Integrand size = 35, antiderivative size = 351 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {(i a-b)^{3/2} (A+i B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d} \]
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Time = 1.83 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4326, 3686, 3730, 3697, 3696, 95, 209, 212} \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {2 \left (35 a^2 A-42 a b B-3 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}+\frac {2 \left (105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}-\frac {(-b+i a)^{3/2} (A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (7 a B+8 A b) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d} \]
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Rule 95
Rule 209
Rule 212
Rule 3686
Rule 3696
Rule 3697
Rule 3730
Rule 4326
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx \\ & = -\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {1}{7} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{2} a (8 A b+7 a B)-\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (6 a A-7 b B) \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{4} a \left (35 a^2 A-3 A b^2-42 a b B\right )+\frac {35}{4} a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+a b (8 A b+7 a B) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{35 a} \\ & = \frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right )+\frac {105}{8} a^2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac {1}{4} a b \left (35 a^2 A-3 A b^2-42 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^2} \\ & = \frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {105}{16} a^3 \left (a^2 A-A b^2-2 a b B\right )-\frac {105}{16} a^3 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^3} \\ & = \frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {1}{2} \left ((a-i b)^2 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left ((a+i b)^2 (A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left ((a-i b)^2 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left ((a+i b)^2 (A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}+\frac {\left ((a-i b)^2 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\left ((a+i b)^2 (A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \\ & = -\frac {(i a-b)^{3/2} (A+i B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{105 a^2 d}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (8 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{35 d}-\frac {2 a A \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d} \\ \end{align*}
Time = 5.53 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.99 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {\cot ^{\frac {7}{2}}(c+d x) \left (35 a^3 b B \sqrt {a+b \tan (c+d x)}+5 a^3 (6 a A-7 b B) \sqrt {a+b \tan (c+d x)}+a \tan (c+d x) \left (105 (-1)^{3/4} a^2 \left ((-a+i b)^{3/2} (A-i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+(a+i b)^{3/2} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right ) \tan ^{\frac {5}{2}}(c+d x)+6 a^2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}-2 a \left (35 a^2 A-3 A b^2-42 a b B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}-2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}\right )\right )}{105 a^3 d} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.45 (sec) , antiderivative size = 2403427, normalized size of antiderivative = 6847.37
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 12268 vs. \(2 (291) = 582\).
Time = 2.42 (sec) , antiderivative size = 12268, normalized size of antiderivative = 34.95 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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