Optimal. Leaf size=110 \[ -24 a b^2 x-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^3+\frac{48 b^3 \sqrt{2 d x^2-d^2 x^4}}{d x}-24 b^3 x \cos ^{-1}\left (d x^2-1\right ) \]
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Rubi [A] time = 0.055042, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4815, 4841, 12, 1588} \[ -24 a b^2 x-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^3+\frac{48 b^3 \sqrt{2 d x^2-d^2 x^4}}{d x}-24 b^3 x \cos ^{-1}\left (d x^2-1\right ) \]
Antiderivative was successfully verified.
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Rule 4815
Rule 4841
Rule 12
Rule 1588
Rubi steps
\begin{align*} \int \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^3 \, dx &=-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^3-\left (24 b^2\right ) \int \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right ) \, dx\\ &=-24 a b^2 x-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^3-\left (24 b^3\right ) \int \cos ^{-1}\left (-1+d x^2\right ) \, dx\\ &=-24 a b^2 x-24 b^3 x \cos ^{-1}\left (-1+d x^2\right )-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^3-\left (24 b^3\right ) \int \frac{2 d x^2}{\sqrt{2 d x^2-d^2 x^4}} \, dx\\ &=-24 a b^2 x-24 b^3 x \cos ^{-1}\left (-1+d x^2\right )-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^3-\left (48 b^3 d\right ) \int \frac{x^2}{\sqrt{2 d x^2-d^2 x^4}} \, dx\\ &=-24 a b^2 x+\frac{48 b^3 \sqrt{2 d x^2-d^2 x^4}}{d x}-24 b^3 x \cos ^{-1}\left (-1+d x^2\right )-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^2}{d x}+x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^3\\ \end{align*}
Mathematica [A] time = 0.135227, size = 162, normalized size = 1.47 \[ \frac{a d x^2 \left (a^2-24 b^2\right )-6 b \left (a^2-8 b^2\right ) \sqrt{d x^2 \left (2-d x^2\right )}+3 b \cos ^{-1}\left (d x^2-1\right ) \left (a^2 d x^2-4 a b \sqrt{-d x^2 \left (d x^2-2\right )}-8 b^2 d x^2\right )+3 b^2 \cos ^{-1}\left (d x^2-1\right )^2 \left (a d x^2-2 b \sqrt{-d x^2 \left (d x^2-2\right )}\right )+b^3 d x^2 \cos ^{-1}\left (d x^2-1\right )^3}{d x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( d{x}^{2}-1 \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5729, size = 328, normalized size = 2.98 \begin{align*} \frac{b^{3} d x^{2} \arccos \left (d x^{2} - 1\right )^{3} + 3 \, a b^{2} d x^{2} \arccos \left (d x^{2} - 1\right )^{2} + 3 \,{\left (a^{2} b - 8 \, b^{3}\right )} d x^{2} \arccos \left (d x^{2} - 1\right ) +{\left (a^{3} - 24 \, a b^{2}\right )} d x^{2} - 6 \, \sqrt{-d^{2} x^{4} + 2 \, d x^{2}}{\left (b^{3} \arccos \left (d x^{2} - 1\right )^{2} + 2 \, a b^{2} \arccos \left (d x^{2} - 1\right ) + a^{2} b - 8 \, b^{3}\right )}}{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acos}{\left (d x^{2} - 1 \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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