Optimal. Leaf size=43 \[ x \text {ArcCos}\left (a x^2\right )+\frac {2 E\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}-\frac {2 F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4925, 12, 313,
227, 1213, 435} \begin {gather*} x \text {ArcCos}\left (a x^2\right )-\frac {2 F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+\frac {2 E\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 227
Rule 313
Rule 435
Rule 1213
Rule 4925
Rubi steps
\begin {align*} \int \cos ^{-1}\left (a x^2\right ) \, dx &=x \cos ^{-1}\left (a x^2\right )+\int \frac {2 a x^2}{\sqrt {1-a^2 x^4}} \, dx\\ &=x \cos ^{-1}\left (a x^2\right )+(2 a) \int \frac {x^2}{\sqrt {1-a^2 x^4}} \, dx\\ &=x \cos ^{-1}\left (a x^2\right )-2 \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx+2 \int \frac {1+a x^2}{\sqrt {1-a^2 x^4}} \, dx\\ &=x \cos ^{-1}\left (a x^2\right )-\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+2 \int \frac {\sqrt {1+a x^2}}{\sqrt {1-a x^2}} \, dx\\ &=x \cos ^{-1}\left (a x^2\right )+\frac {2 E\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}-\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 34, normalized size = 0.79 \begin {gather*} x \text {ArcCos}\left (a x^2\right )+\frac {2}{3} a x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};a^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 65, normalized size = 1.51
method | result | size |
default | \(x \arccos \left (a \,x^{2}\right )-\frac {2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {a}, i\right )-\EllipticE \left (x \sqrt {a}, i\right )\right )}{\sqrt {a}\, \sqrt {-a^{2} x^{4}+1}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 33, normalized size = 0.77 \begin {gather*} \frac {a x^{2} \arccos \left (a x^{2}\right ) - 2 \, \sqrt {-a^{2} x^{4} + 1}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.62, size = 44, normalized size = 1.02 \begin {gather*} \frac {a x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {a^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + x \operatorname {acos}{\left (a x^{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \mathrm {acos}\left (a\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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