Optimal. Leaf size=246 \[ \frac {35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac {35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}+\frac {35}{8} x \left (b^2+c^2\right )^2 \]
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Rubi [A] time = 0.17, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3113, 2637, 2638} \[ \frac {35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac {35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}+\frac {35}{8} x \left (b^2+c^2\right )^2 \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3113
Rubi steps
\begin {align*} \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx &=-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac {1}{4} \left (7 \sqrt {b^2+c^2}\right ) \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3 \, dx\\ &=-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac {1}{12} \left (35 \left (b^2+c^2\right )\right ) \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2 \, dx\\ &=-\frac {35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac {1}{8} \left (35 \left (b^2+c^2\right )^{3/2}\right ) \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right ) \, dx\\ &=\frac {35}{8} \left (b^2+c^2\right )^2 x-\frac {35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}+\frac {1}{8} \left (35 b \left (b^2+c^2\right )^{3/2}\right ) \int \cos (d+e x) \, dx+\frac {1}{8} \left (35 c \left (b^2+c^2\right )^{3/2}\right ) \int \sin (d+e x) \, dx\\ &=\frac {35}{8} \left (b^2+c^2\right )^2 x-\frac {35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}+\frac {35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac {35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\\ \end {align*}
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Mathematica [C] time = 1.42, size = 238, normalized size = 0.97 \[ \frac {168 \left (b^4-c^4\right ) \sin (2 (d+e x))+420 \left (b^2+c^2\right )^2 (d+e x)+672 b (b-i c) (b+i c) \sqrt {b^2+c^2} \sin (d+e x)+32 b \left (b^2-3 c^2\right ) \sqrt {b^2+c^2} \sin (3 (d+e x))-336 b c \left (b^2+c^2\right ) \cos (2 (d+e x))-672 c (b-i c) (b+i c) \sqrt {b^2+c^2} \cos (d+e x)+32 c \left (c^2-3 b^2\right ) \sqrt {b^2+c^2} \cos (3 (d+e x))-12 b c \left (b^2-c^2\right ) \cos (4 (d+e x))+3 \left (b^4-6 b^2 c^2+c^4\right ) \sin (4 (d+e x))}{96 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 221, normalized size = 0.90 \[ -\frac {24 \, {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{4} - 105 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} e x + 48 \, {\left (3 \, b^{3} c + 4 \, b c^{3}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (2 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{3} + {\left (27 \, b^{4} + 6 \, b^{2} c^{2} - 29 \, c^{4}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 32 \, {\left ({\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (b^{2} c + 2 \, c^{3}\right )} \cos \left (e x + d\right ) - {\left (5 \, b^{3} + 6 \, b c^{2} + {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}}{24 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 287, normalized size = 1.17 \[ -\frac {1}{8} \, {\left (b^{3} c - b c^{3}\right )} \cos \left (4 \, x e + 4 \, d\right ) e^{\left (-1\right )} - \frac {1}{3} \, {\left (3 \, \sqrt {b^{2} + c^{2}} b^{2} c - \sqrt {b^{2} + c^{2}} c^{3}\right )} \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - \frac {7}{2} \, {\left (b^{3} c + b c^{3}\right )} \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} - 7 \, {\left (\sqrt {b^{2} + c^{2}} b^{2} c + \sqrt {b^{2} + c^{2}} c^{3}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} + \frac {1}{32} \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} e^{\left (-1\right )} \sin \left (4 \, x e + 4 \, d\right ) + \frac {1}{3} \, {\left (\sqrt {b^{2} + c^{2}} b^{3} - 3 \, \sqrt {b^{2} + c^{2}} b c^{2}\right )} e^{\left (-1\right )} \sin \left (3 \, x e + 3 \, d\right ) + \frac {7}{4} \, {\left (b^{4} - c^{4}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + 7 \, {\left (\sqrt {b^{2} + c^{2}} b^{3} + \sqrt {b^{2} + c^{2}} b c^{2}\right )} e^{\left (-1\right )} \sin \left (x e + d\right ) + \frac {35}{8} \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 514, normalized size = 2.09 \[ \frac {b^{4} \left (e x +d \right )+2 b^{2} c^{2} \left (e x +d \right )+c^{4} \left (e x +d \right )+b^{4} \left (\frac {\left (\cos ^{3}\left (e x +d \right )+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+6 b^{4} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+c^{4} \left (-\frac {\left (\sin ^{3}\left (e x +d \right )+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+6 c^{4} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \left (\cos ^{3}\left (e x +d \right )\right )+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \left (\sin ^{3}\left (e x +d \right )\right )+\frac {4 \sqrt {b^{2}+c^{2}}\, b^{3} \left (2+\cos ^{2}\left (e x +d \right )\right ) \sin \left (e x +d \right )}{3}+6 b^{2} c^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+4 \sqrt {b^{2}+c^{2}}\, b^{3} \sin \left (e x +d \right )-\frac {4 \sqrt {b^{2}+c^{2}}\, c^{3} \left (2+\sin ^{2}\left (e x +d \right )\right ) \cos \left (e x +d \right )}{3}+6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-4 \sqrt {b^{2}+c^{2}}\, c^{3} \cos \left (e x +d \right )-\left (\cos ^{4}\left (e x +d \right )\right ) b^{3} c +6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \left (\cos ^{3}\left (e x +d \right )\right )}{4}+\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )+c^{3} b \left (\sin ^{4}\left (e x +d \right )\right )-6 \left (\cos ^{2}\left (e x +d \right )\right ) b^{3} c -6 \left (\cos ^{2}\left (e x +d \right )\right ) b \,c^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \sin \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 354, normalized size = 1.44 \[ -\frac {b^{3} c \cos \left (e x + d\right )^{4}}{e} + \frac {b c^{3} \sin \left (e x + d\right )^{4}}{e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) + 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} b^{4}}{32 \, e} + \frac {3 \, {\left (4 \, e x + 4 \, d - \sin \left (4 \, e x + 4 \, d\right )\right )} b^{2} c^{2}}{16 \, e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) - 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} c^{4}}{32 \, e} + {\left (b^{2} + c^{2}\right )}^{2} x - 4 \, {\left (b^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} - \frac {3}{2} \, {\left (\frac {4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} {\left (b^{2} + c^{2}\right )} - \frac {4}{3} \, {\left (\frac {3 \, b^{2} c \cos \left (e x + d\right )^{3}}{e} - \frac {3 \, b c^{2} \sin \left (e x + d\right )^{3}}{e} + \frac {{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{e} - \frac {{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{e}\right )} \sqrt {b^{2} + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.32, size = 522, normalized size = 2.12 \[ \frac {35\,\mathrm {atan}\left (\frac {35\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,{\left (b^2+c^2\right )}^2}{4\,\left (\frac {35\,b^4}{4}+\frac {35\,b^2\,c^2}{2}+\frac {35\,c^4}{4}\right )}\right )\,{\left (b^2+c^2\right )}^2}{4\,e}-\frac {35\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )-\frac {e\,x}{2}\right )\,{\left (b^2+c^2\right )}^2}{4\,e}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}+\frac {29\,b^4}{4}-\frac {27\,c^4}{4}-\frac {3\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (24\,b\,c^3+32\,b^3\,c-\left (32\,b^2\,c+8\,c^3\right )\,\sqrt {b^2+c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (64\,b\,c^3+48\,b^3\,c-\left (48\,b^2\,c+40\,c^3\right )\,\sqrt {b^2+c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (24\,b\,c^3+32\,b^3\,c-\left (32\,b^2\,c+\frac {136\,c^3}{3}\right )\,\sqrt {b^2+c^2}\right )-\left (16\,b^2\,c+\frac {40\,c^3}{3}\right )\,\sqrt {b^2+c^2}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^7\,\left (\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}-\frac {29\,b^4}{4}+\frac {27\,c^4}{4}+\frac {3\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\left (\frac {112\,b^3}{3}+56\,b\,c^2\right )\,\sqrt {b^2+c^2}+\frac {21\,b^4}{4}-\frac {35\,c^4}{4}+\frac {21\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (\left (\frac {112\,b^3}{3}+56\,b\,c^2\right )\,\sqrt {b^2+c^2}-\frac {21\,b^4}{4}+\frac {35\,c^4}{4}-\frac {21\,b^2\,c^2}{2}\right )}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 857, normalized size = 3.48 \[ \begin {cases} \frac {3 b^{4} x \sin ^{4}{\left (d + e x \right )}}{8} + \frac {3 b^{4} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{4} + 3 b^{4} x \sin ^{2}{\left (d + e x \right )} + \frac {3 b^{4} x \cos ^{4}{\left (d + e x \right )}}{8} + 3 b^{4} x \cos ^{2}{\left (d + e x \right )} + b^{4} x + \frac {3 b^{4} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{8 e} + \frac {5 b^{4} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{8 e} + \frac {3 b^{4} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {b^{3} c \cos ^{4}{\left (d + e x \right )}}{e} - \frac {6 b^{3} c \cos ^{2}{\left (d + e x \right )}}{e} + \frac {8 b^{3} \sqrt {b^{2} + c^{2}} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {4 b^{3} \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} + \frac {4 b^{3} \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )}}{e} + \frac {3 b^{2} c^{2} x \sin ^{4}{\left (d + e x \right )}}{4} + \frac {3 b^{2} c^{2} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{2} + 6 b^{2} c^{2} x \sin ^{2}{\left (d + e x \right )} + \frac {3 b^{2} c^{2} x \cos ^{4}{\left (d + e x \right )}}{4} + 6 b^{2} c^{2} x \cos ^{2}{\left (d + e x \right )} + 2 b^{2} c^{2} x + \frac {3 b^{2} c^{2} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{4 e} - \frac {3 b^{2} c^{2} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{4 e} - \frac {4 b^{2} c \sqrt {b^{2} + c^{2}} \cos ^{3}{\left (d + e x \right )}}{e} - \frac {4 b^{2} c \sqrt {b^{2} + c^{2}} \cos {\left (d + e x \right )}}{e} + \frac {b c^{3} \sin ^{4}{\left (d + e x \right )}}{e} - \frac {6 b c^{3} \cos ^{2}{\left (d + e x \right )}}{e} + \frac {4 b c^{2} \sqrt {b^{2} + c^{2}} \sin ^{3}{\left (d + e x \right )}}{e} + \frac {4 b c^{2} \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )}}{e} + \frac {3 c^{4} x \sin ^{4}{\left (d + e x \right )}}{8} + \frac {3 c^{4} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{4} + 3 c^{4} x \sin ^{2}{\left (d + e x \right )} + \frac {3 c^{4} x \cos ^{4}{\left (d + e x \right )}}{8} + 3 c^{4} x \cos ^{2}{\left (d + e x \right )} + c^{4} x - \frac {5 c^{4} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{8 e} - \frac {3 c^{4} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{8 e} - \frac {3 c^{4} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {4 c^{3} \sqrt {b^{2} + c^{2}} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {8 c^{3} \sqrt {b^{2} + c^{2}} \cos ^{3}{\left (d + e x \right )}}{3 e} - \frac {4 c^{3} \sqrt {b^{2} + c^{2}} \cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \left (b \cos {\relax (d )} + c \sin {\relax (d )} + \sqrt {b^{2} + c^{2}}\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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