Integrand size = 36, antiderivative size = 148 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {8 \sqrt [4]{-1} a^3 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^3 (5 A-6 i B)}{15 d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 (5 A-9 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{15 d \cot ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.75 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3662, 3674, 3672, 3614, 214} \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {8 \sqrt [4]{-1} a^3 (B+i A) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 (5 A-9 i B) \left (a^3 \cot (c+d x)+i a^3\right )}{15 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {16 a^3 (5 A-6 i B)}{15 d \sqrt {\cot (c+d x)}}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]
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Rule 214
Rule 3614
Rule 3662
Rule 3672
Rule 3674
Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x))^3 (B+A \cot (c+d x))}{\cot ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B (i a+a \cot (c+d x))^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2}{5} \int \frac {(i a+a \cot (c+d x))^2 \left (\frac {1}{2} a (5 i A+9 B)+\frac {1}{2} a (5 A-i B) \cot (c+d x)\right )}{\cot ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B (i a+a \cot (c+d x))^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 (5 A-9 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4}{15} \int \frac {(i a+a \cot (c+d x)) \left (2 a^2 (5 i A+6 B)+a^2 (5 A-3 i B) \cot (c+d x)\right )}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {16 a^3 (5 A-6 i B)}{15 d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 (5 A-9 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4}{15} \int \frac {15 a^3 (i A+B)+15 a^3 (A-i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {16 a^3 (5 A-6 i B)}{15 d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 (5 A-9 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{15 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (120 a^6 (i A+B)^2\right ) \text {Subst}\left (\int \frac {1}{-15 a^3 (i A+B)+15 a^3 (A-i B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {8 \sqrt [4]{-1} a^3 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^3 (5 A-6 i B)}{15 d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 (5 A-9 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{15 d \cot ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Time = 5.94 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.74 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 i a^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-60 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)} \left (-45 i A-60 B+5 (A-3 i B) \tan (c+d x)+3 B \tan ^2(c+d x)\right )\right )}{15 d} \]
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Time = 0.42 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.64
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (4 i A +4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {-\frac {2 i A}{3}-2 B}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {8 i B -6 A}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 i B}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}\right )}{d}\) | \(243\) |
default | \(\frac {a^{3} \left (-\frac {\left (4 i A +4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {-\frac {2 i A}{3}-2 B}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {8 i B -6 A}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 i B}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}\right )}{d}\) | \(243\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (118) = 236\).
Time = 0.27 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.39 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 \, {\left (15 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{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^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - 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^{3}}\right ) - 15 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{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^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - 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^{3}}\right ) - 2 \, {\left ({\left (-25 i \, A - 39 \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, {\left (-10 i \, A - 9 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (25 i \, A + 33 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, {\left (5 i \, A + 6 \, B\right )} a^{3}\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 )}}{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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\[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=- i a^{3} \left (\int i A \sqrt {\cot {\left (c + d x \right )}}\, dx + \int \left (- 3 A \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\right )\, dx + \int A \tan ^{3}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\, dx + \int \left (- 3 B \tan ^{2}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\right )\, dx + \int B \tan ^{4}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\, dx + \int \left (- 3 i A \tan ^{2}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\right )\, dx + \int i B \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\, dx + \int \left (- 3 i B \tan ^{3}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\right )\, dx\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.37 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {15 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - 2 \, {\left (3 i \, B a^{3} - \frac {5 \, {\left (-i \, A - 3 \, B\right )} a^{3}}{\tan \left (d x + c\right )} + \frac {15 \, {\left (3 \, A - 4 i \, B\right )} a^{3}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}}}{15 \, d} \]
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\[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\cot \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int \sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]
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