Integrand size = 25, antiderivative size = 358 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {(i a-b)^{5/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d} \]
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Time = 1.90 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4326, 3646, 3730, 3697, 3696, 95, 209, 212} \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {(-b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d} \]
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Rule 95
Rule 209
Rule 212
Rule 3646
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))^{5/2}}{\tan ^{\frac {11}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {1}{9} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {19 a^2 b}{2}-\frac {9}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac {1}{2} b \left (8 a^2-9 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3}{4} a^2 \left (21 a^2-25 b^2\right )+\frac {63}{4} a b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac {57}{2} a^2 b^2 \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{63 a} \\ & = \frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{8} a^2 b \left (231 a^2-5 b^2\right )+\frac {315}{8} a^3 \left (a^2-3 b^2\right ) \tan (c+d x)+\frac {3}{2} a^2 b \left (21 a^2-25 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{315 a^2} \\ & = \frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{16} a^2 \left (315 a^4-483 a^2 b^2-10 b^4\right )-\frac {945}{16} a^3 b \left (3 a^2-b^2\right ) \tan (c+d x)-\frac {3}{8} a^2 b^2 \left (231 a^2-5 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{945 a^3} \\ & = -\frac {2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {\left (32 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {945}{32} a^4 b \left (3 a^2-b^2\right )-\frac {945}{32} a^5 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{945 a^4} \\ & = -\frac {2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {1}{2} \left ((i a-b)^3 \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 ((i a+b)^3 \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 (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {\left ((i a-b)^3 \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 ((i a+b)^3 \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 (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {\left ((i a-b)^3 \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 ((i a+b)^3 \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)^{5/2} \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)^{5/2} \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 (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d} \\ \end{align*}
Time = 5.70 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.92 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {\cot ^{\frac {9}{2}}(c+d x) \left (-315 \sqrt [4]{-1} a^2 (-a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {9}{2}}(c+d x)-315 \sqrt [4]{-1} a^2 (a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {9}{2}}(c+d x)+\frac {1}{4} \sec ^4(c+d x) \left (-987 a^4+1374 a^2 b^2+30 b^4+4 \left (280 a^4-483 a^2 b^2-10 b^4\right ) \cos (2 (c+d x))+\left (-413 a^4+558 a^2 b^2+10 b^4\right ) \cos (4 (c+d x))+272 a^3 b \sin (2 (c+d x))-10 a b^3 \sin (2 (c+d x))-326 a^3 b \sin (4 (c+d x))+5 a b^3 \sin (4 (c+d x))\right ) \sqrt {a+b \tan (c+d x)}\right )}{315 a^2 d} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 19.34 (sec) , antiderivative size = 9190, normalized size of antiderivative = 25.67
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Leaf count of result is larger than twice the leaf count of optimal. 5248 vs. \(2 (296) = 592\).
Time = 0.87 (sec) , antiderivative size = 5248, normalized size of antiderivative = 14.66 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{11/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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