Optimal. Leaf size=115 \[ -24 a b^2 x+\frac {6 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3-\frac {48 b^3 \sqrt {2 d x^2-d^2 x^4}}{d x}+24 b^3 x \sin ^{-1}\left (1-d x^2\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4814, 4840, 12, 1588} \[ -24 a b^2 x+\frac {6 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3-\frac {48 b^3 \sqrt {2 d x^2-d^2 x^4}}{d x}+24 b^3 x \sin ^{-1}\left (1-d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 1588
Rule 4814
Rule 4840
Rubi steps
\begin {align*} \int \left (a-b \sin ^{-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 \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3-\left (24 b^2\right ) \int \left (a-b \sin ^{-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 \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3+\left (24 b^3\right ) \int \sin ^{-1}\left (1-d x^2\right ) \, dx\\ &=-24 a b^2 x+24 b^3 x \sin ^{-1}\left (1-d x^2\right )+\frac {6 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-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 \sin ^{-1}\left (1-d x^2\right )+\frac {6 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-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 \sin ^{-1}\left (1-d x^2\right )+\frac {6 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3\\ \end {align*}
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Mathematica [A] time = 0.13, size = 166, normalized size = 1.44 \[ \frac {a d x^2 \left (a^2-24 b^2\right )+6 b \left (a^2-8 b^2\right ) \sqrt {d x^2 \left (2-d x^2\right )}-3 b \sin ^{-1}\left (1-d x^2\right ) \left (a^2 d x^2+4 a b \sqrt {-d x^2 \left (d x^2-2\right )}-8 b^2 d x^2\right )+3 b^2 \sin ^{-1}\left (1-d x^2\right )^2 \left (a d x^2+2 b \sqrt {-d x^2 \left (d x^2-2\right )}\right )-b^3 d x^2 \sin ^{-1}\left (1-d x^2\right )^3}{d x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 144, normalized size = 1.25 \[ \frac {b^{3} d x^{2} \arcsin \left (d x^{2} - 1\right )^{3} + 3 \, a b^{2} d x^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 3 \, {\left (a^{2} b - 8 \, b^{3}\right )} d x^{2} \arcsin \left (d x^{2} - 1\right ) + {\left (a^{3} - 24 \, a b^{2}\right )} d x^{2} + 6 \, \sqrt {-d^{2} x^{4} + 2 \, d x^{2}} {\left (b^{3} \arcsin \left (d x^{2} - 1\right )^{2} + 2 \, a b^{2} \arcsin \left (d x^{2} - 1\right ) + a^{2} b - 8 \, b^{3}\right )}}{d x} \]
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 \[ \int {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsin \left (d \,x^{2}-1\right )\right )^{3}\, dx \]
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 \[ b^{3} x \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{3} + 3 \, {\left (x \arcsin \left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {-d x^{2} + 2} d}\right )} a^{2} b + a^{3} x + \int \frac {3 \, {\left (2 \, \sqrt {-d x^{2} + 2} b^{3} \sqrt {d} x \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{2} + {\left (a b^{2} d x^{2} - 2 \, a b^{2}\right )} \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{2}\right )}}{d x^{2} - 2}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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