Optimal. Leaf size=80 \[ \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.0363379, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4653, 260} \[ \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4653
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{x}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.202672, size = 77, normalized size = 0.96 \[ -\frac{\sqrt{d-c^2 d x^2} \left (2 a c x+b \sqrt{1-c^2 x^2} \log \left (c^2 x^2-1\right )+2 b c x \sin ^{-1}(c x)\right )}{2 c d^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.086, size = 177, normalized size = 2.2 \begin{align*}{\frac{ax}{d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}+{\frac{ib\arcsin \left ( cx \right ) }{c{d}^{2} \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}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\ln \left ( 1+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.7018, size = 93, normalized size = 1.16 \begin{align*} -\frac{b c \sqrt{\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{2 \, d} + \frac{b x \arcsin \left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a x}{\sqrt{-c^{2} d x^{2} + d} d} \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 )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{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{a + b \operatorname{asin}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{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{b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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