Optimal. Leaf size=135 \[ 384 b^4 x-\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^4 \]
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Rubi [A]
time = 0.02, 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 = {4898, 8}
\begin {gather*} -\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^4+384 b^4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4898
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 = 0.08, size = 131, normalized size = 0.97 \begin {gather*} \frac {8 b \sqrt {-d x^2 \left (-2+d x^2\right )} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^4-48 b^2 \left (-8 b^2 x+\frac {4 b \sqrt {-d x^2 \left (-2+d x^2\right )} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \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 = 2.61, size = 207, normalized size = 1.53 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \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 = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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