Integrand size = 36, antiderivative size = 138 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {8 \sqrt [4]{-1} a^3 (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 i a^3 A \sqrt {\cot (c+d x)}}{3 d}+\frac {2 i a B (i a+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d} \]
-8*(-1)^(1/4)*a^3*(I*A+B)*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d+2*I*a*B*( I*a+a*cot(d*x+c))^2/d/cot(d*x+c)^(1/2)-16/3*I*a^3*A*cot(d*x+c)^(1/2)/d-2/3 *(A+3*I*B)*(I*a^3+a^3*cot(d*x+c))*cot(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.00 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.82 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 a^3 \left (3 i B-3 i A \cot (c+d x)-9 B \cot (c+d x)-3 A \cot ^2(c+d x)+4 A \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},i \tan (c+d x)\right )+12 B \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},i \tan (c+d x)\right )\right )}{3 d \sqrt {\cot (c+d x)}} \]
(-2*a^3*((3*I)*B - (3*I)*A*Cot[c + d*x] - 9*B*Cot[c + d*x] - 3*A*Cot[c + d *x]^2 + 4*A*Cot[c + d*x]^2*Hypergeometric2F1[-3/2, 1, -1/2, I*Tan[c + d*x] ] + 12*B*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, I*Tan[c + d*x]]))/(3 *d*Sqrt[Cot[c + d*x]])
Time = 0.94 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {3042, 4064, 3042, 4076, 27, 3042, 4077, 27, 3042, 4075, 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.
| \(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{5/2} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+i a)^3 (A \cot (c+d x)+B)}{\cot ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle 2 \int \frac {(\cot (c+d x) a+i a)^2 (a (i A+5 B)+a (A+3 i B) \cot (c+d x))}{2 \sqrt {\cot (c+d x)}}dx+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(\cot (c+d x) a+i a)^2 (a (i A+5 B)+a (A+3 i B) \cot (c+d x))}{\sqrt {\cot (c+d x)}}dx+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (a (i A+5 B)-a (A+3 i B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 4077 |
| \(\displaystyle -\frac {2}{3} \int \frac {2 (\cot (c+d x) a+i a) \left (a^2 (A-3 i B)-2 i a^2 A \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4}{3} \int \frac {(\cot (c+d x) a+i a) \left (a^2 (A-3 i B)-2 i a^2 A \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4}{3} \int \frac {\left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((A-3 i B) a^2+2 i A \tan \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle -\frac {4}{3} \left (\int \frac {3 (i A+B) a^3+3 (A-i B) \cot (c+d x) a^3}{\sqrt {\cot (c+d x)}}dx+\frac {4 i a^3 A \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4}{3} \left (\int \frac {3 a^3 (i A+B)-3 a^3 (A-i B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 i a^3 A \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle -\frac {4}{3} \left (\frac {18 a^6 (B+i A)^2 \int \frac {1}{3 a^3 (A-i B) \cot (c+d x)-3 a^3 (i A+B)}d\sqrt {\cot (c+d x)}}{d}+\frac {4 i a^3 A \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {4}{3} \left (\frac {6 \sqrt [4]{-1} a^3 (B+i A) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {4 i a^3 A \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 (A+3 i B) \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{d \sqrt {\cot (c+d x)}}\) |
(-4*((6*(-1)^(1/4)*a^3*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + ((4*I)*a^3*A*Sqrt[Cot[c + d*x]])/d))/3 + ((2*I)*a*B*(I*a + a*Cot[c + d* x])^2)/(d*Sqrt[Cot[c + d*x]]) - (2*(A + (3*I)*B)*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x]))/(3*d)
3.6.15.3.1 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[((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[(-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[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. 240 vs. \(2 (116 ) = 232\).
Time = 1.21 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.75
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {2 A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-6 i A \sqrt {\cot \left (d x +c \right )}-2 B \sqrt {\cot \left (d x +c \right )}-\frac {\left (-4 i A -4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {2 i B}{\sqrt {\cot \left (d x +c \right )}}\right )}{d}\) | \(241\) |
| default | \(\frac {a^{3} \left (-\frac {2 A \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-6 i A \sqrt {\cot \left (d x +c \right )}-2 B \sqrt {\cot \left (d x +c \right )}-\frac {\left (-4 i A -4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {2 i B}{\sqrt {\cot \left (d x +c \right )}}\right )}{d}\) | \(241\) |
a^3/d*(-2/3*A*cot(d*x+c)^(3/2)-6*I*A*cot(d*x+c)^(1/2)-2*B*cot(d*x+c)^(1/2) -1/4*(-4*I*A-4*B)*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(1+c ot(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*(4*I*B-4*A)*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)) )-2*I*B/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. 409 vs. \(2 (110) = 220\).
Time = 0.26 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.96 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (3 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 3 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) + 2 \, {\left ({\left (5 i \, A + 3 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (i \, A - 3 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, A a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )}}{3 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - d\right )}} \]
-2/3*(3*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^6/d^2)*(d*e^(4*I*d*x + 4*I*c) - d) *log(2*((A - I*B)*a^3*e^(2*I*d*x + 2*I*c) + sqrt(-(I*A^2 + 2*A*B - I*B^2)* a^6/d^2)*(d*e^(2*I*d*x + 2*I*c) - 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^3)) - 3*sqrt(-( I*A^2 + 2*A*B - I*B^2)*a^6/d^2)*(d*e^(4*I*d*x + 4*I*c) - d)*log(2*((A - I* B)*a^3*e^(2*I*d*x + 2*I*c) - sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^6/d^2)*(d*e^( 2*I*d*x + 2*I*c) - 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^3)) + 2*((5*I*A + 3*B)*a^3*e^( 4*I*d*x + 4*I*c) + (I*A - 3*B)*a^3*e^(2*I*d*x + 2*I*c) - 4*I*A*a^3)*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) - d)
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.41 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {-6 i \, B a^{3} \sqrt {\tan \left (d x + c\right )} - 3 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} + \frac {6 \, {\left (-3 i \, A - B\right )} a^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {2 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{3 \, d} \]
1/3*(-6*I*B*a^3*sqrt(tan(d*x + c)) - 3*(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)))) - s qrt(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^3 + 6*(-3*I*A - B)*a^3/sqrt(tan(d*x + c)) - 2*A *a^3/tan(d*x + c)^(3/2))/d
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]