Integrand size = 36, antiderivative size = 297 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) ((6+i) A+(1+4 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {((7-5 i) A+(5+3 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((7+5 i) A-(5-3 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {((-7-5 i) A+(5-3 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d} \]
[Out]
Time = 0.79 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3662, 3676, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) ((6+i) A+(1+4 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {((7-5 i) A+(5+3 i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{4 \sqrt {2} a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {5 (-B+i A) \sqrt {\cot (c+d x)}}{2 a d}+\frac {((7+5 i) A-(5-3 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{8 \sqrt {2} a d}+\frac {((5-3 i) B-(7+5 i) A) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{8 \sqrt {2} a d} \]
[In]
[Out]
Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3662
Rule 3676
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{\frac {5}{2}}(c+d x) (B+A \cot (c+d x))}{i a+a \cot (c+d x)} \, dx \\ & = \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\int \cot ^{\frac {3}{2}}(c+d x) \left (-\frac {5}{2} a (i A-B)+\frac {1}{2} a (7 A+3 i B) \cot (c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\int \sqrt {\cot (c+d x)} \left (-\frac {1}{2} a (7 A+3 i B)-\frac {5}{2} a (i A-B) \cot (c+d x)\right ) \, dx}{2 a^2} \\ & = \frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\int \frac {\frac {5}{2} a (i A-B)-\frac {1}{2} a (7 A+3 i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 a^2} \\ & = \frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {-\frac {5}{2} a (i A-B)+\frac {1}{2} a (7 A+3 i B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d} \\ & = \frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}-\frac {((7+5 i) A-(5-3 i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a d}+\frac {((7-5 i) A+(5+3 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a d} \\ & = \frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((7+5 i) A-(5-3 i) B) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a d}+\frac {((7+5 i) A-(5-3 i) B) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a d}+\frac {((7-5 i) A+(5+3 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 a d}+\frac {((7-5 i) A+(5+3 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 a d} \\ & = \frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((7+5 i) A-(5-3 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}-\frac {((7+5 i) A-(5-3 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {((7-5 i) A+(5+3 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}-\frac {((7-5 i) A+(5+3 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d} \\ & = -\frac {((7-5 i) A+(5+3 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {((7-5 i) A+(5+3 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((7+5 i) A-(5-3 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}-\frac {((7+5 i) A-(5-3 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d} \\ \end{align*}
Time = 3.47 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\frac {\cot ^{\frac {3}{2}}(c+d x) \left (4 i A+8 A \tan (c+d x)+12 i B \tan (c+d x)+15 i A \tan ^2(c+d x)-15 B \tan ^2(c+d x)+3 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x) (-i+\tan (c+d x))+6 \sqrt [4]{-1} (3 A+2 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x) (-i+\tan (c+d x))\right )}{6 a d (-i+\tan (c+d x))} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (246 ) = 492\).
Time = 0.41 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.83
method | result | size |
derivativedivides | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (-12 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+18 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-12 i B \tan \left (d x +c \right ) \sqrt {2}+3 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {2}+15 B \tan \left (d x +c \right )^{2} \sqrt {2}-15 i A \tan \left (d x +c \right )^{2} \sqrt {2}-8 A \tan \left (d x +c \right ) \sqrt {2}\right ) \sqrt {2}}{12 a d \left (-\tan \left (d x +c \right )+i\right )}\) | \(543\) |
default | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (-12 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+18 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-12 i B \tan \left (d x +c \right ) \sqrt {2}+3 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {2}+15 B \tan \left (d x +c \right )^{2} \sqrt {2}-15 i A \tan \left (d x +c \right )^{2} \sqrt {2}-8 A \tan \left (d x +c \right ) \sqrt {2}\right ) \sqrt {2}}{12 a d \left (-\tan \left (d x +c \right )+i\right )}\) | \(543\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (222) = 444\).
Time = 0.27 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.41 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {3 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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^{2} 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 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{2} d^{2}}} \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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^{2} 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 ) - 6 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} \log \left (\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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 {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} + 3 \, A + 2 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 6 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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 {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} - 3 \, A - 2 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - 2 \, {\left ({\left (19 i \, A - 27 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, {\left (19 i \, A - 15 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, A - 3 \, 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}}}{24 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {5}{2}}}{i \, a \tan \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
[In]
[Out]