Optimal. Leaf size=192 \[ \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4647, 4641, 4627, 321, 216} \[ \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 4627
Rule 4641
Rule 4647
Rubi steps
\begin {align*} \int \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 128, normalized size = 0.67 \[ \frac {1}{6} \sqrt {d-c^2 d x^2} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{b c \sqrt {1-c^2 x^2}}-\frac {3 b \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )}{2 c \sqrt {1-c^2 x^2}}+3 x \left (a+b \sin ^{-1}(c x)\right )^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}, x\right ) \]
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 [B] time = 0.20, size = 564, normalized size = 2.94 \[ \frac {x \sqrt {-c^{2} d \,x^{2}+d}\, a^{2}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{6 c \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{4 c \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right )^{2} x^{3}}{2 c^{2} x^{2}-2}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} x}{2 \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} x^{3}}{4 \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x}{4 c^{2} x^{2}-4}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{2}}{2 c^{2} x^{2}-2}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 c \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \sqrt {-c^{2} x^{2}+1}\, x^{2}}{2 c^{2} x^{2}-2}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) x^{3}}{c^{2} x^{2}-1}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{4 c \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{c^{2} x^{2}-1} \]
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 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a^{2} + \sqrt {d} \int {\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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