Integrand size = 36, antiderivative size = 171 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {8 \sqrt [4]{-1} a^3 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {8 a^3 (i A+B) \sqrt {\cot (c+d x)}}{d}+\frac {8 a^3 (23 A-21 i B) \cot ^{\frac {3}{2}}(c+d x)}{105 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2 (11 i A+7 B) \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d} \]
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Time = 0.60 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3662, 3675, 3673, 3609, 3614, 214} \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {8 \sqrt [4]{-1} a^3 (B+i A) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {8 a^3 (23 A-21 i B) \cot ^{\frac {3}{2}}(c+d x)}{105 d}-\frac {2 (7 B+11 i A) \cot ^{\frac {3}{2}}(c+d x) \left (a^3 \cot (c+d x)+i a^3\right )}{35 d}+\frac {8 a^3 (B+i A) \sqrt {\cot (c+d x)}}{d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}{7 d} \]
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Rule 214
Rule 3609
Rule 3614
Rule 3662
Rule 3673
Rule 3675
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^3 (B+A \cot (c+d x)) \, dx \\ & = -\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2}{7} \int \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2 \left (\frac {1}{2} a (3 A-7 i B)-\frac {1}{2} a (11 i A+7 B) \cot (c+d x)\right ) \, dx \\ & = -\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2 (11 i A+7 B) \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac {4}{35} \int \sqrt {\cot (c+d x)} (i a+a \cot (c+d x)) \left (-2 a^2 (6 i A+7 B)-a^2 (23 A-21 i B) \cot (c+d x)\right ) \, dx \\ & = \frac {8 a^3 (23 A-21 i B) \cot ^{\frac {3}{2}}(c+d x)}{105 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2 (11 i A+7 B) \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac {4}{35} \int \sqrt {\cot (c+d x)} \left (35 a^3 (A-i B)-35 a^3 (i A+B) \cot (c+d x)\right ) \, dx \\ & = \frac {8 a^3 (i A+B) \sqrt {\cot (c+d x)}}{d}+\frac {8 a^3 (23 A-21 i B) \cot ^{\frac {3}{2}}(c+d x)}{105 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2 (11 i A+7 B) \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac {4}{35} \int \frac {35 a^3 (i A+B)+35 a^3 (A-i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {8 a^3 (i A+B) \sqrt {\cot (c+d x)}}{d}+\frac {8 a^3 (23 A-21 i B) \cot ^{\frac {3}{2}}(c+d x)}{105 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2 (11 i A+7 B) \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d}+\frac {\left (280 a^6 (i A+B)^2\right ) \text {Subst}\left (\int \frac {1}{-35 a^3 (i A+B)+35 a^3 (A-i B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {8 \sqrt [4]{-1} a^3 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {8 a^3 (i A+B) \sqrt {\cot (c+d x)}}{d}+\frac {8 a^3 (23 A-21 i B) \cot ^{\frac {3}{2}}(c+d x)}{105 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2}{7 d}-\frac {2 (11 i A+7 B) \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.93 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.66 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 a^3 \cot ^{\frac {3}{2}}(c+d x) \left (35 i B+21 i A \cot (c+d x)+63 B \cot (c+d x)+45 A \cot ^2(c+d x)-60 A \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},i \tan (c+d x)\right )-84 B \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},i \tan (c+d x)\right )\right )}{105 d} \]
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Time = 1.31 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.61
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {2 A \cot \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {6 i A \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 B \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-2 i B \cot \left (d x +c \right )^{\frac {3}{2}}+\frac {8 A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+8 i A \sqrt {\cot \left (d x +c \right )}+8 B \sqrt {\cot \left (d x +c \right )}-\frac {\left (4 i A +4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(275\) |
default | \(\frac {a^{3} \left (-\frac {2 A \cot \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {6 i A \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 B \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-2 i B \cot \left (d x +c \right )^{\frac {3}{2}}+\frac {8 A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+8 i A \sqrt {\cot \left (d x +c \right )}+8 B \sqrt {\cot \left (d x +c \right )}-\frac {\left (4 i A +4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(275\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (139) = 278\).
Time = 0.28 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.97 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 \, {\left (105 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 105 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 2 \, {\left ({\left (-319 i \, A - 273 \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, {\left (323 i \, A + 336 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-551 i \, A - 567 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, {\left (41 i \, A + 42 \, B\right )} a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )}}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.27 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {105 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} + \frac {840 \, {\left (i \, A + B\right )} a^{3}}{\sqrt {\tan \left (d x + c\right )}} + \frac {70 \, {\left (4 \, A - 3 i \, B\right )} a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {42 \, {\left (-3 i \, A - B\right )} a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {30 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {7}{2}}}}{105 \, d} \]
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\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]
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