Integrand size = 36, antiderivative size = 307 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) ((4+i) A+(1+6 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) ((4+i) A+(1+6 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {((3-5 i) A+(5+7 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) ((1+4 i) A-(6+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d} \]
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Time = 0.78 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3662, 3677, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) ((4+i) A+(1+6 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) ((4+i) A+(1+6 i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {-B+i A}{2 d \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+i a)}-\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (-B+i A)}{2 a d \sqrt {\cot (c+d x)}}+\frac {((3-5 i) A+(5+7 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{8 \sqrt {2} a d}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) ((1+4 i) A-(6+i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3610
Rule 3615
Rule 3662
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \int \frac {B+A \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (i a+a \cot (c+d x))} \, dx \\ & = \frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {\int \frac {-\frac {1}{2} a (3 A+7 i B)-\frac {5}{2} a (i A-B) \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx}{2 a^2} \\ & = -\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {\int \frac {-\frac {5}{2} a (i A-B)+\frac {1}{2} a (3 A+7 i B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx}{2 a^2} \\ & = -\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {\int \frac {\frac {1}{2} a (3 A+7 i B)+\frac {5}{2} a (i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} a (3 A+7 i B)-\frac {5}{2} a (i A-B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d} \\ & = -\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}-\frac {((3+5 i) A-(5-7 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a d}-\frac {((3-5 i) A+(5+7 i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a d} \\ & = -\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}-\frac {((3+5 i) A-(5-7 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 a d}-\frac {((3+5 i) A-(5-7 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 a d}+\frac {((3-5 i) A+(5+7 i) B) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a d}+\frac {((3-5 i) A+(5+7 i) B) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a d} \\ & = -\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {((3-5 i) A+(5+7 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}-\frac {((3-5 i) A+(5+7 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}-\frac {((3+5 i) A-(5-7 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {((3+5 i) A-(5-7 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d} \\ & = \frac {((3+5 i) A-(5-7 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}-\frac {((3+5 i) A-(5-7 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}-\frac {3 A+7 i B}{6 a d \cot ^{\frac {3}{2}}(c+d x)}-\frac {5 (i A-B)}{2 a d \sqrt {\cot (c+d x)}}+\frac {i A-B}{2 d \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))}+\frac {((3-5 i) A+(5+7 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}-\frac {((3-5 i) A+(5+7 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d} \\ \end{align*}
Time = 5.57 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.58 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-3 (-1)^{3/4} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (-i+\tan (c+d x))+6 \sqrt [4]{-1} (-2 i A+3 B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (-i+\tan (c+d x))+\sqrt {\tan (c+d x)} \left (-15 (A+i B)+4 (-3 i A+2 B) \tan (c+d x)-4 i B \tan ^2(c+d x)\right )\right )}{6 a d (-i+\tan (c+d x))} \]
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Time = 0.41 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (-\frac {i A}{4}-\frac {B}{4}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}+\frac {-2 i A +2 B}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 i B}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {i \left (-\frac {i \left (i A -B \right ) \sqrt {\cot \left (d x +c \right )}}{i+\cot \left (d x +c \right )}+\frac {4 \left (3 i B +2 A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2}}{a d}\) | \(161\) |
default | \(\frac {\frac {4 \left (-\frac {i A}{4}-\frac {B}{4}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}+\frac {-2 i A +2 B}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 i B}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {i \left (-\frac {i \left (i A -B \right ) \sqrt {\cot \left (d x +c \right )}}{i+\cot \left (d x +c \right )}+\frac {4 \left (3 i B +2 A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2}}{a d}\) | \(161\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (222) = 444\).
Time = 0.27 (sec) , antiderivative size = 794, normalized size of antiderivative = 2.59 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=-\frac {3 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a 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}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - 3 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (-i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a 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}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) + 6 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {-4 i \, A^{2} + 12 \, A B + 9 i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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}} \sqrt {\frac {-4 i \, A^{2} + 12 \, A B + 9 i \, B^{2}}{a^{2} d^{2}}} + 2 i \, A - 3 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - 6 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {-4 i \, A^{2} + 12 \, A B + 9 i \, B^{2}}{a^{2} d^{2}}} \log \left (\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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}} \sqrt {\frac {-4 i \, A^{2} + 12 \, A B + 9 i \, B^{2}}{a^{2} d^{2}}} - 2 i \, A + 3 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 2 \, {\left ({\left (27 \, A + 19 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (3 \, A + 19 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (27 \, A + 35 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 \, A - 3 i \, B\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}}}{24 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
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\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=- \frac {i \left (\int \frac {A}{\tan {\left (c + d x \right )} \cot ^{\frac {5}{2}}{\left (c + d x \right )} - i \cot ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \tan {\left (c + d x \right )}}{\tan {\left (c + d x \right )} \cot ^{\frac {5}{2}}{\left (c + d x \right )} - i \cot ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx\right )}{a} \]
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Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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