Optimal. Leaf size=359 \[ -\frac {413312 c^3 \sqrt {1+a^2 x^2}}{128625 a}-\frac {30256 c^3 \left (1+a^2 x^2\right )^{3/2}}{385875 a}-\frac {2664 c^3 \left (1+a^2 x^2\right )^{5/2}}{214375 a}-\frac {6 c^3 \left (1+a^2 x^2\right )^{7/2}}{2401 a}+\frac {4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac {1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {48 c^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac {8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.52, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {5786, 5772,
5798, 267, 5784, 455, 45, 200, 12, 1261, 712, 1813, 1864} \begin {gather*} \frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}-\frac {6 c^3 \left (a^2 x^2+1\right )^{7/2}}{2401 a}-\frac {2664 c^3 \left (a^2 x^2+1\right )^{5/2}}{214375 a}-\frac {30256 c^3 \left (a^2 x^2+1\right )^{3/2}}{385875 a}-\frac {413312 c^3 \sqrt {a^2 x^2+1}}{128625 a}+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac {3 c^3 \left (a^2 x^2+1\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}-\frac {18 c^3 \left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {8 c^3 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {48 c^3 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {4322 c^3 x \sinh ^{-1}(a x)}{1225} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 200
Rule 267
Rule 455
Rule 712
Rule 1261
Rule 1813
Rule 1864
Rule 5772
Rule 5784
Rule 5786
Rule 5798
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^3 \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^3 \, dx-\frac {1}{7} \left (3 a c^3\right ) \int x \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2 \, dx\\ &=-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^3 \, dx+\frac {1}{49} \left (6 c^3\right ) \int \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x) \, dx-\frac {1}{35} \left (18 a c^3\right ) \int x \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2 \, dx\\ &=\frac {6}{49} c^3 x \sinh ^{-1}(a x)+\frac {6}{49} a^2 c^3 x^3 \sinh ^{-1}(a x)+\frac {18}{245} a^4 c^3 x^5 \sinh ^{-1}(a x)+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac {1}{175} \left (36 c^3\right ) \int \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x) \, dx+\frac {1}{35} \left (16 c^3\right ) \int \sinh ^{-1}(a x)^3 \, dx-\frac {1}{49} \left (6 a c^3\right ) \int \frac {x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right )}{35 \sqrt {1+a^2 x^2}} \, dx-\frac {1}{35} \left (24 a c^3\right ) \int x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \, dx\\ &=\frac {402 c^3 x \sinh ^{-1}(a x)}{1225}+\frac {318 a^2 c^3 x^3 \sinh ^{-1}(a x)}{1225}+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac {1}{35} \left (16 c^3\right ) \int \left (1+a^2 x^2\right ) \sinh ^{-1}(a x) \, dx-\frac {\left (6 a c^3\right ) \int \frac {x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right )}{\sqrt {1+a^2 x^2}} \, dx}{1715}-\frac {1}{175} \left (36 a c^3\right ) \int \frac {x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt {1+a^2 x^2}} \, dx-\frac {1}{35} \left (48 a c^3\right ) \int \frac {x \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {962 c^3 x \sinh ^{-1}(a x)}{1225}+\frac {1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {48 c^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac {8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac {1}{35} \left (96 c^3\right ) \int \sinh ^{-1}(a x) \, dx-\frac {\left (3 a c^3\right ) \text {Subst}\left (\int \frac {35+35 a^2 x+21 a^4 x^2+5 a^6 x^3}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )}{1715}-\frac {1}{875} \left (12 a c^3\right ) \int \frac {x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {1+a^2 x^2}} \, dx-\frac {1}{35} \left (16 a c^3\right ) \int \frac {x \left (1+\frac {a^2 x^2}{3}\right )}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac {1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {48 c^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac {8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3-\frac {\left (3 a c^3\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1+a^2 x}}+8 \sqrt {1+a^2 x}+6 \left (1+a^2 x\right )^{3/2}+5 \left (1+a^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{1715}-\frac {1}{875} \left (6 a c^3\right ) \text {Subst}\left (\int \frac {15+10 a^2 x+3 a^4 x^2}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {1}{35} \left (8 a c^3\right ) \text {Subst}\left (\int \frac {1+\frac {a^2 x}{3}}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {1}{35} \left (96 a c^3\right ) \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {960 c^3 \sqrt {1+a^2 x^2}}{343 a}-\frac {16 c^3 \left (1+a^2 x^2\right )^{3/2}}{1715 a}-\frac {36 c^3 \left (1+a^2 x^2\right )^{5/2}}{8575 a}-\frac {6 c^3 \left (1+a^2 x^2\right )^{7/2}}{2401 a}+\frac {4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac {1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {48 c^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac {8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3-\frac {1}{875} \left (6 a c^3\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1+a^2 x}}+4 \sqrt {1+a^2 x}+3 \left (1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )-\frac {1}{35} \left (8 a c^3\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+a^2 x}}+\frac {1}{3} \sqrt {1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {413312 c^3 \sqrt {1+a^2 x^2}}{128625 a}-\frac {30256 c^3 \left (1+a^2 x^2\right )^{3/2}}{385875 a}-\frac {2664 c^3 \left (1+a^2 x^2\right )^{5/2}}{214375 a}-\frac {6 c^3 \left (1+a^2 x^2\right )^{7/2}}{2401 a}+\frac {4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac {1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac {48 c^3 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac {8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 169, normalized size = 0.47 \begin {gather*} \frac {c^3 \left (-2 \sqrt {1+a^2 x^2} \left (22329151+747937 a^2 x^2+134541 a^4 x^4+16875 a^6 x^6\right )+210 a x \left (226905+26495 a^2 x^2+7371 a^4 x^4+1125 a^6 x^6\right ) \sinh ^{-1}(a x)-11025 \sqrt {1+a^2 x^2} \left (2161+757 a^2 x^2+351 a^4 x^4+75 a^6 x^6\right ) \sinh ^{-1}(a x)^2+385875 a x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right ) \sinh ^{-1}(a x)^3\right )}{13505625 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{3} \arcsinh \left (a x \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 276, normalized size = 0.77 \begin {gather*} -\frac {1}{1225} \, {\left (75 \, \sqrt {a^{2} x^{2} + 1} a^{4} c^{3} x^{6} + 351 \, \sqrt {a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 757 \, \sqrt {a^{2} x^{2} + 1} c^{3} x^{2} + \frac {2161 \, \sqrt {a^{2} x^{2} + 1} c^{3}}{a^{2}}\right )} a \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \operatorname {arsinh}\left (a x\right )^{3} - \frac {2}{13505625} \, {\left (16875 \, \sqrt {a^{2} x^{2} + 1} a^{4} c^{3} x^{6} + 134541 \, \sqrt {a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt {a^{2} x^{2} + 1} c^{3} x^{2} + \frac {22329151 \, \sqrt {a^{2} x^{2} + 1} c^{3}}{a^{2}} - \frac {105 \, {\left (1125 \, a^{6} c^{3} x^{7} + 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} + 226905 \, c^{3} x\right )} \operatorname {arsinh}\left (a x\right )}{a}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 248, normalized size = 0.69 \begin {gather*} \frac {385875 \, {\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 11025 \, {\left (75 \, a^{6} c^{3} x^{6} + 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} + 2161 \, c^{3}\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (1125 \, a^{7} c^{3} x^{7} + 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} + 226905 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, {\left (16875 \, a^{6} c^{3} x^{6} + 134541 \, a^{4} c^{3} x^{4} + 747937 \, a^{2} c^{3} x^{2} + 22329151 \, c^{3}\right )} \sqrt {a^{2} x^{2} + 1}}{13505625 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.48, size = 355, normalized size = 0.99 \begin {gather*} \begin {cases} \frac {a^{6} c^{3} x^{7} \operatorname {asinh}^{3}{\left (a x \right )}}{7} + \frac {6 a^{6} c^{3} x^{7} \operatorname {asinh}{\left (a x \right )}}{343} - \frac {3 a^{5} c^{3} x^{6} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{49} - \frac {6 a^{5} c^{3} x^{6} \sqrt {a^{2} x^{2} + 1}}{2401} + \frac {3 a^{4} c^{3} x^{5} \operatorname {asinh}^{3}{\left (a x \right )}}{5} + \frac {702 a^{4} c^{3} x^{5} \operatorname {asinh}{\left (a x \right )}}{6125} - \frac {351 a^{3} c^{3} x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{1225} - \frac {29898 a^{3} c^{3} x^{4} \sqrt {a^{2} x^{2} + 1}}{1500625} + a^{2} c^{3} x^{3} \operatorname {asinh}^{3}{\left (a x \right )} + \frac {1514 a^{2} c^{3} x^{3} \operatorname {asinh}{\left (a x \right )}}{3675} - \frac {757 a c^{3} x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{1225} - \frac {1495874 a c^{3} x^{2} \sqrt {a^{2} x^{2} + 1}}{13505625} + c^{3} x \operatorname {asinh}^{3}{\left (a x \right )} + \frac {4322 c^{3} x \operatorname {asinh}{\left (a x \right )}}{1225} - \frac {2161 c^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{1225 a} - \frac {44658302 c^{3} \sqrt {a^{2} x^{2} + 1}}{13505625 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\mathrm {asinh}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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