Integrand size = 35, antiderivative size = 383 \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {(i a-b)^{3/2} (A+i B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{8 b^{3/2} d}+\frac {(i a+b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}} \]
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Time = 3.14 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {4326, 3688, 3728, 3736, 6857, 65, 223, 212, 95, 211, 214} \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (a^2 (-B)+6 a A b-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {\left (a^3 (-B)+6 a^2 A b-24 a b^2 B-16 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b^{3/2} d}+\frac {(-b+i a)^{3/2} (A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(b+i a)^{3/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}} \]
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Rule 65
Rule 95
Rule 211
Rule 212
Rule 214
Rule 223
Rule 3688
Rule 3728
Rule 3736
Rule 4326
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx \\ & = \frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{3/2} \left (-\frac {a B}{2}-3 b B \tan (c+d x)+\frac {1}{2} (6 A b-a B) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{3 b} \\ & = \frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (-\frac {3}{4} a (2 A b+a B)-6 b (A b+a B) \tan (c+d x)+\frac {3}{4} \left (6 a A b-a^2 B-8 b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{6 b} \\ & = \frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{8} a \left (10 a A b+a^2 B-8 b^2 B\right )-6 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {3}{8} \left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{6 b} \\ & = \frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {-\frac {3}{8} a \left (10 a A b+a^2 B-8 b^2 B\right )-6 b \left (2 a A b+a^2 B-b^2 B\right ) x+\frac {3}{8} \left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{6 b d} \\ & = \frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {3 \left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right )}{8 \sqrt {x} \sqrt {a+b x}}-\frac {6 \left (b \left (a^2 A-A b^2-2 a b B\right )+b \left (2 a A b+a^2 B-b^2 B\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{6 b d} \\ & = \frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {b \left (a^2 A-A b^2-2 a b B\right )+b \left (2 a A b+a^2 B-b^2 B\right ) x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {\left (\left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 b d} \\ & = \frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {i b \left (a^2 A-A b^2-2 a b B\right )-b \left (2 a A b+a^2 B-b^2 B\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {i b \left (a^2 A-A b^2-2 a b B\right )+b \left (2 a A b+a^2 B-b^2 B\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}+\frac {\left (\left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{8 b d} \\ & = \frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}-\frac {\left ((a+i b)^2 (i A-B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {\left ((a-i b)^2 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (\left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b d} \\ & = \frac {\left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{8 b^{3/2} d}+\frac {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}}-\frac {\left ((a+i b)^2 (i A-B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{i-(a+i 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 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{i-(-a+i 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 {\left (6 a^2 A b-16 A b^3-a^3 B-24 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{8 b^{3/2} 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 {\left (6 a A b-a^2 B-8 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{8 b d \sqrt {\cot (c+d x)}}+\frac {(6 A b-a B) (a+b \tan (c+d x))^{3/2}}{12 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{5/2}}{3 b d \sqrt {\cot (c+d x)}} \\ \end{align*}
Time = 6.17 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (24 \sqrt [4]{-1} (-a+i b)^{3/2} b (i A+B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+24 (-1)^{3/4} (a+i b)^{3/2} b (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-3 \left (-6 a A b+a^2 B+8 b^2 B\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}+2 (6 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}-\frac {3 \sqrt {a} \left (-6 a^2 A b+16 A b^3+a^3 B+24 a b^2 B\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}\right )}{24 b d} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.64 (sec) , antiderivative size = 2403527, normalized size of antiderivative = 6275.53
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 12332 vs. \(2 (312) = 624\).
Time = 5.51 (sec) , antiderivative size = 24697, normalized size of antiderivative = 64.48 \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]
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