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 = 0.06, 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 30
Rule 4641
Rule 4647
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*}
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Mathematica [A] time = 0.05, 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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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\pi - \pi c^{2} x^{2}} {\left (b \arcsin \left (c x\right ) + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 101, normalized size = 1.49 \[ \frac {a x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \sqrt {\pi }\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x}{2}-\frac {b c \,x^{2} \sqrt {\pi }}{4}+\frac {b \sqrt {\pi }\, \arcsin \left (c x \right )^{2}}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {\pi } b \int \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} + \frac {1}{2} \, {\left (\sqrt {\pi - \pi c^{2} x^{2}} x + \frac {\sqrt {\pi } \arcsin \left (c x\right )}{c}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {\Pi -\Pi \,c^2\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {\pi } \left (\int a \sqrt {- c^{2} x^{2} + 1}\, dx + \int b \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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