Integrand size = 25, antiderivative size = 305 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {i \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{5/2} d}-\frac {i \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{5/2} d}-\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
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Time = 1.25 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4326, 3650, 3730, 3697, 3696, 95, 209, 212} \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 d \left (a^2+b^2\right )^2 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {i \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac {i \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}} \]
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Rule 95
Rule 209
Rule 212
Rule 3650
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 {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx \\ & = -\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {2 b+\frac {1}{2} a \tan (c+d x)+2 b \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx}{a} \\ & = -\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{4} b \left (9 a^2+8 b^2\right )+\frac {3}{4} a^3 \tan (c+d x)+\frac {1}{2} b \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2+b^2\right )} \\ & = -\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3 a^4 b}{4}+\frac {3}{8} a^3 \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2} \\ & = -\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left (i (a-i b)^2 \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}{2 \left (a^2+b^2\right )^2}+\frac {\left (i (a+i b)^2 \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}{2 \left (a^2+b^2\right )^2} \\ & = -\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left (i (a-i b)^2 \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 \left (a^2+b^2\right )^2 d}+\frac {\left (i (a+i b)^2 \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 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left (i (a-i b)^2 \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 )}{\left (a^2+b^2\right )^2 d}+\frac {\left (i (a+i b)^2 \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 )}{\left (a^2+b^2\right )^2 d} \\ & = \frac {i \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)}}{(i a-b)^{5/2} d}-\frac {i \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)}}{(i a+b)^{5/2} d}-\frac {2 b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}} \\ \end{align*}
Time = 4.21 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (-\frac {3 \sqrt [4]{-1} a \left (\frac {(a+i b)^2 \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}-\frac {(a-i b)^2 \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right ) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right )^2}+\frac {6}{(a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2+4 b^2\right ) \tan (c+d x)}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^4+17 a^2 b^2+8 b^4\right ) \tan (c+d x)}{a^2 \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}\right )}{3 a d} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 18.09 (sec) , antiderivative size = 18151, normalized size of antiderivative = 59.51
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Leaf count of result is larger than twice the leaf count of optimal. 11585 vs. \(2 (253) = 506\).
Time = 2.37 (sec) , antiderivative size = 11585, normalized size of antiderivative = 37.98 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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