Optimal. Leaf size=250 \[ \frac {16576 x}{5625 a^4}+\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{625 a}-\frac {32 x \text {ArcCos}(a x)^2}{25 a^4}-\frac {16 x^3 \text {ArcCos}(a x)^2}{75 a^2}-\frac {12}{125} x^5 \text {ArcCos}(a x)^2-\frac {32 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {ArcCos}(a x)^4 \]
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Rubi [A]
time = 0.44, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4724, 4796,
4768, 4716, 8, 30} \begin {gather*} -\frac {32 x \text {ArcCos}(a x)^2}{25 a^4}+\frac {16576 x}{5625 a^4}-\frac {16 x^3 \text {ArcCos}(a x)^2}{75 a^2}-\frac {4 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{25 a}+\frac {24 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{625 a}+\frac {1088 x^3}{16875 a^2}-\frac {32 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{75 a^5}+\frac {16576 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{5625 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{75 a^3}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{5625 a^3}+\frac {1}{5} x^5 \text {ArcCos}(a x)^4-\frac {12}{125} x^5 \text {ArcCos}(a x)^2+\frac {24 x^5}{3125} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4716
Rule 4724
Rule 4768
Rule 4796
Rubi steps
\begin {align*} \int x^4 \cos ^{-1}(a x)^4 \, dx &=\frac {1}{5} x^5 \cos ^{-1}(a x)^4+\frac {1}{5} (4 a) \int \frac {x^5 \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^4-\frac {12}{25} \int x^4 \cos ^{-1}(a x)^2 \, dx+\frac {16 \int \frac {x^3 \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac {12}{125} x^5 \cos ^{-1}(a x)^2-\frac {16 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^4+\frac {32 \int \frac {x \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{75 a^3}-\frac {16 \int x^2 \cos ^{-1}(a x)^2 \, dx}{25 a^2}-\frac {1}{125} (24 a) \int \frac {x^5 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {24 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac {16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac {12}{125} x^5 \cos ^{-1}(a x)^2-\frac {32 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^4+\frac {24 \int x^4 \, dx}{625}-\frac {32 \int \cos ^{-1}(a x)^2 \, dx}{25 a^4}-\frac {96 \int \frac {x^3 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{625 a}-\frac {32 \int \frac {x^3 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{75 a}\\ &=\frac {24 x^5}{3125}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac {32 x \cos ^{-1}(a x)^2}{25 a^4}-\frac {16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac {12}{125} x^5 \cos ^{-1}(a x)^2-\frac {32 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^4-\frac {64 \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{625 a^3}-\frac {64 \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{225 a^3}-\frac {64 \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}+\frac {32 \int x^2 \, dx}{625 a^2}+\frac {32 \int x^2 \, dx}{225 a^2}\\ &=\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac {32 x \cos ^{-1}(a x)^2}{25 a^4}-\frac {16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac {12}{125} x^5 \cos ^{-1}(a x)^2-\frac {32 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^4+\frac {64 \int 1 \, dx}{625 a^4}+\frac {64 \int 1 \, dx}{225 a^4}+\frac {64 \int 1 \, dx}{25 a^4}\\ &=\frac {16576 x}{5625 a^4}+\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac {32 x \cos ^{-1}(a x)^2}{25 a^4}-\frac {16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac {12}{125} x^5 \cos ^{-1}(a x)^2-\frac {32 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 150, normalized size = 0.60 \begin {gather*} \frac {8 a x \left (31080+680 a^2 x^2+81 a^4 x^4\right )+120 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right ) \text {ArcCos}(a x)-900 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \text {ArcCos}(a x)^2-4500 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {ArcCos}(a x)^3+16875 a^5 x^5 \text {ArcCos}(a x)^4}{84375 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 197, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{4}}{5}-\frac {4 \arccos \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 a^{5} x^{5} \arccos \left (a x \right )^{2}}{125}+\frac {8 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 a^{3} x^{3} \arccos \left (a x \right )^{2}}{75}+\frac {32 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 a x \arccos \left (a x \right )^{2}}{25}+\frac {64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\) | \(197\) |
default | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{4}}{5}-\frac {4 \arccos \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 a^{5} x^{5} \arccos \left (a x \right )^{2}}{125}+\frac {8 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 a^{3} x^{3} \arccos \left (a x \right )^{2}}{75}+\frac {32 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 a x \arccos \left (a x \right )^{2}}{25}+\frac {64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 206, normalized size = 0.82 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right )^{4} - \frac {4}{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 )^{3} + \frac {4}{84375} \, {\left (2 \, a {\left (\frac {15 \, {\left (27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{5}} + \frac {81 \, a^{4} x^{5} + 680 \, a^{2} x^{3} + 31080 \, x}{a^{6}}\right )} - \frac {225 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arccos \left (a x\right )^{2}}{a^{5}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.04, size = 134, normalized size = 0.54 \begin {gather*} \frac {16875 \, a^{5} x^{5} \arccos \left (a x\right )^{4} + 648 \, a^{5} x^{5} + 5440 \, a^{3} x^{3} - 900 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arccos \left (a x\right )^{2} + 248640 \, a x - 60 \, \sqrt {-a^{2} x^{2} + 1} {\left (75 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arccos \left (a x\right )^{3} - 2 \, {\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \arccos \left (a x\right )\right )}}{84375 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.07, size = 248, normalized size = 0.99 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acos}^{4}{\left (a x \right )}}{5} - \frac {12 x^{5} \operatorname {acos}^{2}{\left (a x \right )}}{125} + \frac {24 x^{5}}{3125} - \frac {4 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{25 a} + \frac {24 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{625 a} - \frac {16 x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{75 a^{2}} + \frac {1088 x^{3}}{16875 a^{2}} - \frac {16 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{75 a^{3}} + \frac {1088 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{5625 a^{3}} - \frac {32 x \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{4}} + \frac {16576 x}{5625 a^{4}} - \frac {32 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{75 a^{5}} + \frac {16576 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{5625 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{5}}{80} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 212, normalized size = 0.85 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right )^{4} - \frac {12}{125} \, x^{5} \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )^{3}}{25 \, a} + \frac {24}{3125} \, x^{5} + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{625 \, a} - \frac {16 \, x^{3} \arccos \left (a x\right )^{2}}{75 \, a^{2}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{75 \, a^{3}} + \frac {1088 \, x^{3}}{16875 \, a^{2}} + \frac {1088 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{5625 \, a^{3}} - \frac {32 \, x \arccos \left (a x\right )^{2}}{25 \, a^{4}} - \frac {32 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{75 \, a^{5}} + \frac {16576 \, x}{5625 \, a^{4}} + \frac {16576 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{5625 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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