Optimal. Leaf size=75 \[ -\frac {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 272, 45}
\begin {gather*} -\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4724
Rubi steps
\begin {align*} \int x^4 \cos ^{-1}(a x) \, dx &=\frac {1}{5} x^5 \cos ^{-1}(a x)+\frac {1}{5} a \int \frac {x^5}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{5} x^5 \cos ^{-1}(a x)+\frac {1}{10} a \text {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 \cos ^{-1}(a x)+\frac {1}{10} a \text {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {1-a^2 x}}-\frac {2 \sqrt {1-a^2 x}}{a^4}+\frac {\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \cos ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right )}{75 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 72, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )}{5}-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{25}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{75}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{75}}{a^{5}}\) | \(72\) |
default | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )}{5}-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{25}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{75}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{75}}{a^{5}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 71, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right ) - \frac {1}{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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.48, size = 50, normalized size = 0.67 \begin {gather*} \frac {15 \, a^{5} x^{5} \arccos \left (a x\right ) - {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {-a^{2} x^{2} + 1}}{75 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 75, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acos}{\left (a x \right )}}{5} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{3}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 67, normalized size = 0.89 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{25 \, a} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{75 \, a^{3}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{75 \, 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 = -0.01 \begin {gather*} \int x^4\,\mathrm {acos}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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