Integrand size = 33, antiderivative size = 437 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}-\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
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Time = 1.94 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3662, 3690, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a (A b-a B)}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}-\frac {-3 a^2 B+a A b-2 b^2 B}{b^2 d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)}}+\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {a^{3/2} \left (-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d \left (a^2+b^2\right )^2} \]
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Rule 65
Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3662
Rule 3690
Rule 3715
Rule 3730
Rule 3734
Rubi steps \begin{align*} \text {integral}& = \int \frac {B+A \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2} \, dx \\ & = \frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\int \frac {\frac {1}{2} \left (a A b-3 a^2 B-2 b^2 B\right )-b (A b-a B) \cot (c+d x)+\frac {3}{2} a (A b-a B) \cot ^2(c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))} \, dx}{b \left (a^2+b^2\right )} \\ & = -\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {2 \int \frac {\frac {1}{4} \left (-a^2 A b-2 A b^3+3 a^3 B+4 a b^2 B\right )+\frac {1}{2} b^2 (a A+b B) \cot (c+d x)-\frac {1}{4} a \left (a A b-3 a^2 B-2 b^2 B\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{b^2 \left (a^2+b^2\right )} \\ & = -\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {2 \int \frac {\frac {1}{2} b^2 \left (a^2 A-A b^2+2 a b B\right )+\frac {1}{2} b^2 \left (2 a A b-a^2 B+b^2 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {4 \text {Subst}\left (\int \frac {-\frac {1}{2} b^2 \left (a^2 A-A b^2+2 a b B\right )-\frac {1}{2} b^2 \left (2 a A b-a^2 B+b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 b^2 \left (a^2+b^2\right )^2 d} \\ & = -\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2 \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}-\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \\ & = -\frac {a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}-\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d} \\ & = -\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}-\frac {a A b-3 a^2 B-2 b^2 B}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \\ \end{align*}
Time = 3.63 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.89 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {2 \sqrt {2} \left (a^2 (A-B)+b^2 (-A+B)+2 a b (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{\left (a^2+b^2\right )^2}+\frac {4 a^{3/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right )}+\frac {8 a^{3/2} \left (a^2 A b+3 A b^3-2 a^3 B-4 a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right )^2}-\frac {\sqrt {2} \left (2 a b (-A+B)+a^2 (A+B)-b^2 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{\left (a^2+b^2\right )^2}+\frac {8 B \sqrt {\tan (c+d x)}}{b^2}+\frac {4 a^2 (-A b+a B) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{4 d} \]
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Time = 0.46 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-2 a b A +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 B}{b^{2} \sqrt {\cot \left (d x +c \right )}}-\frac {2 a^{2} \left (\frac {\left (\frac {1}{2} A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {1}{2} B \,a^{3}-\frac {1}{2} B a \,b^{2}\right ) \sqrt {\cot \left (d x +c \right )}}{b +a \cot \left (d x +c \right )}+\frac {\left (A \,a^{2} b +5 A \,b^{3}-3 B \,a^{3}-7 B a \,b^{2}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2} \left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(351\) |
default | \(\frac {-\frac {2 \left (\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-2 a b A +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 B}{b^{2} \sqrt {\cot \left (d x +c \right )}}-\frac {2 a^{2} \left (\frac {\left (\frac {1}{2} A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {1}{2} B \,a^{3}-\frac {1}{2} B a \,b^{2}\right ) \sqrt {\cot \left (d x +c \right )}}{b +a \cot \left (d x +c \right )}+\frac {\left (A \,a^{2} b +5 A \,b^{3}-3 B \,a^{3}-7 B a \,b^{2}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2} \left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(351\) |
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Leaf count of result is larger than twice the leaf count of optimal. 5952 vs. \(2 (396) = 792\).
Time = 35.18 (sec) , antiderivative size = 11930, normalized size of antiderivative = 27.30 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.92 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, B a^{5} - A a^{4} b + 7 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (2 \, B a^{2} b + 2 \, B b^{3} + \frac {3 \, B a^{3} - A a^{2} b + 2 \, B a b^{2}}{\tan \left (d x + c\right )}\right )}}{\frac {a^{2} b^{3} + b^{5}}{\sqrt {\tan \left (d x + c\right )}} + \frac {a^{3} b^{2} + a b^{4}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}}{4 \, d} \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]
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