Integrand size = 36, antiderivative size = 364 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+6 i) A-(29+i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {((5-7 i) A+(28+30 i) B) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{\sqrt {2} a^3 d}+\frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Time = 0.87 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3676, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+6 i) A-(29+i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {((5-7 i) A+(28+30 i) B) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{16 \sqrt {2} a^3 d}-\frac {7 (-4 B+i A) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {(-B+i A) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3676
Rubi steps \begin{align*} \text {integral}& = \frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\tan ^{\frac {5}{2}}(c+d x) \left (\frac {7}{2} a (i A-B)+\frac {1}{2} a (A+13 i B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2} \\ & = \frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-5 a^2 (2 A+5 i B)+a^2 (4 i A-31 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4} \\ & = \frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {\int \sqrt {\tan (c+d x)} \left (-21 a^3 (i A-4 B)-15 a^3 (A+6 i B) \tan (c+d x)\right ) \, dx}{48 a^6} \\ & = \frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {\int \frac {15 a^3 (A+6 i B)-21 a^3 (i A-4 B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{48 a^6} \\ & = \frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {15 a^3 (A+6 i B)-21 a^3 (i A-4 B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{24 a^6 d} \\ & = \frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {((5+7 i) A-(28-30 i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{16 a^3 d}-\frac {((5-7 i) A+(28+30 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{16 a^3 d} \\ & = \frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {((5+7 i) A-(28-30 i) B) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 \sqrt {2} a^3 d}+\frac {((5+7 i) A-(28-30 i) B) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 \sqrt {2} a^3 d}-\frac {((5-7 i) A+(28+30 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 a^3 d}-\frac {((5-7 i) A+(28+30 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 a^3 d} \\ & = \frac {((5+7 i) A-(28-30 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {((5+7 i) A-(28-30 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {((5-7 i) A+(28+30 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {((5-7 i) A+(28+30 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^3 d} \\ & = \frac {((5-7 i) A+(28+30 i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^3 d}-\frac {((5-7 i) A+(28+30 i) B) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {((5+7 i) A-(28-30 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {((5+7 i) A-(28-30 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {5 (A+6 i B) \sqrt {\tan (c+d x)}}{8 a^3 d}+\frac {(i A-B) \tan ^{\frac {7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(2 A+5 i B) \tan ^{\frac {5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac {7 (i A-4 B) \tan ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )} \\ \end{align*}
Time = 4.60 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.61 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {\sec ^3(c+d x) \left (-12 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))+12 (-1)^{3/4} (6 A+29 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))+2 i ((9 A+33 i B) \cos (c+d x)+21 (A+7 i B) \cos (3 (c+d x))+2 i (19 A+97 i B+(19 A+145 i B) \cos (2 (c+d x))) \sin (c+d x)) \sqrt {\tan (c+d x)}\right )}{96 a^3 d (-i+\tan (c+d x))^3} \]
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Time = 0.06 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.50
method | result | size |
derivativedivides | \(\frac {2 i B \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {i \left (\frac {-i \left (9 i A -20 B \right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )+\left (-\frac {38 i A}{3}+\frac {98 B}{3}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\left (-14 i B -5 A \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{\left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {2 \left (29 i B +6 A \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}\right )}{8}+\frac {4 \left (\frac {i A}{16}+\frac {B}{16}\right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}}{d \,a^{3}}\) | \(181\) |
default | \(\frac {2 i B \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {i \left (\frac {-i \left (9 i A -20 B \right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )+\left (-\frac {38 i A}{3}+\frac {98 B}{3}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\left (-14 i B -5 A \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{\left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {2 \left (29 i B +6 A \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}\right )}{8}+\frac {4 \left (\frac {i A}{16}+\frac {B}{16}\right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}}{d \,a^{3}}\) | \(181\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (271) = 542\).
Time = 0.28 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.88 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{3} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{6} d^{2}}} - {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - 3 \, a^{3} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{6} d^{2}}} - {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) + 3 \, a^{3} d \sqrt {\frac {36 i \, A^{2} - 348 \, A B - 841 i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {36 i \, A^{2} - 348 \, A B - 841 i \, B^{2}}{a^{6} d^{2}}} + 6 i \, A - 29 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 3 \, a^{3} d \sqrt {\frac {36 i \, A^{2} - 348 \, A B - 841 i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {36 i \, A^{2} - 348 \, A B - 841 i \, B^{2}}{a^{6} d^{2}}} - 6 i \, A + 29 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 2 \, {\left (2 \, {\left (10 \, A + 73 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (14 \, A + 41 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (5 \, A + 8 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
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Timed out. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.95 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.46 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} {\left (-6 i \, A + 29 \, B\right )} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} + \frac {\left (i - 1\right ) \, \sqrt {2} {\left (-i \, A - B\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} + \frac {2 i \, B \sqrt {\tan \left (d x + c\right )}}{a^{3} d} + \frac {-27 i \, A \tan \left (d x + c\right )^{\frac {5}{2}} + 60 \, B \tan \left (d x + c\right )^{\frac {5}{2}} - 38 \, A \tan \left (d x + c\right )^{\frac {3}{2}} - 98 i \, B \tan \left (d x + c\right )^{\frac {3}{2}} + 15 i \, A \sqrt {\tan \left (d x + c\right )} - 42 \, B \sqrt {\tan \left (d x + c\right )}}{24 \, a^{3} d {\left (\tan \left (d x + c\right ) - i\right )}^{3}} \]
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Time = 11.53 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\mathrm {atan}\left (\frac {8\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {A^2\,9{}\mathrm {i}}{64\,a^6\,d^2}}}{3\,A}\right )\,\sqrt {\frac {A^2\,9{}\mathrm {i}}{64\,a^6\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {16\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {A^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}}{A}\right )\,\sqrt {-\frac {A^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {B^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}\,16{}\mathrm {i}}{B}\right )\,\sqrt {\frac {B^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {B^2\,841{}\mathrm {i}}{256\,a^6\,d^2}}\,16{}\mathrm {i}}{29\,B}\right )\,\sqrt {-\frac {B^2\,841{}\mathrm {i}}{256\,a^6\,d^2}}\,2{}\mathrm {i}+\frac {\frac {5\,A\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{8\,a^3\,d}-\frac {9\,A\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}}{8\,a^3\,d}+\frac {A\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,19{}\mathrm {i}}{12\,a^3\,d}}{-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1}-\frac {\frac {49\,B\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{12\,a^3\,d}-\frac {B\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,7{}\mathrm {i}}{4\,a^3\,d}+\frac {B\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,5{}\mathrm {i}}{2\,a^3\,d}}{-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1}+\frac {B\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,2{}\mathrm {i}}{a^3\,d} \]
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