Optimal. Leaf size=39 \[ -\frac{\text{Si}\left (2 \sin ^{-1}(a+b x)\right )}{b}-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)} \]
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Rubi [A] time = 0.114454, antiderivative size = 39, normalized size of antiderivative = 1., 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 = {4807, 4659, 4635, 4406, 12, 3299} \[ -\frac{\text{Si}\left (2 \sin ^{-1}(a+b x)\right )}{b}-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 4807
Rule 4659
Rule 4635
Rule 4406
Rule 12
Rule 3299
Rubi steps
\begin{align*} \int \frac{\sqrt{1-a^2-2 a b x-b^2 x^2}}{\sin ^{-1}(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{x}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)}-\frac{\text{Si}\left (2 \sin ^{-1}(a+b x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0580095, size = 46, normalized size = 1.18 \[ \frac{a^2-\sin ^{-1}(a+b x) \text{Si}\left (2 \sin ^{-1}(a+b x)\right )+2 a b x+b^2 x^2-1}{b \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 42, normalized size = 1.1 \begin{align*} -{\frac{2\,{\it Si} \left ( 2\,\arcsin \left ( bx+a \right ) \right ) \arcsin \left ( bx+a \right ) +\cos \left ( 2\,\arcsin \left ( bx+a \right ) \right ) +1}{2\,b\arcsin \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b^{2} x^{2} + 2 \, a b x - 2 \, b \arctan \left (b x + a, \sqrt{b x + a + 1} \sqrt{-b x - a + 1}\right ) \int \frac{b x + a}{\arctan \left (b x + a, \sqrt{b x + a + 1} \sqrt{-b x - a + 1}\right )}\,{d x} + a^{2} - 1}{b \arctan \left (b x + a, \sqrt{b x + a + 1} \sqrt{-b x - a + 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{\arcsin \left (b x + a\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{\operatorname{asin}^{2}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27118, size = 59, normalized size = 1.51 \begin{align*} -\frac{\operatorname{Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b} + \frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b \arcsin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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