Integrand size = 36, antiderivative size = 237 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d} \]
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Time = 0.43 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3662, 3677, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {(-B+i A) \sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3662
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \int \frac {B+A \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))} \, dx \\ & = \frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+\frac {\int \frac {\frac {1}{2} a (A-3 i B)-\frac {1}{2} a (i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 a^2} \\ & = \frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} a (A-3 i B)+\frac {1}{2} a (i A-B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d} \\ & = \frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+-\frac {\left (\left (\frac {1}{4}+\frac {i}{4}\right ) (A-(2+i) B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d}+-\frac {\left (\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d} \\ & = \frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+\frac {\left (\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {\left (\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+-\frac {\left (\left (\frac {1}{8}-\frac {i}{8}\right ) (A+(2-i) B)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d}+-\frac {\left (\left (\frac {1}{8}-\frac {i}{8}\right ) (A+(2-i) B)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d} \\ & = \frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}+-\frac {\left (\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {\left (\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-i) B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d} \\ & = \frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (A+(2-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 ) (A+(2-i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (A-(2+i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d} \\ \end{align*}
Time = 3.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.57 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left ((A+i B) \sqrt {\tan (c+d x)}+\sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (-i+\tan (c+d x))-2 \sqrt [4]{-1} B \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (-i+\tan (c+d x))\right )}{2 a d (-i+\tan (c+d x))} \]
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Time = 0.43 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (\frac {i A}{4}+\frac {B}{4}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}+\frac {i \left (-\frac {i \left (i A -B \right ) \sqrt {\cot \left (d x +c \right )}}{i+\cot \left (d x +c \right )}+\frac {4 i B \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2}}{a d}\) | \(127\) |
default | \(\frac {\frac {4 \left (\frac {i A}{4}+\frac {B}{4}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}+\frac {i \left (-\frac {i \left (i A -B \right ) \sqrt {\cot \left (d x +c \right )}}{i+\cot \left (d x +c \right )}+\frac {4 i B \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2}}{a d}\) | \(127\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (178) = 356\).
Time = 0.27 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.41 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {{\left (a d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \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 ) - a d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \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 ) - 2 \, a d \sqrt {\frac {i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \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 {i \, B^{2}}{a^{2} d^{2}}} + B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 2 \, a d \sqrt {\frac {i \, B^{2}}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \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 {i \, B^{2}}{a^{2} d^{2}}} - B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 2 \, {\left ({\left (A + i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - A - 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}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a d} \]
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\[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=- \frac {i \left (\int \frac {A}{\tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - i \sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \frac {B \tan {\left (c + d x \right )}}{\tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - i \sqrt {\cot {\left (c + d x \right )}}}\, dx\right )}{a} \]
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Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )} \sqrt {\cot \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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