Integrand size = 28, antiderivative size = 174 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {(1+i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {26 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{15 d}-\frac {2 i \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{15 d}-\frac {2 \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d} \]
(-1-I)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2))*a^ (1/2)*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d-2/15*I*cot(d*x+c)^(3/2)*(a+I*a*t an(d*x+c))^(1/2)/d-2/5*cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d+26/15*c ot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d
Time = 0.98 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}+2 \left (-13+i \cot (c+d x)+3 \cot ^2(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}\right )}{15 d} \]
-1/15*(Sqrt[Cot[c + d*x]]*(15*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d* x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[I*a*Tan[c + d*x]] + 2*(-13 + I*Cot[c + d*x] + 3*Cot[c + d*x]^2)*Sqrt[a + I*a*Tan[c + d*x]]))/d
Time = 1.12 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.11, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4729, 3042, 4044, 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.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} \sqrt {a+i a \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan ^{\frac {7}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4044 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 \int \frac {(i a-4 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{2 \tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {(i a-4 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {(i a-4 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^{5/2}}dx}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {2 \int -\frac {\sqrt {i \tan (c+d x) a+a} \left (2 i \tan (c+d x) a^2+13 a^2\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (2 i \tan (c+d x) a^2+13 a^2\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (2 i \tan (c+d x) a^2+13 a^2\right )}{\tan (c+d x)^{3/2}}dx}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {\frac {2 \int \frac {15 i a^3 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {26 a^2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {15 i a^2 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {26 a^2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {15 i a^2 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {26 a^2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {\frac {30 a^4 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {26 a^2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {\frac {(15+15 i) a^{5/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {26 a^2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*Sqrt[a + I*a*Tan[c + d*x]])/(5* d*Tan[c + d*x]^(5/2)) + ((((-2*I)/3)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[ c + d*x]^(3/2)) - (((15 + 15*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[ c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (26*a^2*Sqrt[a + I*a*Tan[c + d *x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a))/(5*a))
3.8.51.3.1 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)*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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*(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*(c^2 + d^2)*(n + 1)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d , e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
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. 802 vs. \(2 (139 ) = 278\).
Time = 37.49 (sec) , antiderivative size = 803, normalized size of antiderivative = 4.61
-1/60/d*csc(d*x+c)*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x +c)))^(7/2)*(1-cos(d*x+c))*(-a*(2*I*(csc(d*x+c)-cot(d*x+c))-csc(d*x+c)^2*( 1-cos(d*x+c))^2+1)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1))^(1/2)*(3*csc(d*x+c)^ 5*(1-cos(d*x+c))^5-5*I*csc(d*x+c)^4*(1-cos(d*x+c))^4+30*I*csc(d*x+c)^2*2^( 1/2)*(1-cos(d*x+c))^2*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc (d*x+c))^(1/2)*2^(1/2)+1)+30*I*csc(d*x+c)^2*2^(1/2)*(1-cos(d*x+c))^2*(cot( d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-1)+1 5*I*csc(d*x+c)^2*2^(1/2)*(1-cos(d*x+c))^2*(cot(d*x+c)-csc(d*x+c))^(1/2)*ln (-((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+csc(d*x+c)-cot(d*x+c)-1)/(cot(d*x +c)-csc(d*x+c)+(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1))-15*csc(d*x+c)^2*2 ^(1/2)*(1-cos(d*x+c))^2*(cot(d*x+c)-csc(d*x+c))^(1/2)*ln(-(cot(d*x+c)-csc( d*x+c)+(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1)/((cot(d*x+c)-csc(d*x+c))^( 1/2)*2^(1/2)+csc(d*x+c)-cot(d*x+c)-1))-30*csc(d*x+c)^2*2^(1/2)*(1-cos(d*x+ c))^2*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2 ^(1/2)+1)-30*csc(d*x+c)^2*2^(1/2)*(1-cos(d*x+c))^2*(cot(d*x+c)-csc(d*x+c)) ^(1/2)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-1)-60*csc(d*x+c)^3*(1- cos(d*x+c))^3+60*I*csc(d*x+c)^2*(1-cos(d*x+c))^2+5*csc(d*x+c)-5*cot(d*x+c) -3*I)/(cot(d*x+c)-csc(d*x+c)+I)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1)^3*2^(1/2 )
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (130) = 260\).
Time = 0.27 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.18 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {8 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (17 \, e^{\left (5 i \, d x + 5 i \, c\right )} - 20 \, e^{\left (3 i \, d x + 3 i \, c\right )} + 15 \, e^{\left (i \, d x + i \, c\right )}\right )} - 15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {8 i \, a}{d^{2}}} \log \left (-{\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} - 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{60 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
1/60*(8*sqrt(2)*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))*(17*e^(5*I*d*x + 5*I*c) - 20*e^(3*I*d* x + 3*I*c) + 15*e^(I*d*x + I*c)) - 15*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I* d*x + 2*I*c) + d)*sqrt(8*I*a/d^2)*log((sqrt(2)*(d*e^(2*I*d*x + 2*I*c) - 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))*sqrt(8*I*a/d^2) + 4*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + 15*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(8*I* a/d^2)*log(-(sqrt(2)*(d*e^(2*I*d*x + 2*I*c) - 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))*sqrt( 8*I*a/d^2) - 4*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)))/(d*e^(4*I*d*x + 4*I *c) - 2*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (130) = 260\).
Time = 0.47 (sec) , antiderivative size = 1134, normalized size of antiderivative = 6.52 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Too large to display} \]
-1/30*(3*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((-(10*I + 10)*cos(3*d*x + 3*c) + (13*I + 13)*cos(d*x + c) - (10*I - 10)*sin(3*d*x + 3*c) + (13*I - 13)*sin(d*x + c))*cos(3/2*arctan2(sin(2*d* x + 2*c), cos(2*d*x + 2*c) - 1)) + ((10*I - 10)*cos(3*d*x + 3*c) - (13*I - 13)*cos(d*x + c) - (10*I + 10)*sin(3*d*x + 3*c) + (13*I + 13)*sin(d*x + c ))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + 15* (2*((I - 1)*cos(2*d*x + 2*c)^2 + (I - 1)*sin(2*d*x + 2*c)^2 - (2*I - 2)*co s(2*d*x + 2*c) + I - 1)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d *x + 2*c) - 1)) + 2*sin(d*x + c), 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c) ^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2 *d*x + 2*c) - 1)) + 2*cos(d*x + c)) + ((I + 1)*cos(2*d*x + 2*c)^2 + (I + 1 )*sin(2*d*x + 2*c)^2 - (2*I + 2)*cos(2*d*x + 2*c) + I + 1)*log(4*cos(d*x + c)^2 + 4*sin(d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2 *c) - 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/ 4)*(cos(d*x + c)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))))) *(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/...
\[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]