Optimal. Leaf size=135 \[ -\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4+384 b^4 x \]
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Rubi [A] time = 0.03, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4814, 8} \[ -\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4+384 b^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4814
Rubi steps
\begin {align*} \int \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4 \, dx &=\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4-\left (48 b^2\right ) \int \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2 \, dx\\ &=-\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx\\ &=384 b^4 x-\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4\\ \end {align*}
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Mathematica [A] time = 0.13, size = 131, normalized size = 0.97 \[ -48 b^2 \left (\frac {4 b \sqrt {-d x^2 \left (d x^2-2\right )} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2-8 b^2 x\right )+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^4+\frac {8 b \sqrt {-d x^2 \left (d x^2-2\right )} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^3}{d x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 207, normalized size = 1.53 \[ \frac {b^{4} d x^{2} \arcsin \left (d x^{2} - 1\right )^{4} + 4 \, a b^{3} d x^{2} \arcsin \left (d x^{2} - 1\right )^{3} + 6 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 4 \, {\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arcsin \left (d x^{2} - 1\right ) + {\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 8 \, {\left (b^{4} \arcsin \left (d x^{2} - 1\right )^{3} + 3 \, a b^{3} \arcsin \left (d x^{2} - 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arcsin \left (d x^{2} - 1\right )\right )} \sqrt {-d^{2} x^{4} + 2 \, d x^{2}}}{d x} \]
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 \[ \int {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsin \left (d \,x^{2}-1\right )\right )^{4}\, dx \]
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 \[ b^{4} x \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{4} + 4 \, {\left (x \arcsin \left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {-d x^{2} + 2} d}\right )} a^{3} b + a^{4} x + \int \frac {2 \, {\left (4 \, \sqrt {-d x^{2} + 2} b^{4} \sqrt {d} x \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{3} + 2 \, {\left (a b^{3} d x^{2} - 2 \, a b^{3}\right )} \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{3} + 3 \, {\left (a^{2} b^{2} d x^{2} - 2 \, a^{2} b^{2}\right )} \arctan \left (d x^{2} - 1, \sqrt {-d x^{2} + 2} \sqrt {d} x\right )^{2}\right )}}{d x^{2} - 2}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^4 \,d x \]
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 \[ \int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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