Integrand size = 36, antiderivative size = 284 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((2+i) A+(7-2 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((1+2 i) A+(2-7 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d} \]
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Time = 0.83 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((2+i) A+(7-2 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac {((1-3 i) A-(9-5 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((1+2 i) A+(2-7 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {(-B+i A) \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2} \]
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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))^2} \, dx \\ & = \frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \frac {\frac {1}{2} a (A-7 i B)-\frac {3}{2} a (i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{4 a^2} \\ & = \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \frac {-\frac {3}{2} a^2 (i A+3 B)-\frac {1}{2} a^2 (A+5 i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{8 a^4} \\ & = \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\text {Subst}\left (\int \frac {\frac {3}{2} a^2 (i A+3 B)+\frac {1}{2} a^2 (A+5 i B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a^4 d} \\ & = \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+2 i) A+(2-7 i) B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d} \\ & = \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 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 )}{32 \sqrt {2} a^2 d}+\frac {((1-3 i) A-(9-5 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 )}{32 \sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d} \\ & = \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}-\frac {((1-3 i) A-(9-5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((1+3 i) A+(9+5 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d} \\ & = -\frac {((1+3 i) A+(9+5 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}-\frac {((1-3 i) A-(9-5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d} \\ \end{align*}
Time = 4.38 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.69 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-2 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt [4]{-1} (A-7 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-\sqrt {\tan (c+d x)} (A+5 i B+(3 i A-7 B) \tan (c+d x))\right )}{8 a^2 d (-i+\tan (c+d x))^2} \]
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Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.51
method | result | size |
derivativedivides | \(\frac {-\frac {\left (-\frac {5 i B}{2}-\frac {A}{2}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {7 B}{2}-\frac {3 i A}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}-\frac {\left (-7 i B +A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \left (\sqrt {2}+i \sqrt {2}\right )}-\frac {i \left (i A +B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}}{a^{2} d}\) | \(144\) |
default | \(\frac {-\frac {\left (-\frac {5 i B}{2}-\frac {A}{2}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {7 B}{2}-\frac {3 i A}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}-\frac {\left (-7 i B +A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \left (\sqrt {2}+i \sqrt {2}\right )}-\frac {i \left (i A +B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}}{a^{2} d}\) | \(144\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (209) = 418\).
Time = 0.26 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.35 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {{\left (2 \, a^{2} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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^{4} 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^{2} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {2 \, {\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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^{4} 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^{2} d \sqrt {\frac {i \, A^{2} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} + A - 7 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - a^{2} d \sqrt {\frac {i \, A^{2} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} - A + 7 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + 2 \, {\left (2 \, {\left (i \, A - 3 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (3 i \, A - 7 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A - 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 (-4 i \, d x - 4 i \, c\right )}}{32 \, a^{2} d} \]
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\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {A}{\tan ^{2}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \tan {\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx}{a^{2}} \]
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Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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