Integrand size = 36, antiderivative size = 146 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {8 \sqrt [4]{-1} a^3 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {16 a^3 (6 A-5 i B) \sqrt {\cot (c+d x)}}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}{5 d}-\frac {2 (9 i A+5 B) \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{15 d} \]
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Time = 0.54 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3662, 3675, 3673, 3614, 214} \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {8 \sqrt [4]{-1} a^3 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {16 a^3 (6 A-5 i B) \sqrt {\cot (c+d x)}}{15 d}-\frac {2 (5 B+9 i A) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}{5 d} \]
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Rule 214
Rule 3614
Rule 3662
Rule 3673
Rule 3675
Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x))^3 (B+A \cot (c+d x))}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a A \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}{5 d}-\frac {2}{5} \int \frac {(i a+a \cot (c+d x))^2 \left (\frac {1}{2} a (A-5 i B)-\frac {1}{2} a (9 i A+5 B) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a A \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}{5 d}-\frac {2 (9 i A+5 B) \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{15 d}+\frac {4}{15} \int \frac {(i a+a \cot (c+d x)) \left (-a^2 (3 i A+5 B)-2 a^2 (6 A-5 i B) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {16 a^3 (6 A-5 i B) \sqrt {\cot (c+d x)}}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}{5 d}-\frac {2 (9 i A+5 B) \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{15 d}+\frac {4}{15} \int \frac {15 a^3 (A-i B)-15 a^3 (i A+B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {16 a^3 (6 A-5 i B) \sqrt {\cot (c+d x)}}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}{5 d}-\frac {2 (9 i A+5 B) \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{15 d}+\frac {\left (120 a^6 (A-i B)^2\right ) \text {Subst}\left (\int \frac {1}{-15 a^3 (A-i B)-15 a^3 (i A+B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {8 \sqrt [4]{-1} a^3 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {16 a^3 (6 A-5 i B) \sqrt {\cot (c+d x)}}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}{5 d}-\frac {2 (9 i A+5 B) \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{15 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.77 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 a^3 \sqrt {\cot (c+d x)} \left (-15 i B-5 i A \cot (c+d x)-15 B \cot (c+d x)-9 A \cot ^2(c+d x)+12 A \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},i \tan (c+d x)\right )+20 B \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (c+d x)\right )\right )}{15 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (123 ) = 246\).
Time = 1.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.73
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {2 A \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-2 i A \cot \left (d x +c \right )^{\frac {3}{2}}-\frac {2 B \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-6 i B \sqrt {\cot \left (d x +c \right )}+8 A \sqrt {\cot \left (d x +c \right )}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i A -4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(252\) |
default | \(\frac {a^{3} \left (-\frac {2 A \cot \left (d x +c \right )^{\frac {5}{2}}}{5}-2 i A \cot \left (d x +c \right )^{\frac {3}{2}}-\frac {2 B \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-6 i B \sqrt {\cot \left (d x +c \right )}+8 A \sqrt {\cot \left (d x +c \right )}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i A -4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(252\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (118) = 236\).
Time = 0.27 (sec) , antiderivative size = 449, normalized size of antiderivative = 3.08 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 2 \, {\left ({\left (39 \, A - 25 i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 3 \, {\left (19 \, A - 15 i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, {\left (6 \, A - 5 i \, B\right )} a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )}}{15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.37 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {15 \, {\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} + \frac {30 \, {\left (4 \, A - 3 i \, B\right )} a^{3}}{\sqrt {\tan \left (d x + c\right )}} + \frac {10 \, {\left (-3 i \, A - B\right )} a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {6 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{15 \, d} \]
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\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]
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