Integrand size = 23, antiderivative size = 500 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d} \]
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Time = 0.92 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3647, 3728, 3712, 3566, 722, 1108, 648, 632, 212, 642} \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 722
Rule 1108
Rule 3566
Rule 3647
Rule 3712
Rule 3728
Rubi steps \begin{align*} \text {integral}& = \frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {2 \int \frac {\tan (c+d x) \left (-2 a-\frac {5}{2} b \tan (c+d x)-2 a \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{5 b} \\ & = -\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {4 \int \frac {2 a^2+\frac {1}{4} \left (8 a^2-15 b^2\right ) \tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{15 b^2} \\ & = \frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\int \frac {1}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{d} \\ & = \frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}-x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = \frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}-\frac {b \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = -\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}-\frac {b \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d}-\frac {b \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d} \\ & = \frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d} \\ \end{align*}
Time = 3.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.37 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {-\frac {15 \sqrt {-b^2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}+\frac {-\frac {15 \left (-b^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}+2 \sqrt {a+b \tan (c+d x)} \left (8 a^2-15 b^2-4 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{b^2}}{15 b d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1640\) vs. \(2(409)=818\).
Time = 0.11 (sec) , antiderivative size = 1641, normalized size of antiderivative = 3.28
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1641\) |
default | \(\text {Expression too large to display}\) | \(1641\) |
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Time = 0.26 (sec) , antiderivative size = 789, normalized size of antiderivative = 1.58 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {15 \, b^{3} d \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b + {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + b^{2} d\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - 15 \, b^{3} d \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b - {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + b^{2} d\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - 15 \, b^{3} d \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b + {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - b^{2} d\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) + 15 \, b^{3} d \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} b - {\left ({\left (a^{3} + a b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - b^{2} d\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) + 4 \, {\left (3 \, b^{2} \tan \left (d x + c\right )^{2} - 4 \, a b \tan \left (d x + c\right ) + 8 \, a^{2} - 15 \, b^{2}\right )} \sqrt {b \tan \left (d x + c\right ) + a}}{30 \, b^{3} d} \]
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\[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\tan ^{4}{\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{4}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]
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Time = 9.21 (sec) , antiderivative size = 791, normalized size of antiderivative = 1.58 \[ \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\left (\frac {4\,a^2}{b^3\,d}-\frac {2\,\left (a^2+b^2\right )}{b^3\,d}\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+\frac {\ln \left (16\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+16\,b^3\,d\,\sqrt {-\frac {1}{d^2\,\left (a-b\,1{}\mathrm {i}\right )}}-\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a-b\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{a\,d^2-b\,d^2\,1{}\mathrm {i}}}}{2}-\ln \left (-16\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+16\,b^3\,d\,\sqrt {-\frac {1}{d^2\,\left (a-b\,1{}\mathrm {i}\right )}}+\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a-b\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{4\,\left (a\,d^2-b\,d^2\,1{}\mathrm {i}\right )}}+\frac {2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{5\,b^3\,d}-\frac {4\,a\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{3\,b^3\,d}+\mathrm {atan}\left (-\frac {b^2\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{-\frac {64\,a\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^4\,d^2\,64{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {128\,a\,b^3\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{-\frac {256\,a^3\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a\,b^5\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^6\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^2\,b^4\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {a^2\,b^2\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{-\frac {256\,a^3\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a\,b^5\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^6\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^2\,b^4\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}\right )\,\sqrt {-\frac {a-b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,2{}\mathrm {i} \]
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