Optimal. Leaf size=63 \[ -\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{2 b}+\frac {\text {ArcSin}(a+b x)^2}{4 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4891, 4741,
4737, 30} \begin {gather*} \frac {\sqrt {1-(a+b x)^2} (a+b x) \text {ArcSin}(a+b x)}{2 b}+\frac {\text {ArcSin}(a+b x)^2}{4 b}-\frac {(a+b x)^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4737
Rule 4741
Rule 4891
Rubi steps
\begin {align*} \int \sqrt {1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b}-\frac {\text {Subst}(\int x \, dx,x,a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b}+\frac {\sin ^{-1}(a+b x)^2}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 64, normalized size = 1.02 \begin {gather*} \frac {-b x (2 a+b x)+2 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \text {ArcSin}(a+b x)+\text {ArcSin}(a+b x)^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 96, normalized size = 1.52
method | result | size |
default | \(\frac {2 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -b^{2} x^{2}+2 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -2 a b x +\arcsin \left (b x +a \right )^{2}-a^{2}}{4 b}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs.
\(2 (55) = 110\).
time = 0.49, size = 240, normalized size = 3.81 \begin {gather*} -\frac {1}{4} \, {\left (x^{2} + \frac {2 \, a x}{b} - \frac {2 \, \arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{2}} - \frac {\arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b - \frac {1}{2} \, {\left (\frac {a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x - \frac {{\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b}\right )} \arcsin \left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.37, size = 63, normalized size = 1.00 \begin {gather*} -\frac {b^{2} x^{2} + 2 \, a b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right ) - \arcsin \left (b x + a\right )^{2}}{4 \, b} \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 \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )} \operatorname {asin}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 79, normalized size = 1.25 \begin {gather*} \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{2 \, b} + \frac {\arcsin \left (b x + a\right )^{2}}{4 \, b} - \frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{4 \, b} - \frac {1}{8 \, b} \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 = -0.02 \begin {gather*} \int \mathrm {asin}\left (a+b\,x\right )\,\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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