Optimal. Leaf size=161 \[ \frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}+\frac {(c+d x)^5}{10 d}-\frac {3 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {3 d^4 \cos (a+b x) \sin (a+b x)}{4 b^5}-\frac {3 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cos (a+b x) \sin (a+b x)}{2 b} \]
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Rubi [A]
time = 0.07, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 32, 2715,
8} \begin {gather*} \frac {3 d^4 \sin (a+b x) \cos (a+b x)}{4 b^5}-\frac {3 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}-\frac {3 d^2 (c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b^3}+\frac {d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}+\frac {(c+d x)^5}{10 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rubi steps
\begin {align*} \int (c+d x)^4 \cos ^2(a+b x) \, dx &=\frac {d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \cos (a+b x) \sin (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^4 \, dx-\frac {\left (3 d^2\right ) \int (c+d x)^2 \cos ^2(a+b x) \, dx}{b^2}\\ &=\frac {(c+d x)^5}{10 d}-\frac {3 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {d (c+d x)^3 \cos ^2(a+b x)}{b^2}-\frac {3 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cos (a+b x) \sin (a+b x)}{2 b}-\frac {\left (3 d^2\right ) \int (c+d x)^2 \, dx}{2 b^2}+\frac {\left (3 d^4\right ) \int \cos ^2(a+b x) \, dx}{2 b^4}\\ &=-\frac {d (c+d x)^3}{2 b^2}+\frac {(c+d x)^5}{10 d}-\frac {3 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {3 d^4 \cos (a+b x) \sin (a+b x)}{4 b^5}-\frac {3 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cos (a+b x) \sin (a+b x)}{2 b}+\frac {\left (3 d^4\right ) \int 1 \, dx}{4 b^4}\\ &=\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}+\frac {(c+d x)^5}{10 d}-\frac {3 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {3 d^4 \cos (a+b x) \sin (a+b x)}{4 b^5}-\frac {3 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cos (a+b x) \sin (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 132, normalized size = 0.82 \begin {gather*} \frac {8 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )+20 b d (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+10 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \sin (2 (a+b x))}{80 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1026\) vs.
\(2(147)=294\).
time = 0.18, size = 1027, normalized size = 6.38
method | result | size |
risch | \(\frac {d^{4} x^{5}}{10}+\frac {c \,d^{3} x^{4}}{2}+d^{2} c^{2} x^{3}+d \,c^{3} x^{2}+\frac {c^{4} x}{2}+\frac {c^{5}}{10 d}+\frac {d \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 b x +2 a \right )}{4 b^{4}}+\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+8 b^{4} c^{3} d x +2 c^{4} b^{4}-6 b^{2} d^{4} x^{2}-12 b^{2} c \,d^{3} x -6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \sin \left (2 b x +2 a \right )}{8 b^{5}}\) | \(227\) |
norman | \(\frac {\frac {d^{4} x^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {\left (2 c^{4} b^{4}-6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b^{5}}+\frac {\left (2 c^{4} b^{4}+6 b^{2} c^{2} d^{2}-3 d^{4}\right ) x}{4 b^{4}}+\frac {d^{2} \left (2 b^{2} c^{2}+d^{2}\right ) x^{3}}{2 b^{2}}+\frac {2 c d \left (2 b^{2} c^{2}-3 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {c \,d^{3} x^{4}}{2}-\frac {3 d^{2} \left (2 b^{2} c^{2}-d^{2}\right ) x^{2} \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{3}}+\frac {d^{2} \left (2 b^{2} c^{2}-3 d^{2}\right ) x^{3} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}-\frac {4 c \,d^{3} x^{3} \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {d^{2} \left (2 b^{2} c^{2}+d^{2}\right ) x^{3} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{2}}+\frac {c d \left (2 b^{2} c^{2}+3 d^{2}\right ) x^{2}}{2 b^{2}}+\frac {3 d^{2} \left (2 b^{2} c^{2}-d^{2}\right ) x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {4 c \,d^{3} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {d^{4} x^{5}}{10}-\frac {2 c d \left (2 b^{2} c^{2}-3 d^{2}\right ) x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{3}}+\frac {c d \left (2 b^{2} c^{2}-9 d^{2}\right ) x^{2} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {c d \left (2 b^{2} c^{2}+3 d^{2}\right ) x^{2} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{2}}+\frac {\left (2 c^{4} b^{4}-18 b^{2} c^{2} d^{2}+9 d^{4}\right ) x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{4}}+\frac {d^{4} x^{5} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{10}+\frac {2 \left (-2 b^{2} c^{3} d +3 c \,d^{3}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{4}}+\frac {d^{4} x^{5} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5}-\frac {\left (2 c^{4} b^{4}-6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{5}}+\frac {\left (2 c^{4} b^{4}+6 b^{2} c^{2} d^{2}-3 d^{4}\right ) x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b^{4}}+c \,d^{3} x^{4} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {d^{4} x^{4} \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {c \,d^{3} x^{4} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{2}}\) | \(743\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1027\) |
default | \(\text {Expression too large to display}\) | \(1027\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 717 vs.
\(2 (147) = 294\).
time = 0.33, size = 717, normalized size = 4.45 \begin {gather*} \frac {10 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} c^{4} - \frac {40 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a c^{3} d}{b} + \frac {60 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} c^{2} d^{2}}{b^{2}} - \frac {40 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a^{3} c d^{3}}{b^{3}} + \frac {10 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a^{4} d^{4}}{b^{4}} + \frac {20 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} c^{3} d}{b} - \frac {60 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} a c^{2} d^{2}}{b^{2}} + \frac {60 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} a^{2} c d^{3}}{b^{3}} - \frac {20 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} a^{3} d^{4}}{b^{4}} + \frac {10 \, {\left (4 \, {\left (b x + a\right )}^{3} + 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d^{2}}{b^{2}} - \frac {20 \, {\left (4 \, {\left (b x + a\right )}^{3} + 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{3}}{b^{3}} + \frac {10 \, {\left (4 \, {\left (b x + a\right )}^{3} + 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{4}}{b^{4}} + \frac {10 \, {\left (2 \, {\left (b x + a\right )}^{4} + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{3}}{b^{3}} - \frac {10 \, {\left (2 \, {\left (b x + a\right )}^{4} + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{4}}{b^{4}} + \frac {{\left (4 \, {\left (b x + a\right )}^{5} + 10 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 5 \, {\left (2 \, {\left (b x + a\right )}^{4} - 6 \, {\left (b x + a\right )}^{2} + 3\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{4}}{b^{4}}}{40 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 287, normalized size = 1.78 \begin {gather*} \frac {2 \, b^{5} d^{4} x^{5} + 10 \, b^{5} c d^{3} x^{4} + 10 \, {\left (2 \, b^{5} c^{2} d^{2} - b^{3} d^{4}\right )} x^{3} + 10 \, {\left (2 \, b^{5} c^{3} d - 3 \, b^{3} c d^{3}\right )} x^{2} + 10 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 5 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 2 \, b^{4} c^{4} - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \, {\left (2 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 5 \, {\left (2 \, b^{5} c^{4} - 6 \, b^{3} c^{2} d^{2} + 3 \, b d^{4}\right )} x}{20 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 660 vs.
\(2 (156) = 312\).
time = 0.56, size = 660, normalized size = 4.10 \begin {gather*} \begin {cases} \frac {c^{4} x \sin ^{2}{\left (a + b x \right )}}{2} + \frac {c^{4} x \cos ^{2}{\left (a + b x \right )}}{2} + c^{3} d x^{2} \sin ^{2}{\left (a + b x \right )} + c^{3} d x^{2} \cos ^{2}{\left (a + b x \right )} + c^{2} d^{2} x^{3} \sin ^{2}{\left (a + b x \right )} + c^{2} d^{2} x^{3} \cos ^{2}{\left (a + b x \right )} + \frac {c d^{3} x^{4} \sin ^{2}{\left (a + b x \right )}}{2} + \frac {c d^{3} x^{4} \cos ^{2}{\left (a + b x \right )}}{2} + \frac {d^{4} x^{5} \sin ^{2}{\left (a + b x \right )}}{10} + \frac {d^{4} x^{5} \cos ^{2}{\left (a + b x \right )}}{10} + \frac {c^{4} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {2 c^{3} d x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} + \frac {3 c^{2} d^{2} x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} + \frac {2 c d^{3} x^{3} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} - \frac {c^{3} d \sin ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {3 c^{2} d^{2} x \sin ^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 c^{2} d^{2} x \cos ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c d^{3} x^{2} \sin ^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 c d^{3} x^{2} \cos ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{4} x^{3} \sin ^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {d^{4} x^{3} \cos ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c^{2} d^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{3}} - \frac {3 c d^{3} x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{b^{3}} - \frac {3 d^{4} x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{3}} + \frac {3 c d^{3} \sin ^{2}{\left (a + b x \right )}}{2 b^{4}} + \frac {3 d^{4} x \sin ^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {3 d^{4} x \cos ^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac {3 d^{4} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 222, normalized size = 1.38 \begin {gather*} \frac {1}{10} \, d^{4} x^{5} + \frac {1}{2} \, c d^{3} x^{4} + c^{2} d^{2} x^{3} + c^{3} d x^{2} + \frac {1}{2} \, c^{4} x + \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \cos \left (2 \, b x + 2 \, a\right )}{4 \, b^{5}} + \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 349, normalized size = 2.17 \begin {gather*} \frac {\frac {15\,d^4\,\sin \left (2\,a+2\,b\,x\right )}{2}+10\,b^5\,c^4\,x+5\,b^4\,c^4\,\sin \left (2\,a+2\,b\,x\right )+2\,b^5\,d^4\,x^5+10\,b^3\,c^3\,d\,\cos \left (2\,a+2\,b\,x\right )+20\,b^5\,c^3\,d\,x^2+10\,b^5\,c\,d^3\,x^4-15\,b^2\,c^2\,d^2\,\sin \left (2\,a+2\,b\,x\right )+10\,b^3\,d^4\,x^3\,\cos \left (2\,a+2\,b\,x\right )+20\,b^5\,c^2\,d^2\,x^3-15\,b^2\,d^4\,x^2\,\sin \left (2\,a+2\,b\,x\right )+5\,b^4\,d^4\,x^4\,\sin \left (2\,a+2\,b\,x\right )-15\,b\,c\,d^3\,\cos \left (2\,a+2\,b\,x\right )-15\,b\,d^4\,x\,\cos \left (2\,a+2\,b\,x\right )+30\,b^4\,c^2\,d^2\,x^2\,\sin \left (2\,a+2\,b\,x\right )-30\,b^2\,c\,d^3\,x\,\sin \left (2\,a+2\,b\,x\right )+20\,b^4\,c^3\,d\,x\,\sin \left (2\,a+2\,b\,x\right )+30\,b^3\,c^2\,d^2\,x\,\cos \left (2\,a+2\,b\,x\right )+30\,b^3\,c\,d^3\,x^2\,\cos \left (2\,a+2\,b\,x\right )+20\,b^4\,c\,d^3\,x^3\,\sin \left (2\,a+2\,b\,x\right )}{20\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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