Integrand size = 33, antiderivative size = 534 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
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Time = 1.54 (sec) , antiderivative size = 534, 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} \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{4 b^2 d \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\sqrt {a} \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{5/2} d \left (a^2+b^2\right )^3} \]
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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)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^3} \, dx \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {\int \frac {\frac {1}{2} \left (-a A b-3 a^2 B-4 b^2 B\right )-2 b (A b-a B) \cot (c+d x)+\frac {3}{2} a (A b-a B) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )} \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (a^3 A b+9 a A b^3+3 a^4 B+3 a^2 b^2 B+8 b^4 B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \cot (c+d x)+\frac {1}{4} a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \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-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\int \frac {-2 b^2 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )-2 b^2 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 b^2 \left (a^2+b^2\right )^3}+\frac {\left (a \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{8 b^2 \left (a^2+b^2\right )^3} \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {2 b^2 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )+2 b^2 \left (3 a^2 A b-A b^3-a^3 B+3 a 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 )^3 d}+\frac {\left (a \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{8 b^2 \left (a^2+b^2\right )^3 d} \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\left (a \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 )^3 d} \\ & = -\frac {\sqrt {a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 )^3 d} \\ & = -\frac {\sqrt {a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 )^3 d} \\ & = -\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d} \\ \end{align*}
Time = 6.53 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \left (\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{4 \left (a^2+b^2\right )^3}-\frac {3 \sqrt {a} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 b^{5/2} \left (a^2+b^2\right )}+\frac {\sqrt {a} \left (a^2 A b+3 A b^3-2 a^3 B-4 a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 b^{5/2} \left (a^2+b^2\right )^2}+\frac {\sqrt {a} \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \left (\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{8 \left (a^2+b^2\right )^3}-\frac {a^2 (A b-a B) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {3 a (A b-a B) \sqrt {\tan (c+d x)}}{8 b^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {a \left (a^2 A b+3 A b^3-2 a^3 B-4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{2 b^2 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{d} \]
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Time = 0.50 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {a \left (A \,a^{4} b -6 A \,a^{2} b^{3}-7 A \,b^{5}+3 B \,a^{5}+14 B \,a^{3} b^{2}+11 B a \,b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}}{8 b^{2}}-\frac {\left (A \,a^{4} b +10 A \,a^{2} b^{3}+9 A \,b^{5}-5 B \,a^{5}-18 B \,a^{3} b^{2}-13 B a \,b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 b}}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (A \,a^{4} b +18 A \,a^{2} b^{3}-15 A \,b^{5}+3 B \,a^{5}+6 B \,a^{3} b^{2}+35 B a \,b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(451\) |
default | \(\frac {-\frac {2 a \left (\frac {\frac {a \left (A \,a^{4} b -6 A \,a^{2} b^{3}-7 A \,b^{5}+3 B \,a^{5}+14 B \,a^{3} b^{2}+11 B a \,b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}}{8 b^{2}}-\frac {\left (A \,a^{4} b +10 A \,a^{2} b^{3}+9 A \,b^{5}-5 B \,a^{5}-18 B \,a^{3} b^{2}-13 B a \,b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 b}}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (A \,a^{4} b +18 A \,a^{2} b^{3}-15 A \,b^{5}+3 B \,a^{5}+6 B \,a^{3} b^{2}+35 B a \,b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(451\) |
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Leaf count of result is larger than twice the leaf count of optimal. 8923 vs. \(2 (482) = 964\).
Time = 106.83 (sec) , antiderivative size = 17872, normalized size of antiderivative = 33.47 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.04 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, B a^{6} + A a^{5} b + 6 \, B a^{4} b^{2} + 18 \, A a^{3} b^{3} + 35 \, B a^{2} b^{4} - 15 \, A a b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\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^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\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^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\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^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {\frac {5 \, B a^{4} b - A a^{3} b^{2} + 13 \, B a^{2} b^{3} - 9 \, A a b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {3 \, B a^{5} + A a^{4} b + 11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8} + \frac {2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )}}{\tan \left (d x + c\right )} + \frac {a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]
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