Optimal. Leaf size=69 \[ 24 x+\frac {24 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{a}-12 x \text {ArcCos}(a x)^2-\frac {4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{a}+x \text {ArcCos}(a x)^4 \]
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Rubi [A]
time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4768, 8}
\begin {gather*} -\frac {4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{a}+\frac {24 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{a}+x \text {ArcCos}(a x)^4-12 x \text {ArcCos}(a x)^2+24 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4716
Rule 4768
Rubi steps
\begin {align*} \int \cos ^{-1}(a x)^4 \, dx &=x \cos ^{-1}(a x)^4+(4 a) \int \frac {x \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4-12 \int \cos ^{-1}(a x)^2 \, dx\\ &=-12 x \cos ^{-1}(a x)^2-\frac {4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4-(24 a) \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {24 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}-12 x \cos ^{-1}(a x)^2-\frac {4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x+\frac {24 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}-12 x \cos ^{-1}(a x)^2-\frac {4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 69, normalized size = 1.00 \begin {gather*} 24 x+\frac {24 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{a}-12 x \text {ArcCos}(a x)^2-\frac {4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^3}{a}+x \text {ArcCos}(a x)^4 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 67, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {a x \arccos \left (a x \right )^{4}-4 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arccos \left (a x \right )^{2}+24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
default | \(\frac {a x \arccos \left (a x \right )^{4}-4 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arccos \left (a x \right )^{2}+24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 74, normalized size = 1.07 \begin {gather*} x \arccos \left (a x\right )^{4} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arccos \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x + \frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 55, normalized size = 0.80 \begin {gather*} \frac {a x \arccos \left (a x\right )^{4} - 12 \, a x \arccos \left (a x\right )^{2} + 24 \, a x - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 70, normalized size = 1.01 \begin {gather*} \begin {cases} x \operatorname {acos}^{4}{\left (a x \right )} - 12 x \operatorname {acos}^{2}{\left (a x \right )} + 24 x - \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{a} + \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x}{16} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 65, normalized size = 0.94 \begin {gather*} x \arccos \left (a x\right )^{4} - 12 \, x \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} + 24 \, x + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 63, normalized size = 0.91 \begin {gather*} \left \{\begin {array}{cl} \frac {x\,\pi ^4}{16} & \text {\ if\ \ }a=0\\ x\,\left ({\mathrm {acos}\left (a\,x\right )}^4-12\,{\mathrm {acos}\left (a\,x\right )}^2+24\right )+\frac {\sqrt {1-a^2\,x^2}\,\left (24\,\mathrm {acos}\left (a\,x\right )-4\,{\mathrm {acos}\left (a\,x\right )}^3\right )}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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