Optimal. Leaf size=45 \[ a x+\frac {2 b \sqrt {2 d x^2-d^2 x^4}}{d x}-b x \text {ArcSin}\left (1-d x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4924, 12, 1602}
\begin {gather*} a x+b (-x) \text {ArcSin}\left (1-d x^2\right )+\frac {2 b \sqrt {2 d x^2-d^2 x^4}}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1602
Rule 4924
Rubi steps
\begin {align*} \int \left (a-b \sin ^{-1}\left (1-d x^2\right )\right ) \, dx &=a x-b \int \sin ^{-1}\left (1-d x^2\right ) \, dx\\ &=a x-b x \sin ^{-1}\left (1-d x^2\right )+b \int -\frac {2 d x^2}{\sqrt {2 d x^2-d^2 x^4}} \, dx\\ &=a x-b x \sin ^{-1}\left (1-d x^2\right )-(2 b d) \int \frac {x^2}{\sqrt {2 d x^2-d^2 x^4}} \, dx\\ &=a x+\frac {2 b \sqrt {2 d x^2-d^2 x^4}}{d x}-b x \sin ^{-1}\left (1-d x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.96 \begin {gather*} a x+\frac {2 b \sqrt {-d x^2 \left (-2+d x^2\right )}}{d x}-b x \text {ArcSin}\left (1-d x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 45, normalized size = 1.00
method | result | size |
default | \(a x +b \left (x \arcsin \left (d \,x^{2}-1\right )-\frac {2 x \left (d \,x^{2}-2\right )}{\sqrt {-d^{2} x^{4}+2 d \,x^{2}}}\right )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 45, normalized size = 1.00 \begin {gather*} {\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 )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.41, size = 48, normalized size = 1.07 \begin {gather*} \frac {b d x^{2} \arcsin \left (d x^{2} - 1\right ) + a d x^{2} + 2 \, \sqrt {-d^{2} x^{4} + 2 \, d x^{2}} b}{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 )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 50, normalized size = 1.11 \begin {gather*} {\left (x \arcsin \left (d x^{2} - 1\right ) - \frac {2 \, \sqrt {2} \mathrm {sgn}\left (x\right )}{\sqrt {d}} + \frac {2 \, \sqrt {-d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\left (x\right )}\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 39, normalized size = 0.87 \begin {gather*} a\,x+b\,x\,\mathrm {asin}\left (d\,x^2-1\right )+\frac {2\,b\,\sqrt {1-{\left (d\,x^2-1\right )}^2}}{d\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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