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{32 a^3}-\frac{5 \int \frac{x^2 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}+\frac{5 \int x^3 \, dx}{216 a^2}+\frac{5 \int x^3 \, dx}{96 a^2}\\ &=\frac{65 x^4}{3456 a^2}+\frac{x^6}{324}+\frac{245 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}+\frac{65 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac{5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \cos ^{-1}(a x)^2-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{5 \cos ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4-\frac{5 \int \frac{\cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{144 a^5}-\frac{5 \int \frac{\cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{64 a^5}-\frac{5 \int \frac{\cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{16 a^5}+\frac{5 \int x \, dx}{144 a^4}+\frac{5 \int x \, dx}{64 a^4}+\frac{5 \int x \, dx}{16 a^4}\\ &=\frac{245 x^2}{1152 a^4}+\frac{65 x^4}{3456 a^2}+\frac{x^6}{324}+\frac{245 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}+\frac{65 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}+\frac{245 \cos ^{-1}(a x)^2}{1152 a^6}-\frac{5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{1}{18} x^6 \cos ^{-1}(a x)^2-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{5 \cos ^{-1}(a x)^4}{96 a^6}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.087507, size = 167, normalized size = 0.59 \[ \frac{a^2 x^2 \left (32 a^4 x^4+195 a^2 x^2+2205\right )+108 \left (16 a^6 x^6-5\right ) \cos ^{-1}(a x)^4-144 a x \sqrt{1-a^2 x^2} \left (8 a^4 x^4+10 a^2 x^2+15\right ) \cos ^{-1}(a x)^3-9 \left (64 a^6 x^6+120 a^4 x^4+360 a^2 x^2-245\right ) \cos ^{-1}(a x)^2+6 a x \sqrt{1-a^2 x^2} \left (32 a^4 x^4+130 a^2 x^2+735\right ) \cos ^{-1}(a x)}{10368 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 318, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{6}} \left ({\frac{{a}^{6}{x}^{6} \left ( \arccos \left ( ax \right ) \right ) ^{4}}{6}}-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{3}}{72} \left ( 8\,{a}^{5}{x}^{5}\sqrt{-{a}^{2}{x}^{2}+1}+10\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}+15\,ax\sqrt{-{a}^{2}{x}^{2}+1}+15\,\arccos \left ( ax \right ) \right ) }-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{2}{a}^{6}{x}^{6}}{18}}+{\frac{\arccos \left ( ax \right ) }{432} \left ( 8\,{a}^{5}{x}^{5}\sqrt{-{a}^{2}{x}^{2}+1}+10\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}+15\,ax\sqrt{-{a}^{2}{x}^{2}+1}+15\,\arccos \left ( ax \right ) \right ) }-{\frac{245\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}{1152}}+{\frac{{x}^{6}{a}^{6}}{324}}+{\frac{65\,{a}^{4}{x}^{4}}{3456}}+{\frac{245\,{a}^{2}{x}^{2}}{1152}}-{\frac{5\,{a}^{4}{x}^{4} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{48}}+{\frac{5\,\arccos \left ( ax \right ) }{192} \left ( 2\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}+3\,ax\sqrt{-{a}^{2}{x}^{2}+1}+3\,\arccos \left ( ax \right ) \right ) }-{\frac{5\,{a}^{2}{x}^{2} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{16}}+{\frac{5\,\arccos \left ( ax \right ) }{16} \left ( ax\sqrt{-{a}^{2}{x}^{2}+1}+\arccos \left ( ax \right ) \right ) }-{\frac{5}{32}}+{\frac{5\, \left ( \arccos \left ( ax \right ) \right ) ^{4}}{32}} \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{1}{6} \, x^{6} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{4} - 2 \, a \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x^{6} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3}}{3 \,{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45637, size = 383, normalized size = 1.36 \begin{align*} \frac{32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \,{\left (16 \, a^{6} x^{6} - 5\right )} \arccos \left (a x\right )^{4} + 2205 \, a^{2} x^{2} - 9 \,{\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \arccos \left (a x\right )^{2} - 6 \, \sqrt{-a^{2} x^{2} + 1}{\left (24 \,{\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \arccos \left (a x\right )^{3} -{\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \arccos \left (a x\right )\right )}}{10368 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.1654, size = 275, normalized size = 0.98 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acos}^{4}{\left (a x \right )}}{6} - \frac{x^{6} \operatorname{acos}^{2}{\left (a x \right )}}{18} + \frac{x^{6}}{324} - \frac{x^{5} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{9 a} + \frac{x^{5} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{54 a} - \frac{5 x^{4} \operatorname{acos}^{2}{\left (a x \right )}}{48 a^{2}} + \frac{65 x^{4}}{3456 a^{2}} - \frac{5 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{36 a^{3}} + \frac{65 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{864 a^{3}} - \frac{5 x^{2} \operatorname{acos}^{2}{\left (a x \right )}}{16 a^{4}} + \frac{245 x^{2}}{1152 a^{4}} - \frac{5 x \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{24 a^{5}} + \frac{245 x \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{576 a^{5}} - \frac{5 \operatorname{acos}^{4}{\left (a x \right )}}{96 a^{6}} + \frac{245 \operatorname{acos}^{2}{\left (a x \right )}}{1152 a^{6}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{6}}{96} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18754, size = 331, normalized size = 1.17 \begin{align*} \frac{1}{6} \, x^{6} \arccos \left (a x\right )^{4} - \frac{1}{18} \, x^{6} \arccos \left (a x\right )^{2} - \frac{\sqrt{-a^{2} x^{2} + 1} x^{5} \arccos \left (a x\right )^{3}}{9 \, a} + \frac{1}{324} \, x^{6} + \frac{\sqrt{-a^{2} x^{2} + 1} x^{5} \arccos \left (a x\right )}{54 \, a} - \frac{5 \, x^{4} \arccos \left (a x\right )^{2}}{48 \, a^{2}} - \frac{5 \, \sqrt{-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{3}}{36 \, a^{3}} + \frac{65 \, x^{4}}{3456 \, a^{2}} + \frac{65 \, \sqrt{-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )}{864 \, a^{3}} - \frac{5 \, x^{2} \arccos \left (a x\right )^{2}}{16 \, a^{4}} - \frac{5 \, \sqrt{-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{24 \, a^{5}} + \frac{245 \, x^{2}}{1152 \, a^{4}} - \frac{5 \, \arccos \left (a x\right )^{4}}{96 \, a^{6}} + \frac{245 \, \sqrt{-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{576 \, a^{5}} + \frac{245 \, \arccos \left (a x\right )^{2}}{1152 \, a^{6}} - \frac{9485}{82944 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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