Integrand size = 36, antiderivative size = 103 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {4 \sqrt [4]{-1} a^2 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (5 i A+3 B) \sqrt {\cot (c+d x)}}{3 d}-\frac {2 A \sqrt {\cot (c+d x)} \left (i a^2+a^2 \cot (c+d x)\right )}{3 d} \]
[Out]
Time = 0.52 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3662, 3675, 3673, 3614, 214} \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {4 \sqrt [4]{-1} a^2 (B+i A) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (3 B+5 i A) \sqrt {\cot (c+d x)}}{3 d}-\frac {2 A \sqrt {\cot (c+d x)} \left (a^2 \cot (c+d x)+i a^2\right )}{3 d} \]
[In]
[Out]
Rule 214
Rule 3614
Rule 3662
Rule 3673
Rule 3675
Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x))^2 (B+A \cot (c+d x))}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 A \sqrt {\cot (c+d x)} \left (i a^2+a^2 \cot (c+d x)\right )}{3 d}-\frac {2}{3} \int \frac {(i a+a \cot (c+d x)) \left (\frac {1}{2} a (A-3 i B)-\frac {1}{2} a (5 i A+3 B) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a^2 (5 i A+3 B) \sqrt {\cot (c+d x)}}{3 d}-\frac {2 A \sqrt {\cot (c+d x)} \left (i a^2+a^2 \cot (c+d x)\right )}{3 d}-\frac {2}{3} \int \frac {3 a^2 (i A+B)+3 a^2 (A-i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {2 a^2 (5 i A+3 B) \sqrt {\cot (c+d x)}}{3 d}-\frac {2 A \sqrt {\cot (c+d x)} \left (i a^2+a^2 \cot (c+d x)\right )}{3 d}-\frac {\left (12 a^4 (i A+B)^2\right ) \text {Subst}\left (\int \frac {1}{-3 a^2 (i A+B)+3 a^2 (A-i B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = -\frac {4 \sqrt [4]{-1} a^2 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (5 i A+3 B) \sqrt {\cot (c+d x)}}{3 d}-\frac {2 A \sqrt {\cot (c+d x)} \left (i a^2+a^2 \cot (c+d x)\right )}{3 d} \\ \end{align*}
Time = 3.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) \left (A+3 (2 i A+B) \tan (c+d x)-6 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{3 d} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (86 ) = 172\).
Time = 0.70 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.23
method | result | size |
derivativedivides | \(-\frac {a^{2} \left (\frac {2 A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+4 i A \sqrt {\cot \left (d x +c \right )}+2 B \sqrt {\cot \left (d x +c \right )}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(230\) |
default | \(-\frac {a^{2} \left (\frac {2 A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+4 i A \sqrt {\cot \left (d x +c \right )}+2 B \sqrt {\cot \left (d x +c \right )}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(230\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (83) = 166\).
Time = 0.25 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.83 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {3 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 3 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) + 2 \, {\left ({\left (7 i \, A + 3 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-5 i \, A - 3 \, B\right )} a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (83) = 166\).
Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.75 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {3 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} - \frac {12 \, {\left (-2 i \, A - B\right )} a^{2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {4 \, A a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{6 \, d} \]
[In]
[Out]
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
[In]
[Out]