Optimal. Leaf size=46 \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}} \]
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Rubi [A] time = 0.0843548, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4807, 4643, 4641} \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}} \]
Antiderivative was successfully verified.
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Rule 4807
Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a+b x)}{\sqrt{\left (1-a^2\right ) c-2 a b c x-b^2 c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{c-c x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sqrt{1-(a+b x)^2} \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{b \sqrt{c-c (a+b x)^2}}\\ &=\frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}}\\ \end{align*}
Mathematica [A] time = 0.0414277, size = 54, normalized size = 1.17 \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{-c \left (a^2+2 a b x+b^2 x^2-1\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 80, normalized size = 1.7 \begin{align*} -{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}{2\,b \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}-1 \right ) c}\sqrt{-c \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}-1 \right ) }\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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} c x^{2} - 2 \, a b c x -{\left (a^{2} - 1\right )} c} \arcsin \left (b x + a\right )}{b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} - 1\right )} c}, 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{\operatorname{asin}{\left (a + b x \right )}}{\sqrt{- c \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (b x + a\right )}{\sqrt{-b^{2} c x^{2} - 2 \, a b c x -{\left (a^{2} - 1\right )} c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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