Optimal. Leaf size=110 \[ -\frac{c \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.110228, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4693, 29, 4641} \[ -\frac{c \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4693
Rule 29
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x} \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{c \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt{1-c^2 x^2}}+\frac{b c \sqrt{d-c^2 d x^2} \log (x)}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.328402, size = 142, normalized size = 1.29 \[ -\frac{a \sqrt{-d \left (c^2 x^2-1\right )}}{x}+a c \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{-d \left (c^2 x^2-1\right )}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-\frac{b c \sqrt{d \left (1-c^2 x^2\right )} \left (\frac{2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}-2 \log (c x)+\sin ^{-1}(c x)^2\right )}{2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.172, size = 308, normalized size = 2.8 \begin{align*} -{\frac{a}{dx} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-a{c}^{2}x\sqrt{-{c}^{2}d{x}^{2}+d}-{a{c}^{2}d\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b \left ( \arcsin \left ( cx \right ) \right ) ^{2}c}{2\,{c}^{2}{x}^{2}-2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{ib\arcsin \left ( cx \right ) c}{{c}^{2}{x}^{2}-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) x{c}^{2}}{{c}^{2}{x}^{2}-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b\arcsin \left ( cx \right ) }{ \left ({c}^{2}{x}^{2}-1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{{c}^{2}{x}^{2}-1}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{2}}, 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{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{x^{2}}\, 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{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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