Optimal. Leaf size=66 \[ \frac{b c \sqrt{1-c^2 x^2} \log (x)}{\sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x} \]
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Rubi [A] time = 0.0904784, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {4681, 29} \[ \frac{b c \sqrt{1-c^2 x^2} \log (x)}{\sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x} \]
Antiderivative was successfully verified.
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Rule 4681
Rule 29
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^2 \sqrt{d-c^2 d x^2}} \, dx &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}+\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}+\frac{b c \sqrt{1-c^2 x^2} \log (x)}{\sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.124217, size = 69, normalized size = 1.05 \[ \frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d \sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.148, size = 216, normalized size = 3.3 \begin{align*} -{\frac{a}{dx}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{ib\arcsin \left ( cx \right ) c}{d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) x{c}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b\arcsin \left ( cx \right ) }{ \left ({c}^{2}{x}^{2}-1 \right ) dx}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\ln \left ( \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) } \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 [A] time = 2.09764, size = 487, normalized size = 7.38 \begin{align*} \left [\frac{b c \sqrt{d} x \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} - d}{c^{2} x^{4} - x^{2}}\right ) - 2 \, \sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{2 \, d x}, \frac{b c \sqrt{-d} x \arctan \left (\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{2} + 1\right )} \sqrt{-d}}{c^{2} d x^{4} -{\left (c^{2} + 1\right )} d x^{2} + d}\right ) - \sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{d 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{a + b \operatorname{asin}{\left (c x \right )}}{x^{2} \sqrt{- d \left (c x - 1\right ) \left (c 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{b \arcsin \left (c x\right ) + a}{\sqrt{-c^{2} d x^{2} + d} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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