Integrand size = 36, antiderivative size = 130 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {4 \sqrt [4]{-1} a^2 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i B \left (i a^2+a^2 \cot (c+d x)\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]
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Time = 0.42 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3662, 3674, 3672, 3610, 3614, 214} \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {4 \sqrt [4]{-1} a^2 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (B+i A)}{d \sqrt {\cot (c+d x)}}+\frac {2 i B \left (a^2 \cot (c+d x)+i a^2\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]
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Rule 214
Rule 3610
Rule 3614
Rule 3662
Rule 3672
Rule 3674
Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x))^2 (B+A \cot (c+d x))}{\cot ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 i B \left (i a^2+a^2 \cot (c+d x)\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2}{5} \int \frac {(i a+a \cot (c+d x)) \left (\frac {1}{2} a (5 i A+7 B)+\frac {1}{2} a (5 A-3 i B) \cot (c+d x)\right )}{\cot ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 i B \left (i a^2+a^2 \cot (c+d x)\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2}{5} \int \frac {5 a^2 (i A+B)+5 a^2 (A-i B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i B \left (i a^2+a^2 \cot (c+d x)\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2}{5} \int \frac {5 a^2 (A-i B)-5 a^2 (i A+B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i B \left (i a^2+a^2 \cot (c+d x)\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {\left (20 a^4 (A-i B)^2\right ) \text {Subst}\left (\int \frac {1}{-5 a^2 (A-i B)-5 a^2 (i A+B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {4 \sqrt [4]{-1} a^2 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (5 A-7 i B)}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i B \left (i a^2+a^2 \cot (c+d x)\right )}{5 d \cot ^{\frac {5}{2}}(c+d x)} \\ \end{align*}
Time = 4.87 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.75 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 a^2 \left (30 (i A+B)+\frac {30 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{\sqrt {\tan (c+d x)}}-5 (A-2 i B) \tan (c+d x)-3 B \tan ^2(c+d x)\right )}{15 d \sqrt {\cot (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (109 ) = 218\).
Time = 0.38 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.87
method | result | size |
derivativedivides | \(-\frac {a^{2} \left (-\frac {2 \left (2 i A +2 B \right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 \left (2 i B -A \right )}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 B}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(243\) |
default | \(-\frac {a^{2} \left (-\frac {2 \left (2 i A +2 B \right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 \left (2 i B -A \right )}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 B}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(243\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (104) = 208\).
Time = 0.26 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.88 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=-\frac {15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{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^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{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^{2}}\right ) - 15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{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^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{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^{2}}\right ) - 2 \, {\left ({\left (35 \, A - 43 i \, B\right )} a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (25 \, A - 11 i \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (35 \, A - 31 i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (25 \, A - 23 i \, B\right )} a^{2}\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}}}{15 \, {\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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\[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=- a^{2} \left (\int \left (- \frac {A}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \frac {A \tan ^{2}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {B \tan {\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \frac {B \tan ^{3}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {2 i A \tan {\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \left (- \frac {2 i B \tan ^{2}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx\right ) \]
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Time = 0.31 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.53 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=-\frac {4 \, {\left (3 \, B a^{2} + \frac {5 \, {\left (A - 2 i \, B\right )} a^{2}}{\tan \left (d x + c\right )} - \frac {30 \, {\left (i \, A + B\right )} a^{2}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 15 \, {\left (2 \, \sqrt {2} {\left (-\left (i - 1\right ) \, A - \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i - 1\right ) \, A - \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2}}{30 \, d} \]
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\[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]
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