Optimal. Leaf size=299 \[ \frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}+\frac {i \sqrt {3} \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac {i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
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Rubi [A]
time = 0.29, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3642, 3681,
3562, 59, 631, 210, 31, 3680} \begin {gather*} -\frac {i \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}+\frac {i \sqrt {3} \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac {i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac {\sqrt [3]{a} x}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 59
Rule 210
Rule 631
Rule 3562
Rule 3642
Rule 3680
Rule 3681
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx &=-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac {\int \cot (c+d x) \left (\frac {i a}{3}-\frac {2}{3} a \tan (c+d x)\right ) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{a}\\ &=-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac {i \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{3 a}-\int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac {(i a) \text {Subst}\left (\int \frac {1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{3 d}+\frac {(i a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\left (i \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}+\frac {\left (3 i \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\left (i a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}+\frac {\left (3 i a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}\\ &=\frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac {i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}+\frac {\left (i \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{d}-\frac {\left (3 i \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{2^{2/3} d}\\ &=\frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt [3]{a} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} d}+\frac {i \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} \log (\tan (c+d x))}{6 d}+\frac {i \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}\\ \end {align*}
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Mathematica [F]
time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 261, normalized size = 0.87 \begin {gather*} \frac {i \, {\left (\frac {6 \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} - \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {3 \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {6 \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} - \frac {2 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} + \frac {4 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} + \frac {12 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{a \tan \left (d x + c\right )}\right )} a}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 555 vs. \(2 (213) = 426\).
time = 0.57, size = 555, normalized size = 1.86 \begin {gather*} -\frac {2 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - {\left ({\left (i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {3} d + d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (\sqrt {3} d + i \, d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) - {\left ({\left (-i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {3} d + d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - {\left (\sqrt {3} d - i \, d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) - 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - 2 i \, d \left (\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) - {\left ({\left (i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {3} d + d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - \frac {3}{2} \, {\left (\sqrt {3} d + i \, d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}}\right ) - {\left ({\left (-i \, \sqrt {3} d - d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {3} d + d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + \frac {3}{2} \, {\left (\sqrt {3} d - i \, d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}}\right ) - 2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + 3 i \, d \left (-\frac {i \, a}{27 \, d^{3}}\right )^{\frac {1}{3}}\right )}{2 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.20, size = 806, normalized size = 2.70 \begin {gather*} \ln \left (\left (\left (a^7\,d^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,81{}\mathrm {i}-1458\,a^7\,d^6\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{2/3}+a^8\,d^3\,225{}\mathrm {i}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}+90\,a^8\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}+\ln \left (\left (\left (a^7\,d^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,81{}\mathrm {i}-1458\,a^7\,d^6\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{2/3}+a^8\,d^3\,225{}\mathrm {i}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}+90\,a^8\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}+\frac {\ln \left (90\,a^8\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a^8\,d^3\,225{}\mathrm {i}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^7\,d^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,81{}\mathrm {i}-729\,a^7\,d^6\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{2/3}}{4}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (90\,a^8\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a^8\,d^3\,225{}\mathrm {i}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^7\,d^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,81{}\mathrm {i}+729\,a^7\,d^6\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{2/3}}{4}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a\,1{}\mathrm {i}}{4\,d^3}\right )}^{1/3}}{2}+\frac {\ln \left (90\,a^8\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a^8\,d^3\,225{}\mathrm {i}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^7\,d^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,81{}\mathrm {i}-729\,a^7\,d^6\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{2/3}}{4}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (90\,a^8\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a^8\,d^3\,225{}\mathrm {i}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^7\,d^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,81{}\mathrm {i}+729\,a^7\,d^6\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{2/3}}{4}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a\,1{}\mathrm {i}}{27\,d^3}\right )}^{1/3}}{2}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d\,\mathrm {tan}\left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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