Integrand size = 34, antiderivative size = 103 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {2 \sqrt [4]{-1} a (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a (A-i B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a (i A+B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d} \]
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Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3662, 3673, 3609, 3614, 214} \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {2 \sqrt [4]{-1} a (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a (B+i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 a (A-i B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d} \]
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Rule 214
Rule 3609
Rule 3614
Rule 3662
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \int \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x)) (B+A \cot (c+d x)) \, dx \\ & = -\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \cot ^{\frac {3}{2}}(c+d x) (-a (A-i B)+a (i A+B) \cot (c+d x)) \, dx \\ & = -\frac {2 a (i A+B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \sqrt {\cot (c+d x)} (-a (i A+B)-a (A-i B) \cot (c+d x)) \, dx \\ & = \frac {2 a (A-i B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a (i A+B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \frac {a (A-i B)-a (i A+B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {2 a (A-i B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a (i A+B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {\left (2 a^2 (A-i B)^2\right ) \text {Subst}\left (\int \frac {1}{-a (A-i B)-a (i A+B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {2 \sqrt [4]{-1} a (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a (A-i B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a (i A+B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a A \cot ^{\frac {5}{2}}(c+d x)}{5 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.53 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {2 a \cot ^{\frac {3}{2}}(c+d x) \left (3 A \cot (c+d x)+5 (i A+B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (c+d x)\right )\right )}{15 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (84 ) = 168\).
Time = 0.61 (sec) , antiderivative size = 551, normalized size of antiderivative = 5.35
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right ) \left (30 i A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 i B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 i B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+15 i B \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+15 i A \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+40 i A \tan \left (d x +c \right )-30 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}-30 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}-15 A \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+15 B \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 i A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}-120 A \tan \left (d x +c \right )^{2}+120 i B \tan \left (d x +c \right )^{2}+40 B \tan \left (d x +c \right )+24 A \right )}{60 d}\) | \(551\) |
default | \(-\frac {a \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right ) \left (30 i A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 i B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 i B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+15 i B \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+15 i A \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+40 i A \tan \left (d x +c \right )-30 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}-30 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}-15 A \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+15 B \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}+30 i A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {5}{2}}-120 A \tan \left (d x +c \right )^{2}+120 i B \tan \left (d x +c \right )^{2}+40 B \tan \left (d x +c \right )+24 A \right )}{60 d}\) | \(551\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (81) = 162\).
Time = 0.27 (sec) , antiderivative size = 434, normalized size of antiderivative = 4.21 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) - 15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) - 4 \, {\left ({\left (23 \, A - 20 i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, {\left (4 \, A - 5 i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (13 \, A - 10 i \, B\right )} a\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{30 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (81) = 162\).
Time = 0.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.84 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {15 \, {\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a + \frac {120 \, {\left (A - i \, B\right )} a}{\sqrt {\tan \left (d x + c\right )}} + \frac {40 \, {\left (-i \, A - B\right )} a}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {24 \, A a}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{60 \, d} \]
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\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
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