Integrand size = 38, antiderivative size = 297 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {(4+4 i) a^{5/2} (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {8 a^2 (197 A-195 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {8 a^2 (59 i A+60 B) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{315 d}+\frac {2 a^2 (46 A-45 i B) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (4 i A+3 B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \] Output:
(4+4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan( d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d-8/315*a^2*(197*A-195*I* B)*cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d+8/315*a^2*(59*I*A+60*B)*cot (d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d+2/105*a^2*(46*A-45*I*B)*cot(d*x+c )^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/21*a^2*(4*I*A+3*B)*cot(d*x+c)^(7/2)*( a+I*a*tan(d*x+c))^(1/2)/d-2/9*a*A*cot(d*x+c)^(9/2)*(a+I*a*tan(d*x+c))^(3/2 )/d
Time = 12.35 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.19 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {\left (4 \sqrt {2} (A-i B) e^{-3 i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {\frac {i \left (1+e^{2 i (c+d x)}\right )}{-1+e^{2 i (c+d x)}}} \text {arctanh}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )+\frac {\sqrt {\cot (c+d x)} \csc ^4(c+d x) \sqrt {\sec (c+d x)} (\cos (2 c)-i \sin (2 c)) (-2331 A+2205 i B+12 (251 A-260 i B) \cos (2 (c+d x))+(-961 A+915 i B) \cos (4 (c+d x))+282 i A \sin (2 (c+d x))+390 B \sin (2 (c+d x))-331 i A \sin (4 (c+d x))-285 B \sin (4 (c+d x)))}{1260 (\cos (d x)+i \sin (d x))^2}\right ) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{d \sec ^{\frac {7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \] Input:
Integrate[Cot[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
Output:
(((4*Sqrt[2]*(A - I*B)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x)) /(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2 *I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/ E^((3*I)*(c + d*x)) + (Sqrt[Cot[c + d*x]]*Csc[c + d*x]^4*Sqrt[Sec[c + d*x] ]*(Cos[2*c] - I*Sin[2*c])*(-2331*A + (2205*I)*B + 12*(251*A - (260*I)*B)*C os[2*(c + d*x)] + (-961*A + (915*I)*B)*Cos[4*(c + d*x)] + (282*I)*A*Sin[2* (c + d*x)] + 390*B*Sin[2*(c + d*x)] - (331*I)*A*Sin[4*(c + d*x)] - 285*B*S in[4*(c + d*x)]))/(1260*(Cos[d*x] + I*Sin[d*x])^2))*(a + I*a*Tan[c + d*x]) ^(5/2)*(A + B*Tan[c + d*x]))/(d*Sec[c + d*x]^(7/2)*(A*Cos[c + d*x] + B*Sin [c + d*x]))
Time = 2.09 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.08, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 4729, 3042, 4076, 27, 3042, 4076, 27, 3042, 4081, 27, 3042, 4081, 27, 3042, 4081, 27, 3042, 4027, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{11/2} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{5/2} (A+B \tan (c+d x))}{\tan (c+d x)^{11/2}}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2}{9} \int \frac {3 (i \tan (c+d x) a+a)^{3/2} (a (4 i A+3 B)-a (2 A-3 i B) \tan (c+d x))}{2 \tan ^{\frac {9}{2}}(c+d x)}dx-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \int \frac {(i \tan (c+d x) a+a)^{3/2} (a (4 i A+3 B)-a (2 A-3 i B) \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)}dx-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \int \frac {(i \tan (c+d x) a+a)^{3/2} (a (4 i A+3 B)-a (2 A-3 i B) \tan (c+d x))}{\tan (c+d x)^{9/2}}dx-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {2}{7} \int -\frac {\sqrt {i \tan (c+d x) a+a} \left ((46 A-45 i B) a^2+(38 i A+39 B) \tan (c+d x) a^2\right )}{2 \tan ^{\frac {7}{2}}(c+d x)}dx-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-\frac {1}{7} \int \frac {\sqrt {i \tan (c+d x) a+a} \left ((46 A-45 i B) a^2+(38 i A+39 B) \tan (c+d x) a^2\right )}{\tan ^{\frac {7}{2}}(c+d x)}dx-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (-\frac {1}{7} \int \frac {\sqrt {i \tan (c+d x) a+a} \left ((46 A-45 i B) a^2+(38 i A+39 B) \tan (c+d x) a^2\right )}{\tan (c+d x)^{7/2}}dx-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 \int \frac {2 \sqrt {i \tan (c+d x) a+a} \left (a^3 (59 i A+60 B)-a^3 (46 A-45 i B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (a^3 (59 i A+60 B)-a^3 (46 A-45 i B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (a^3 (59 i A+60 B)-a^3 (46 A-45 i B) \tan (c+d x)\right )}{\tan (c+d x)^{5/2}}dx}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (\frac {2 \int -\frac {\sqrt {i \tan (c+d x) a+a} \left ((197 A-195 i B) a^4+2 (59 i A+60 B) \tan (c+d x) a^4\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left ((197 A-195 i B) a^4+2 (59 i A+60 B) \tan (c+d x) a^4\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left ((197 A-195 i B) a^4+2 (59 i A+60 B) \tan (c+d x) a^4\right )}{\tan (c+d x)^{3/2}}dx}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {\frac {2 \int \frac {315 a^5 (i A+B) \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {2 a^4 (197 A-195 i B) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {315 a^4 (B+i A) \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 a^4 (197 A-195 i B) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {315 a^4 (B+i A) \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 a^4 (197 A-195 i B) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {-\frac {630 i a^6 (B+i A) \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {2 a^4 (197 A-195 i B) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 (46 A-45 i B) \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (-\frac {2 a^3 (60 B+59 i A) \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\frac {(315-315 i) a^{9/2} (B+i A) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a^4 (197 A-195 i B) \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}\right )}{5 a}\right )-\frac {2 a^2 (3 B+4 i A) \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
Input:
Int[Cot[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x]
Output:
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*a*A*(a + I*a*Tan[c + d*x])^(3/2 ))/(9*d*Tan[c + d*x]^(9/2)) + ((-2*a^2*((4*I)*A + 3*B)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) + ((2*a^2*(46*A - (45*I)*B)*Sqrt[a + I*a *Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (4*((-2*a^3*((59*I)*A + 60*B)*S qrt[a + I*a*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (((315 - 315*I)*a^(9 /2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Ta n[c + d*x]]])/d - (2*a^4*(197*A - (195*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/( d*Sqrt[Tan[c + d*x]]))/(3*a)))/(5*a))/7)/3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-a^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Simp[a/(d*(b*c + a*d)*(n + 1)) Int[ (a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b *d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (244 ) = 488\).
Time = 1.83 (sec) , antiderivative size = 893, normalized size of antiderivative = 3.01
| method | result | size |
| derivativedivides | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {11}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2} \left (-1576 A \sqrt {i a}\, \tan \left (d x +c \right )^{4} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-315 i \sqrt {i a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}+630 i \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}+480 B \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-270 i B \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+1260 B \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}+1260 i A \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}-315 \sqrt {i a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}+276 A \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+1560 i B \sqrt {i a}\, \tan \left (d x +c \right )^{4} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-630 \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}+472 i A \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-190 i A \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-90 B \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-70 A \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{315 d \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(893\) |
| default | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {11}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2} \left (-1576 A \sqrt {i a}\, \tan \left (d x +c \right )^{4} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-315 i \sqrt {i a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}+630 i \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}+480 B \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-270 i B \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+1260 B \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}+1260 i A \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}-315 \sqrt {i a}\, \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{5}+276 A \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{2} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+1560 i B \sqrt {i a}\, \tan \left (d x +c \right )^{4} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}-630 \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \tan \left (d x +c \right )^{5}+472 i A \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right )^{3} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-190 i A \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-90 B \sqrt {-i a}\, \sqrt {i a}\, \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-70 A \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{315 d \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(893\) |
Input:
int(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_ RETURNVERBOSE)
Output:
1/315/d*(1/tan(d*x+c))^(11/2)*tan(d*x+c)*(a*(1+I*tan(d*x+c)))^(1/2)*a^2*(- 1576*A*(I*a)^(1/2)*tan(d*x+c)^4*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I* a)^(1/2)-315*I*(I*a)^(1/2)*2^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c )*(1+I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c) ^5+630*I*(-I*a)^(1/2)*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d* x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1/2))*a*tan(d*x+c)^5+480*B*(-I*a)^(1/2) *(I*a)^(1/2)*tan(d*x+c)^3*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-270*I*B*(I *a)^(1/2)*tan(d*x+c)^2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)+ 1260*B*(-I*a)^(1/2)*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+ c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1/2))*a*tan(d*x+c)^5+1260*I*A*(-I*a)^(1/2 )*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^( 1/2)+a)/(I*a)^(1/2))*a*tan(d*x+c)^5-315*(I*a)^(1/2)*2^(1/2)*ln((2*2^(1/2)* (-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(ta n(d*x+c)+I))*a*tan(d*x+c)^5+276*A*(-I*a)^(1/2)*(I*a)^(1/2)*tan(d*x+c)^2*(a *tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+1560*I*B*(I*a)^(1/2)*tan(d*x+c)^4*(a*t an(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)-630*(-I*a)^(1/2)*ln(1/2*(2* I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a )^(1/2))*a*tan(d*x+c)^5+472*I*A*(I*a)^(1/2)*tan(d*x+c)^3*(a*tan(d*x+c)*(1+ I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)-190*I*A*(I*a)^(1/2)*tan(d*x+c)*(a*tan(d* x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)-90*B*(-I*a)^(1/2)*(I*a)^(1/2)...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (229) = 458\).
Time = 0.10 (sec) , antiderivative size = 629, normalized size of antiderivative = 2.12 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, a lgorithm="fricas")
Output:
2/315*(315*sqrt(2)*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d *x + 2*I*c) + d)*log(4*((A - I*B)*a^3*e^(I*d*x + I*c) - sqrt(-(-I*A^2 - 2* A*B + I*B^2)*a^5/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))) *e^(-I*d*x - I*c)/((-I*A - B)*a^2)) - 315*sqrt(2)*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^( 4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(4*((A - I*B)*a^3*e^(I* d*x + I*c) - sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^5/d^2)*(-I*d*e^(2*I*d*x + 2* I*c) + I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a^2)) - 2*sq rt(2)*(2*(323*A - 300*I*B)*a^2*e^(9*I*d*x + 9*I*c) - 27*(61*A - 65*I*B)*a^ 2*e^(7*I*d*x + 7*I*c) + 63*(37*A - 35*I*B)*a^2*e^(5*I*d*x + 5*I*c) - 1365* (A - I*B)*a^2*e^(3*I*d*x + 3*I*c) + 315*(A - I*B)*a^2*e^(I*d*x + I*c))*sqr t(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d* x + 2*I*c) - 1)))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e ^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(11/2)*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4543 vs. \(2 (229) = 458\).
Time = 4.79 (sec) , antiderivative size = 4543, normalized size of antiderivative = 15.30 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, a lgorithm="maxima")
Output:
-1/1260*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((5040*((I + 1)*A - (I - 1)*B)*a^2*cos(7*d*x + 7*c) + 16800*(-(I + 1 )*A + (I - 1)*B)*a^2*cos(5*d*x + 5*c) + 20496*((I + 1)*A - (I - 1)*B)*a^2* cos(3*d*x + 3*c) + (-(9071*I + 9071)*A + (8841*I - 8841)*B)*a^2*cos(d*x + c) + 5040*((I - 1)*A + (I + 1)*B)*a^2*sin(7*d*x + 7*c) + 16800*(-(I - 1)*A - (I + 1)*B)*a^2*sin(5*d*x + 5*c) + 20496*((I - 1)*A + (I + 1)*B)*a^2*sin (3*d*x + 3*c) + (-(9071*I - 9071)*A - (8841*I + 8841)*B)*a^2*sin(d*x + c)) *cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 8*((-(121*I + 121)*A + (75*I - 75)*B)*a^2*cos(d*x + c) + (-(121*I - 121)*A - (75*I + 75) *B)*a^2*sin(d*x + c) + ((-(121*I + 121)*A + (75*I - 75)*B)*a^2*cos(d*x + c ) + (-(121*I - 121)*A - (75*I + 75)*B)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^ 2 + ((-(121*I + 121)*A + (75*I - 75)*B)*a^2*cos(d*x + c) + (-(121*I - 121) *A - (75*I + 75)*B)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 + 630*(((I + 1)*A - (I - 1)*B)*a^2*cos(2*d*x + 2*c)^2 + ((I + 1)*A - (I - 1)*B)*a^2*sin(2*d *x + 2*c)^2 + 2*(-(I + 1)*A + (I - 1)*B)*a^2*cos(2*d*x + 2*c) + ((I + 1)*A - (I - 1)*B)*a^2)*cos(3*d*x + 3*c) + 2*(((121*I + 121)*A - (75*I - 75)*B) *a^2*cos(d*x + c) + ((121*I - 121)*A + (75*I + 75)*B)*a^2*sin(d*x + c))*co s(2*d*x + 2*c) + 630*(((I - 1)*A + (I + 1)*B)*a^2*cos(2*d*x + 2*c)^2 + ((I - 1)*A + (I + 1)*B)*a^2*sin(2*d*x + 2*c)^2 + 2*(-(I - 1)*A - (I + 1)*B)*a ^2*cos(2*d*x + 2*c) + ((I - 1)*A + (I + 1)*B)*a^2)*sin(3*d*x + 3*c))*co...
Exception generated. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, a lgorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{11/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \] Input:
int(cot(c + d*x)^(11/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2) ,x)
Output:
int(cot(c + d*x)^(11/2)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2) , x)
\[ \int \cot ^{\frac {11}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )^{3}d x \right ) b -\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )^{2}d x \right ) a +2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )^{2}d x \right ) b i +2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )d x \right ) a i +\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{5}d x \right ) a \right ) \] Input:
int(cot(d*x+c)^(11/2)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)
Output:
sqrt(a)*a**2*( - int(sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d*x))*cot(c + d *x)**5*tan(c + d*x)**3,x)*b - int(sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d* x))*cot(c + d*x)**5*tan(c + d*x)**2,x)*a + 2*int(sqrt(tan(c + d*x)*i + 1)* sqrt(cot(c + d*x))*cot(c + d*x)**5*tan(c + d*x)**2,x)*b*i + 2*int(sqrt(tan (c + d*x)*i + 1)*sqrt(cot(c + d*x))*cot(c + d*x)**5*tan(c + d*x),x)*a*i + int(sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d*x))*cot(c + d*x)**5*tan(c + d* x),x)*b + int(sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d*x))*cot(c + d*x)**5, x)*a)