Optimal. Leaf size=15 \[ -\frac{1}{2 b \sin ^{-1}(a+b x)^2} \]
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Rubi [A] time = 0.0703002, antiderivative size = 15, 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} \[ -\frac{1}{2 b \sin ^{-1}(a+b x)^2} \]
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)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1}{2 b \sin ^{-1}(a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0145088, size = 15, normalized size = 1. \[ -\frac{1}{2 b \sin ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 82.6449, size = 45, normalized size = 3. \begin{align*} -\frac{1}{2 \, b \arctan \left (b x + a, \sqrt{b x + a + 1} \sqrt{-b x - a + 1}\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06257, size = 38, normalized size = 2.53 \begin{align*} -\frac{1}{2 \, b \arcsin \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.26528, size = 29, normalized size = 1.93 \begin{align*} \begin{cases} - \frac{1}{2 b \operatorname{asin}^{2}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{1 - a^{2}} \operatorname{asin}^{3}{\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 = 1.29891, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{2 \, b \arcsin \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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