Optimal. Leaf size=13 \[ -\frac{1}{b \sin ^{-1}(a+b x)} \]
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Rubi [A] time = 0.0752961, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {4807, 4641} \[ -\frac{1}{b \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 4807
Rule 4641
Rubi steps
\begin{align*} \int \frac{1}{\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{1}{\sqrt{1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1}{b \sin ^{-1}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0143502, size = 13, normalized size = 1. \[ -\frac{1}{b \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b\arcsin \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.68438, size = 45, normalized size = 3.46 \begin{align*} -\frac{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 [A] time = 2.14064, size = 32, normalized size = 2.46 \begin{align*} -\frac{1}{b \arcsin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58624, size = 26, normalized size = 2. \begin{align*} \begin{cases} - \frac{1}{b \operatorname{asin}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{1 - a^{2}} \operatorname{asin}^{2}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24989, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{b \arcsin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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