Integrand size = 28, antiderivative size = 188 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{a^{5/2} d}+\frac {i}{5 d \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}}+\frac {i}{10 a d \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}-\frac {i}{20 a^2 d \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}} \]
(-1/8-1/8*I)*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)/a^(5/2)/d-1/20*I/a^2/d/cot(d*x+c)^(1 /2)/(a+I*a*tan(d*x+c))^(1/2)+1/5*I/d/cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^( 5/2)+1/10*I/a/d/cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(3/2)
Time = 1.79 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {i \left (\frac {5 \sqrt {2} a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {i a \tan (c+d x)}}-\frac {2 \left (a+5 a \cot ^2(c+d x)\right )}{(i+\cot (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}\right )}{40 a^3 d \sqrt {\cot (c+d x)}} \]
((-1/40*I)*((5*Sqrt[2]*a*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*(a + 5*a*Cot[c + d*x]^2)) /((I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])))/(a^3*d*Sqrt[Cot[c + d *x]])
Time = 1.02 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {3042, 4729, 3042, 4040, 3042, 4079, 27, 3042, 4079, 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.
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {\tan (c+d x)}}{(i \tan (c+d x) a+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {\tan (c+d x)}}{(i \tan (c+d x) a+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 4040 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\int \frac {i a-4 a \tan (c+d x)}{\sqrt {\tan (c+d x)} (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\int \frac {i a-4 a \tan (c+d x)}{\sqrt {\tan (c+d x)} (i \tan (c+d x) a+a)^{3/2}}dx}{10 a^2}\right )\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\int \frac {3 \left (3 i a^2-2 a^2 \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\int \frac {3 i a^2-2 a^2 \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}dx}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\int \frac {3 i a^2-2 a^2 \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}dx}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\frac {\int \frac {5 i a^3 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a^2}+\frac {i a^2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\frac {5}{2} i a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+\frac {i a^2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\frac {5}{2} i a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+\frac {i a^2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\frac {5 a^3 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}+\frac {i a^2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {i \sqrt {\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac {\frac {\frac {\left (\frac {5}{2}+\frac {5 i}{2}\right ) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {i a^2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}}{2 a^2}-\frac {i a \sqrt {\tan (c+d x)}}{d (a+i a \tan (c+d x))^{3/2}}}{10 a^2}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((I/5)*Sqrt[Tan[c + d*x]])/(d*(a + I*a*Tan[c + d*x])^(5/2)) - (((-I)*a*Sqrt[Tan[c + d*x]])/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (((5/2 + (5*I)/2)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[T an[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (I*a^2*Sqrt[Tan[c + d*x]])/ (d*Sqrt[a + I*a*Tan[c + d*x]]))/(2*a^2))/(10*a^2))
3.8.84.3.1 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_)*Sqrt[(c_.) + (d_.)*tan[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[(-b)*(a + b*Tan[e + f*x])^m*(Sqrt[c + d *Tan[e + f*x]]/(2*a*f*m)), x] + Simp[1/(4*a^2*m) Int[(a + b*Tan[e + f*x]) ^(m + 1)*(Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x]/Sqrt[c + d*Ta n[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] & & EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && IntegersQ[2*m]
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*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
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. 445 vs. \(2 (147 ) = 294\).
Time = 1.87 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.37
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (15 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{2}\left (d x +c \right )\right )-5 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )-5 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a +15 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )-4 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (\tan ^{2}\left (d x +c \right )\right )-20 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{80 d \sqrt {\frac {1}{\tan \left (d x +c \right )}}\, a^{4} \left (1+i \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) \left (-\tan \left (d x +c \right )+i\right )^{3} \sqrt {-i a}}\) | \(446\) |
| default | \(-\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (15 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{2}\left (d x +c \right )\right )-5 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )-5 i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a +15 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )-4 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (\tan ^{2}\left (d x +c \right )\right )-20 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{80 d \sqrt {\frac {1}{\tan \left (d x +c \right )}}\, a^{4} \left (1+i \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) \left (-\tan \left (d x +c \right )+i\right )^{3} \sqrt {-i a}}\) | \(446\) |
-1/80/d*(a*(1+I*tan(d*x+c)))^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*( 15*I*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*tan(d*x+c)*a)/(tan(d*x+c)+I))*a*tan(d*x+c)^2-5*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*tan(d*x+c)*a )/(tan(d*x+c)+I))*a*tan(d*x+c)^3-5*I*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*tan(d*x+c)*a)/(tan(d*x+c)+I))*a+ 15*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*tan(d*x+c)*a)/(tan(d*x+c)+I))*a*tan(d*x+c)-4*(a*tan(d*x+c)*(1+I*ta n(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^2-20*(a*tan(d*x+c)*(1+I*tan(d*x+c )))^(1/2)*(-I*a)^(1/2))/(1/tan(d*x+c))^(1/2)/a^4/(1+I*tan(d*x+c))/tan(d*x+ c)/(-tan(d*x+c)+I)^3/(-I*a)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (136) = 272\).
Time = 0.27 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.95 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {{\left (10 \, a^{3} d \sqrt {\frac {i}{8 \, a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (4 \, {\left (2 \, \sqrt {2} {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{8 \, a^{5} d^{2}}} + i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 10 \, a^{3} d \sqrt {\frac {i}{8 \, a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (-4 \, {\left (2 \, \sqrt {2} {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{8 \, a^{5} d^{2}}} - i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{40 \, a^{3} d} \]
-1/40*(10*a^3*d*sqrt(1/8*I/(a^5*d^2))*e^(5*I*d*x + 5*I*c)*log(4*(2*sqrt(2) *(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqr t((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(1/8*I/(a^5*d ^2)) + I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 10*a^3*d*sqrt(1/8*I/(a^5*d ^2))*e^(5*I*d*x + 5*I*c)*log(-4*(2*sqrt(2)*(a^3*d*e^(2*I*d*x + 2*I*c) - a^ 3*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(1/8*I/(a^5*d^2)) - I*a*e^(I*d*x + I*c))*e^(- I*d*x - I*c)) - 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^(6*I*d*x + 6*I*c) + 2*e^(4* I*d*x + 4*I*c) - 2*e^(2*I*d*x + 2*I*c) - 1))*e^(-5*I*d*x - 5*I*c)/(a^3*d)
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}} \sqrt {\cot {\left (c + d x \right )}}}\, dx \]
Exception generated. \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {1}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]