Integrand size = 28, antiderivative size = 140 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {(1-i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]
(-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-2/3*cot(d*x+c)^(3/2)*(a+I*a*tan( d*x+c))^(1/2)/d-2/3*I*cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d
Time = 0.66 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (3 i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}-2 (i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}\right )}{3 d} \]
(Sqrt[Cot[c + d*x]]*((3*I)*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*(I + Cot[c + d*x] )*Sqrt[a + I*a*Tan[c + d*x]]))/(3*d)
Time = 0.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {3042, 4729, 3042, 4044, 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.
| \(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{5/2} \sqrt {a+i a \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 4044 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 \int \frac {(i a-2 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {(i a-2 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {(i a-2 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^{3/2}}dx}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {2 \int -\frac {3 a^2 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-3 a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-3 a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {6 i a^3 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {(3-3 i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*Sqrt[a + I*a*Tan[c + d*x]])/(3* d*Tan[c + d*x]^(3/2)) + (((-3 + 3*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt [Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((2*I)*a*Sqrt[a + I*a*Tan [c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a))
3.8.52.3.1 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. 727 vs. \(2 (112 ) = 224\).
Time = 37.18 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.20
| method | result | size |
| default | \(\frac {\csc \left (d x +c \right ) \left (-\frac {\csc \left (d x +c \right ) \left (1-\cos \left (d x +c \right )\right )^{2}-\sin \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right ) \sqrt {-\frac {a \left (2 i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right )}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}}\, \left (3 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \ln \left (-\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1}{\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+\csc \left (d x +c \right )-\cot \left (d x +c \right )-1}\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+6 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+6 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+6 \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+6 \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+3 \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+\csc \left (d x +c \right )-\cot \left (d x +c \right )-1}{\cot \left (d x +c \right )-\csc \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1}\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+6 i \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-2 \left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3}-2 i+6 \csc \left (d x +c \right )-6 \cot \left (d x +c \right )\right ) \sqrt {2}}{12 d \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )+i\right ) {\left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right )}^{2}}\) | \(728\) |
1/12/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)))^(5/2)*(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*I*2^(1/2)*(c ot(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)-co t(d*x+c)-1))*(csc(d*x+c)-cot(d*x+c))+6*I*2^(1/2)*(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)*(csc(d*x+c)-cot(d*x+c ))+6*I*2^(1/2)*(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)*(csc(d*x+c)-cot(d*x+c))+6*2^(1/2)*(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)*(csc(d*x+c)-cot(d *x+c))+6*2^(1/2)*(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)*(csc(d*x+c)-cot(d*x+c))+3*2^(1/2)*(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))*(csc (d*x+c)-cot(d*x+c))+6*I*csc(d*x+c)^2*(1-cos(d*x+c))^2-2*csc(d*x+c)^3*(1-co s(d*x+c))^3-2*I+6*csc(d*x+c)-6*cot(d*x+c))/(cot(d*x+c)-csc(d*x+c)+I)/(csc( d*x+c)^2*(1-cos(d*x+c))^2-1)^2*2^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (104) = 208\).
Time = 0.26 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.33 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {-16 i \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (3 i \, d x + 3 i \, c\right )} - 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{12 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
1/12*(-16*I*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))*e^(3*I*d*x + 3*I*c) - 3*(d*e^(2*I* d*x + 2*I*c) - d)*sqrt(-8*I*a/d^2)*log((sqrt(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))*sqrt(-8*I*a/d^2) + 4*I*a*e^(I*d*x + I*c))*e^(-I* d*x - I*c)) + 3*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(-8*I*a/d^2)*log((sqrt(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))*sqrt(-8*I*a/d^2) + 4* I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)))/(d*e^(2*I*d*x + 2*I*c) - d)
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (104) = 208\).
Time = 0.44 (sec) , antiderivative size = 961, normalized size of antiderivative = 6.86 \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Too large to display} \]
1/6*(2*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((-(3*I - 3)*cos(3*d*x + 3*c) - (I - 1)*cos(d*x + c) + (3*I + 3)*sin(3 *d*x + 3*c) + (I + 1)*sin(d*x + c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos( 2*d*x + 2*c) - 1)) + (-(3*I + 3)*cos(3*d*x + 3*c) - (I + 1)*cos(d*x + c) - (3*I - 3)*sin(3*d*x + 3*c) - (I - 1)*sin(d*x + c))*sin(3/2*arctan2(sin(2* d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a) + 3*(2*(-(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)*a rctan2(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*arc tan2(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*ar ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))))*(cos(2*d*x + 2*c)^2 + si n(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a) + 2*(((-(I - 1...
\[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]