Optimal. Leaf size=172 \[ \frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.17, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612,
3611} \begin {gather*} \frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx &=\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {b+a \tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx}{a^2+b^2}\\ &=\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^4}\\ &=\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.24, size = 319, normalized size = 1.85 \begin {gather*} \frac {a \left (\frac {3 i \log (i-\tan (c+d x))}{(a+i b)^4}-\frac {3 i \log (i+\tan (c+d x))}{(a-i b)^4}-\frac {24 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {6 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{6 b d}-\frac {\frac {\log (i-\tan (c+d x))}{(i a-b)^3}-\frac {\log (i+\tan (c+d x))}{(i a+b)^3}-\frac {2 b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac {b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {4 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 191, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2}-b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(191\) |
default | \(\frac {\frac {\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2}-b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(191\) |
norman | \(\frac {\frac {a \left (a^{2} b^{3}-3 b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b}+\frac {a \left (11 a^{4} b^{3}-14 b^{5} a^{2}-b^{7}\right )}{6 b^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {\left (5 a^{4} b^{3}-12 b^{5} a^{2}-b^{7}\right ) \tan \left (d x +c \right )}{2 d \,b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(541\) |
risch | \(\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}+\frac {2 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {12 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {2 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {2 i a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {12 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 b \left (-9 i a^{5}-13 i a \,b^{4}+6 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}+18 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-9 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+9 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+22 i a^{3} b^{2}-9 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+24 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+18 a^{4} b -26 a^{2} b^{3}\right )}{3 \left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} d \left (i a +b \right )^{4}}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {6 a^{2} b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(792\) |
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. 394 vs.
\(2 (168) = 336\).
time = 0.51, size = 394, normalized size = 2.29 \begin {gather*} \frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {11 \, a^{5} - 14 \, a^{3} b^{2} - a b^{4} + 6 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs.
\(2 (168) = 336\).
time = 1.27, size = 528, normalized size = 3.07 \begin {gather*} \frac {27 \, a^{5} b^{2} - 18 \, a^{3} b^{4} - a b^{6} - {\left (11 \, a^{4} b^{3} - 30 \, a^{2} b^{5} + 3 \, b^{7} - 24 \, {\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 24 \, {\left (a^{6} b - a^{4} b^{3}\right )} d x - 3 \, {\left (9 \, a^{5} b^{2} - 26 \, a^{3} b^{4} + 9 \, a b^{6} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (6 \, a^{6} b - 23 \, a^{4} b^{3} + 16 \, a^{2} b^{5} + b^{7} - 24 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs.
\(2 (168) = 336\).
time = 0.83, size = 401, normalized size = 2.33 \begin {gather*} \frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {11 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 11 \, b^{7} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{2} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{4} \tan \left (d x + c\right )^{2} + 15 \, a b^{6} \tan \left (d x + c\right )^{2} + 48 \, a^{6} b \tan \left (d x + c\right ) - 219 \, a^{4} b^{3} \tan \left (d x + c\right ) - 6 \, a^{2} b^{5} \tan \left (d x + c\right ) - 3 \, b^{7} \tan \left (d x + c\right ) + 22 \, a^{7} - 69 \, a^{5} b^{2} - 4 \, a^{3} b^{4} - a b^{6}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.45, size = 355, normalized size = 2.06 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {-11\,a^5+14\,a^3\,b^2+a\,b^4}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-5\,a^4\,b+12\,a^2\,b^3+b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^4-a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{{\left (a^2+b^2\right )}^2}-\frac {8\,b^2}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^4}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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