Integrand size = 28, antiderivative size = 353 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f}+\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2} \]
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Time = 0.74 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3640, 3677, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^2 d^{5/2} f}+\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{5/2} f}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}-\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {1}{4 d f (a+i a \tan (e+f x))^2 (d \tan (e+f x))^{3/2}} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3610
Rule 3615
Rule 3640
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+\frac {\int \frac {\frac {11 a d}{2}-\frac {7}{2} i a d \tan (e+f x)}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))} \, dx}{4 a^2 d} \\ & = \frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+\frac {\int \frac {\frac {49 a^2 d^2}{2}-\frac {45}{2} i a^2 d^2 \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx}{8 a^4 d^2} \\ & = -\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+\frac {\int \frac {-\frac {45}{2} i a^2 d^3-\frac {49}{2} a^2 d^3 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{8 a^4 d^4} \\ & = -\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+\frac {\int \frac {-\frac {49}{2} a^2 d^4+\frac {45}{2} i a^2 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{8 a^4 d^6} \\ & = -\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+\frac {\text {Subst}\left (\int \frac {-\frac {49}{2} a^2 d^5+\frac {45}{2} i a^2 d^4 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 a^4 d^6 f} \\ & = -\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+-\frac {\left (\frac {49}{16}+\frac {45 i}{16}\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 d^2 f}+-\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 d^2 f} \\ & = -\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}+\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}+-\frac {\left (\frac {49}{32}-\frac {45 i}{32}\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 d^2 f}+-\frac {\left (\frac {49}{32}-\frac {45 i}{32}\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 d^2 f} \\ & = \frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2}+-\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f}+\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f} \\ & = \frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{16}-\frac {45 i}{16}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{5/2} f}+\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {\left (\frac {49}{32}+\frac {45 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{5/2} f}-\frac {49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac {45 i}{8 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.71 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=\frac {\sec ^2(e+f x) \left (94 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-i \tan (e+f x)\right ) (\cos (2 (e+f x))+i \sin (2 (e+f x)))+4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (e+f x)\right ) (\cos (2 (e+f x))+i \sin (2 (e+f x)))-3 (11+11 \cos (2 (e+f x))+9 i \sin (2 (e+f x)))\right )}{48 a^2 d f (d \tan (e+f x))^{3/2} (-i+\tan (e+f x))^2} \]
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Time = 0.88 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.42
method | result | size |
derivativedivides | \(\frac {2 d^{3} \left (-\frac {1}{3 d^{4} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 i}{d^{5} \sqrt {d \tan \left (f x +e \right )}}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d^{5} \sqrt {i d}}+\frac {\frac {-\frac {13 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}-\frac {15 d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {47 i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d^{5}}\right )}{f \,a^{2}}\) | \(149\) |
default | \(\frac {2 d^{3} \left (-\frac {1}{3 d^{4} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 i}{d^{5} \sqrt {d \tan \left (f x +e \right )}}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d^{5} \sqrt {i d}}+\frac {\frac {-\frac {13 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}-\frac {15 d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {47 i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d^{5}}\right )}{f \,a^{2}}\) | \(149\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (263) = 526\).
Time = 0.27 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=\frac {12 \, {\left (a^{2} d^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, a^{2} d^{3} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {i}{16 \, a^{4} d^{5} f^{2}}} \log \left (-2 \, {\left (4 \, {\left (a^{2} d^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d^{3} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {i}{16 \, a^{4} d^{5} f^{2}}} + i \, d e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) - 12 \, {\left (a^{2} d^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, a^{2} d^{3} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {i}{16 \, a^{4} d^{5} f^{2}}} \log \left (2 \, {\left (4 \, {\left (a^{2} d^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d^{3} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {i}{16 \, a^{4} d^{5} f^{2}}} - i \, d e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) - 12 \, {\left (a^{2} d^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, a^{2} d^{3} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {\frac {2209 i}{64 \, a^{4} d^{5} f^{2}}} \log \left (-\frac {{\left (8 \, {\left (a^{2} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d^{2} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {2209 i}{64 \, a^{4} d^{5} f^{2}}} + 47 i\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{2} d^{2} f}\right ) + 12 \, {\left (a^{2} d^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, a^{2} d^{3} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {\frac {2209 i}{64 \, a^{4} d^{5} f^{2}}} \log \left (\frac {{\left (8 \, {\left (a^{2} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d^{2} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {2209 i}{64 \, a^{4} d^{5} f^{2}}} - 47 i\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{2} d^{2} f}\right ) - \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (202 \, e^{\left (8 i \, f x + 8 i \, e\right )} - 103 \, e^{\left (6 i \, f x + 6 i \, e\right )} - 269 \, e^{\left (4 i \, f x + 4 i \, e\right )} + 39 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3\right )}}{48 \, {\left (a^{2} d^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, a^{2} d^{3} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \]
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\[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=- \frac {\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (e + f x \right )} - 2 i \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan {\left (e + f x \right )} - \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{2}} \]
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Exception generated. \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 1.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=-\frac {1}{24} \, d^{3} {\left (\frac {6 i \, \sqrt {2} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{2} d^{\frac {11}{2}} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} - \frac {141 i \, \sqrt {2} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{2} d^{\frac {11}{2}} f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 \, {\left (-13 i \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) - 15 \, \sqrt {d \tan \left (f x + e\right )} d\right )}}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2} a^{2} d^{5} f} + \frac {16 \, {\left (-6 i \, d \tan \left (f x + e\right ) + d\right )}}{\sqrt {d \tan \left (f x + e\right )} a^{2} d^{6} f \tan \left (f x + e\right )}\right )} \]
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Time = 7.60 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.59 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx=-\mathrm {atan}\left (8\,a^2\,d^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {1{}\mathrm {i}}{64\,a^4\,d^5\,f^2}}\right )\,\sqrt {-\frac {1{}\mathrm {i}}{64\,a^4\,d^5\,f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {16\,a^2\,d^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {2209{}\mathrm {i}}{256\,a^4\,d^5\,f^2}}}{47}\right )\,\sqrt {\frac {2209{}\mathrm {i}}{256\,a^4\,d^5\,f^2}}\,2{}\mathrm {i}-\frac {\frac {2\,d}{3\,a^2\,f}+\frac {221\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2}{24\,a^2\,f}+\frac {d\,{\mathrm {tan}\left (e+f\,x\right )}^3\,45{}\mathrm {i}}{8\,a^2\,f}-\frac {d\,\mathrm {tan}\left (e+f\,x\right )\,8{}\mathrm {i}}{3\,a^2\,f}}{d^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}-{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}+d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,2{}\mathrm {i}} \]
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