Optimal. Leaf size=111 \[ -\frac {(a+b x) \sqrt {1-(a+b x)^2}}{4 b}+\frac {\text {ArcSin}(a+b x)}{4 b}-\frac {(a+b x)^2 \text {ArcSin}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{2 b}+\frac {\text {ArcSin}(a+b x)^3}{6 b} \]
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Rubi [A]
time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4891, 4741,
4737, 4723, 327, 222} \begin {gather*} \frac {\text {ArcSin}(a+b x)^3}{6 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{2 b}-\frac {(a+b x)^2 \text {ArcSin}(a+b x)}{2 b}+\frac {\text {ArcSin}(a+b x)}{4 b}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 4723
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)^2 \, dx &=\frac {\text {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b}-\frac {\text {Subst}\left (\int x \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac {(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac {\sin ^{-1}(a+b x)^3}{6 b}+\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{4 b}-\frac {(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac {\sin ^{-1}(a+b x)^3}{6 b}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{4 b}\\ &=-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{4 b}+\frac {\sin ^{-1}(a+b x)}{4 b}-\frac {(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac {\sin ^{-1}(a+b x)^3}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 116, normalized size = 1.05 \begin {gather*} \frac {-3 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2}-3 \left (-1+2 a^2+4 a b x+2 b^2 x^2\right ) \text {ArcSin}(a+b x)+6 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \text {ArcSin}(a+b x)^2+2 \text {ArcSin}(a+b x)^3}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 179, normalized size = 1.61
method | result | size |
default | \(\frac {6 \arcsin \left (b x +a \right )^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -6 \arcsin \left (b x +a \right ) b^{2} x^{2}+6 \arcsin \left (b x +a \right )^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -12 \arcsin \left (b x +a \right ) a b x +2 \arcsin \left (b x +a \right )^{3}-6 \arcsin \left (b x +a \right ) a^{2}-3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +3 \arcsin \left (b x +a \right )}{12 b}\) | \(179\) |
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 = 7.71, size = 91, normalized size = 0.82 \begin {gather*} \frac {2 \, \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (2 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2} - b x - a\right )}}{12 \, 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}^{2}{\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 = 0.46, size = 125, normalized size = 1.13 \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}}{2 \, b} + \frac {\arcsin \left (b x + a\right )^{3}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )}{2 \, b} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{4 \, b} - \frac {\arcsin \left (b x + a\right )}{4 \, 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 = -0.01 \begin {gather*} \int {\mathrm {asin}\left (a+b\,x\right )}^2\,\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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