Optimal. Leaf size=16 \[ \frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c} \]
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Rubi [A] time = 0.0480692, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {4639} \[ \frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 4639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c}\\ \end{align*}
Mathematica [A] time = 0.047529, size = 16, normalized size = 1. \[ \frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\arcsin \left ( cx \right ) \right ) }{bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50592, size = 22, normalized size = 1.38 \begin{align*} \frac{\log \left (b \arcsin \left (c x\right ) + a\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93635, size = 42, normalized size = 2.62 \begin{align*} \frac{\log \left (-b \arcsin \left (c x\right ) - a\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.3367, size = 43, normalized size = 2.69 \begin{align*} \frac{\log \left (b^{2} \Im \left ( \arcsin \left (c x\right ) \right )^{2} +{\left (b \Re \left ( \arcsin \left (c x\right ) \right ) + a\right )}^{2}\right )}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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