Integrand size = 36, antiderivative size = 128 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {4 \sqrt [4]{-1} a^2 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {4 a^2 (A-i B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (7 i A+5 B) \cot ^{\frac {3}{2}}(c+d x)}{15 d}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (i a^2+a^2 \cot (c+d x)\right )}{5 d} \]
4*(-1)^(1/4)*a^2*(A-I*B)*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-2/15*a^2*( 7*I*A+5*B)*cot(d*x+c)^(3/2)/d-2/5*A*cot(d*x+c)^(3/2)*(I*a^2+a^2*cot(d*x+c) )/d+4*a^2*(A-I*B)*cot(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.58 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.58 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {2 a^2 \sqrt {\cot (c+d x)} \left (\cot (c+d x) (5 (2 i A+B)+3 A \cot (c+d x))-30 (A-i B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},i \tan (c+d x)\right )\right )}{15 d} \]
(-2*a^2*Sqrt[Cot[c + d*x]]*(Cot[c + d*x]*(5*((2*I)*A + B) + 3*A*Cot[c + d* x]) - 30*(A - I*B)*Hypergeometric2F1[-1/2, 1, 1/2, I*Tan[c + d*x]]))/(15*d )
Time = 0.84 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4064, 3042, 4077, 27, 3042, 4075, 3042, 4011, 3042, 4016, 221}
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.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2 (A \cot (c+d x)+B)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4077 |
| \(\displaystyle -\frac {2}{5} \int \frac {1}{2} \sqrt {\cot (c+d x)} (\cot (c+d x) a+i a) (a (3 A-5 i B)-a (7 i A+5 B) \cot (c+d x))dx-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \int \sqrt {\cot (c+d x)} (\cot (c+d x) a+i a) (a (3 A-5 i B)-a (7 i A+5 B) \cot (c+d x))dx-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a (3 A-5 i B)+a (7 i A+5 B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle \frac {1}{5} \left (-\int \sqrt {\cot (c+d x)} \left (10 (i A+B) a^2+10 (A-i B) \cot (c+d x) a^2\right )dx-\frac {2 a^2 (5 B+7 i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (10 a^2 (i A+B)-10 a^2 (A-i B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 a^2 (5 B+7 i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {10 a^2 (i A+B) \cot (c+d x)-10 a^2 (A-i B)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a^2 (5 B+7 i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {20 a^2 (A-i B) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {-10 (A-i B) a^2-10 (i A+B) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 (5 B+7 i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {20 a^2 (A-i B) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle \frac {1}{5} \left (-\frac {200 a^4 (A-i B)^2 \int \frac {1}{10 (A-i B) a^2+10 (i A+B) \cot (c+d x) a^2}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 B+7 i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {20 a^2 (A-i B) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{5} \left (\frac {20 \sqrt [4]{-1} a^2 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 (5 B+7 i A) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {20 a^2 (A-i B) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 A \cot ^{\frac {3}{2}}(c+d x) \left (a^2 \cot (c+d x)+i a^2\right )}{5 d}\) |
(-2*A*Cot[c + d*x]^(3/2)*(I*a^2 + a^2*Cot[c + d*x]))/(5*d) + ((20*(-1)^(1/ 4)*a^2*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (20*a^2*(A - I*B)*Sqrt[Cot[c + d*x]])/d - (2*a^2*((7*I)*A + 5*B)*Cot[c + d*x]^(3/2))/(3 *d))/5
3.6.8.3.1 Defintions of rubi rules used
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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2*(c^2/f) Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b *Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp [g^(m + n) Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c *Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !Integer Q[p] && IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B *d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f* x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -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[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan [e + f*x])^n*Simp[a*A*d*(m + n) + B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && !LtQ[n, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (108 ) = 216\).
Time = 0.72 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.98
| method | result | size |
| derivativedivides | \(-\frac {a^{2} \left (\frac {2 A \cot \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {4 i A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 B \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+4 i B \sqrt {\cot \left (d x +c \right )}-4 A \sqrt {\cot \left (d x +c \right )}+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(253\) |
| default | \(-\frac {a^{2} \left (\frac {2 A \cot \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {4 i A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 B \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+4 i B \sqrt {\cot \left (d x +c \right )}-4 A \sqrt {\cot \left (d x +c \right )}+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(253\) |
-a^2/d*(2/5*A*cot(d*x+c)^(5/2)+4/3*I*A*cot(d*x+c)^(3/2)+2/3*B*cot(d*x+c)^( 3/2)+4*I*B*cot(d*x+c)^(1/2)-4*A*cot(d*x+c)^(1/2)+1/4*(2*A-2*I*B)*2^(1/2)*( ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)-2^(1/2)*cot(d*x+c )^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x +c)^(1/2)))+1/4*(-2*I*A-2*B)*2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^ (1/2))/(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x +c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (104) = 208\).
Time = 0.26 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.50 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 2 \, {\left ({\left (43 \, A - 35 i \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, {\left (9 \, A - 10 i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (23 \, A - 25 i \, B\right )} a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-1/15*(15*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/d^2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2*((A - I*B)*a^2*e^(2*I*d*x + 2*I*c) - s qrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqr t((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2* I*c)/((-I*A - B)*a^2)) - 15*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/d^2)*(d*e^( 4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2*((A - I*B)*a^2*e^(2* I*d*x + 2*I*c) - sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^4/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1) ))*e^(-2*I*d*x - 2*I*c)/((-I*A - B)*a^2)) - 2*((43*A - 35*I*B)*a^2*e^(4*I* d*x + 4*I*c) - 6*(9*A - 10*I*B)*a^2*e^(2*I*d*x + 2*I*c) + (23*A - 25*I*B)* a^2)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(d*e^(4* I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.55 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {15 \, {\left (2 \, \sqrt {2} {\left (-\left (i - 1\right ) \, A - \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i - 1\right ) \, A - \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} - \frac {120 \, {\left (A - i \, B\right )} a^{2}}{\sqrt {\tan \left (d x + c\right )}} - \frac {20 \, {\left (-2 i \, A - B\right )} a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {12 \, A a^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{30 \, d} \]
-1/30*(15*(2*sqrt(2)*(-(I - 1)*A - (I + 1)*B)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(-(I - 1)*A - (I + 1)*B)*arctan(-1/2* sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt(2)*(-(I + 1)*A + (I - 1)* B)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*(-(I + 1 )*A + (I - 1)*B)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))*a^ 2 - 120*(A - I*B)*a^2/sqrt(tan(d*x + c)) - 20*(-2*I*A - B)*a^2/tan(d*x + c )^(3/2) + 12*A*a^2/tan(d*x + c)^(5/2))/d
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]