Optimal. Leaf size=341 \[ -\frac {i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 d^3}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b^2 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}+\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )} \]
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Rubi [A] time = 0.42, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {4703, 4655, 4657, 4181, 2531, 2282, 6589, 4677, 206, 199} \[ -\frac {i b \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {i b \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac {b^2 \text {PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \text {PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 d^3}+\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b^2 \tanh ^{-1}(c x)}{6 c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 2282
Rule 2531
Rule 4181
Rule 4655
Rule 4657
Rule 4677
Rule 4703
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}-\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {b^2 \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{6 c^2 d^3}+\frac {b \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}-\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{8 c^2 d^2}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^3 d^3}+\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{12 c^2 d^3}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{4 c^2 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}+\frac {b \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}-\frac {b \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}\\ &=\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}-\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {i b \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac {b^2 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}\\ \end {align*}
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Mathematica [A] time = 4.65, size = 446, normalized size = 1.31 \[ \frac {\frac {6 a^2 c x}{c^2 x^2-1}+\frac {12 a^2 c x}{\left (c^2 x^2-1\right )^2}+3 a^2 \log (1-c x)-3 a^2 \log (c x+1)+\frac {a b \left (\sqrt {1-c^2 x^2}+12 \sin ^{-1}(c x) \left (c^3 x^3-\left (c^2 x^2-1\right )^2 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+\left (c^2 x^2-1\right )^2 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+c x\right )-4 \cos \left (2 \sin ^{-1}(c x)\right )+3 \cos \left (3 \sin ^{-1}(c x)\right )-\cos \left (4 \sin ^{-1}(c x)\right )-3\right )}{\left (c^2 x^2-1\right )^2}-12 i a b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )+12 i a b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )+\frac {b^2 \left (2 \sin ^{-1}(c x) \left (\sqrt {1-c^2 x^2}+3 \cos \left (3 \sin ^{-1}(c x)\right )\right )-3 \left (\sin \left (3 \sin ^{-1}(c x)\right )-7 c x\right ) \sin ^{-1}(c x)^2+2 \left (c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )\right )}{2 \left (c^2 x^2-1\right )^2}+4 b^2 \left (-3 i \sin ^{-1}(c x) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )+3 i \sin ^{-1}(c x) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )+3 \text {Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )-3 \text {Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )-2 \tanh ^{-1}(c x)+3 i \sin ^{-1}(c x)^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )\right )}{48 c^3 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{2} x^{2} \arcsin \left (c x\right )^{2} + 2 \, a b x^{2} \arcsin \left (c x\right ) + a^{2} x^{2}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 894, normalized size = 2.62 \[ \frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}-\frac {i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}-\frac {a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}+\frac {a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}-\frac {i a b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}+\frac {i a b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}+\frac {a b \arcsin \left (c x \right ) x^{3}}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2} x}{8 c^{2} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{12 c^{3} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{12 c^{3} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {a^{2}}{16 c^{3} d^{3} \left (c x -1\right )}-\frac {a^{2}}{16 c^{3} d^{3} \left (c x +1\right )^{2}}+\frac {a^{2}}{16 c^{3} d^{3} \left (c x +1\right )}+\frac {a^{2}}{16 c^{3} d^{3} \left (c x -1\right )^{2}}-\frac {a^{2} \ln \left (c x +1\right )}{16 c^{3} d^{3}}+\frac {a^{2} \ln \left (c x -1\right )}{16 c^{3} d^{3}}-\frac {b^{2} x^{3}}{12 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b^{2} \polylog \left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}-\frac {b^{2} \polylog \left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4 c^{3} d^{3}}+\frac {b^{2} x}{12 c^{2} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2} x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 c^{3} d^{3}}+\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 c^{3} d^{3}}+\frac {i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3 c^{3} d^{3}}-\frac {a b \,x^{2} \sqrt {-c^{2} x^{2}+1}}{4 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {a b \arcsin \left (c x \right ) x}{4 c^{2} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2}}{4 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \]
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 \[ \frac {1}{16} \, a^{2} {\left (\frac {2 \, {\left (c^{2} x^{3} + x\right )}}{c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{3} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{3} d^{3}}\right )} - \frac {{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} \log \left (c x + 1\right ) - {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} \log \left (-c x + 1\right ) - 2 \, {\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (c^{7} d^{3} x^{4} - 2 \, c^{5} d^{3} x^{2} + c^{3} d^{3}\right )} \int \frac {16 \, a b c^{2} x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left ({\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (c x + 1\right ) - {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \, {\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{8} d^{3} x^{6} - 3 \, c^{6} d^{3} x^{4} + 3 \, c^{4} d^{3} x^{2} - c^{2} d^{3}}\,{d x}}{16 \, {\left (c^{7} d^{3} x^{4} - 2 \, c^{5} d^{3} x^{2} + c^{3} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a^{2} x^{2}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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