Optimal. Leaf size=68 \[ \frac{1}{2} x \sqrt{\pi -\pi c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi } \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c}-\frac{1}{4} \sqrt{\pi } b c x^2 \]
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Rubi [A] time = 0.0585542, antiderivative size = 116, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4647, 4641, 30} \[ \frac{1}{2} x \sqrt{\pi -\pi c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi -\pi c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{b c x^2 \sqrt{\pi -\pi c^2 x^2}}{4 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4647
Rule 4641
Rule 30
Rubi steps
\begin{align*} \int \sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{2} x \sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi -c^2 \pi x^2} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b c \sqrt{\pi -c^2 \pi x^2}\right ) \int x \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c x^2 \sqrt{\pi -c^2 \pi x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0508784, size = 87, normalized size = 1.28 \[ \frac{\sqrt{\pi } \left (a^2+2 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left (a+b c x \sqrt{1-c^2 x^2}\right )-b^2 c^2 x^2+b^2 \sin ^{-1}(c x)^2\right )}{4 b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 101, normalized size = 1.5 \begin{align*}{\frac{ax}{2}\sqrt{-\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{a\pi }{2}\arctan \left ({x\sqrt{\pi \,{c}^{2}}{\frac{1}{\sqrt{-\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{b\sqrt{\pi }\arcsin \left ( cx \right ) x}{2}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{bc{x}^{2}\sqrt{\pi }}{4}}+{\frac{b\sqrt{\pi } \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{4\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{\pi } b \int \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\,{d x} + \frac{1}{2} \,{\left (\sqrt{\pi - \pi c^{2} x^{2}} x + \frac{\pi \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{\pi c^{2}}}\right )} a \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 (\sqrt{\pi - \pi c^{2} x^{2}}{\left (b \arcsin \left (c x\right ) + a\right )}, 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*} \sqrt{\pi } \left (\int a \sqrt{- c^{2} x^{2} + 1}\, dx + \int b \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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