Integrand size = 28, antiderivative size = 175 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {\sqrt [4]{-1} \sqrt {a} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(1-i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\sqrt {a+i a \tan (c+d x)}}{d \sqrt {\cot (c+d x)}} \]
-(-1)^(1/4)*arctan((-1)^(3/4)*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+(-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+(a+I*a*tan(d*x+c))^(1/2)/d/cot(d*x+c)^(1/2)
Time = 1.08 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-(-1)^{3/4} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}+a (1+i \tan (c+d x)) \tan (c+d x)+i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d \sqrt {a+i a \tan (c+d x)}} \]
(Sqrt[Cot[c + d*x]]*(-((-1)^(3/4)*a*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]] *Sqrt[1 + I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]) + a*(1 + I*Tan[c + d*x])*Tan [c + d*x] + 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]]*Sqrt[a + I*a*Tan[c + d*x]]))/(d*Sq rt[a + I*a*Tan[c + d*x]])
Time = 0.94 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94, 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, 4043, 27, 3042, 4038, 3042, 4027, 218, 4082, 65, 216}
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.
| \(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan ^{\frac {3}{2}}(c+d x) \sqrt {i \tan (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan (c+d x)^{3/2} \sqrt {i \tan (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4043 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\int \frac {(i \tan (c+d x) a+a)^{3/2}}{2 \sqrt {\tan (c+d x)}}dx}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\int \frac {(i \tan (c+d x) a+a)^{3/2}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\int \frac {(i \tan (c+d x) a+a)^{3/2}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )\) |
| \(\Big \downarrow \) 4038 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+i \int \frac {\sqrt {i \tan (c+d x) a+a} (\tan (c+d x) a+i a)}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx+i \int \frac {\sqrt {i \tan (c+d x) a+a} (\tan (c+d x) a+i a)}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {i \int \frac {\sqrt {i \tan (c+d x) a+a} (\tan (c+d x) a+i a)}{\sqrt {\tan (c+d x)}}dx-\frac {4 i a^3 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}}{2 a}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {i \int \frac {\sqrt {i \tan (c+d x) a+a} (\tan (c+d x) a+i a)}{\sqrt {\tan (c+d x)}}dx+\frac {(2-2 i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a}\right )\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {(2-2 i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {a^2 \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}}{2 a}\right )\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {(2-2 i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a^2 \int \frac {1}{1-\frac {i a \tan (c+d x)}{i \tan (c+d x) a+a}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}}{2 a}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\frac {2 \sqrt [4]{-1} a^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}+\frac {(2-2 i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/2*((2*(-1)^(1/4)*a^(3/2)*ArcTan[ ((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ( (2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I* a*Tan[c + d*x]]])/d)/a + (Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d )
3.8.56.3.1 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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_)])^(3/2)/Sqrt[(c_.) + (d_.)*tan[(e_ .) + (f_.)*(x_)]], x_Symbol] :> Simp[2*a Int[Sqrt[a + b*Tan[e + f*x]]/Sqr t[c + d*Tan[e + f*x]], x], x] + Simp[b/a Int[(b + a*Tan[e + f*x])*(Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[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]
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*(m + n - 1))), x] - Simp[1/(a*(m + n - 1)) Int[(a + b *Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 2)*Simp[d*(b*c*m + a*d*(-1 + n)) - a*c^2*(m + n - 1) + d*(b*d*m - a*c*(m + 2*n - 2))*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] && GtQ[n, 1] && NeQ[m + n - 1, 0] && (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[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 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. 653 vs. \(2 (140 ) = 280\).
Time = 36.95 (sec) , antiderivative size = 654, normalized size of antiderivative = 3.74
| method | result | size |
| default | \(\frac {\left (2 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \cos \left (d x +c \right )+i \cos \left (d x +c \right ) \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+1\right ) \sqrt {2}-i \cos \left (d x +c \right ) \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-1\right ) \sqrt {2}+2 i \cos \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\right ) \sqrt {2}+2 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-4 i \cos \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )-4 i \cos \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )-2 i \cos \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 )-2 \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\, \sin \left (d x +c \right )-\cos \left (d x +c \right ) \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+1\right ) \sqrt {2}+\cos \left (d x +c \right ) \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-1\right ) \sqrt {2}+2 \cos \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\right ) \sqrt {2}-4 \cos \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )-4 \cos \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )-2 \cos \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 )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \cos \left (d x +c \right ) \sqrt {2}}{4 d \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )-1\right ) \cot \left (d x +c \right )^{\frac {3}{2}} \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )+1\right )}\) | \(654\) |
1/4/d*(2*I*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*cos(d*x+c)+I*cos(d*x+c)*l n((cot(d*x+c)-csc(d*x+c))^(1/2)+1)*2^(1/2)-I*cos(d*x+c)*ln((cot(d*x+c)-csc (d*x+c))^(1/2)-1)*2^(1/2)+2*I*cos(d*x+c)*arctan((cot(d*x+c)-csc(d*x+c))^(1 /2))*2^(1/2)+2*I*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)-4*I*cos(d*x+c)*arct an((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1)-4*I*cos(d*x+c)*arctan((cot(d*x +c)-csc(d*x+c))^(1/2)*2^(1/2)-1)-2*I*cos(d*x+c)*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))-2*(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)*s in(d*x+c)-cos(d*x+c)*ln((cot(d*x+c)-csc(d*x+c))^(1/2)+1)*2^(1/2)+cos(d*x+c )*ln((cot(d*x+c)-csc(d*x+c))^(1/2)-1)*2^(1/2)+2*cos(d*x+c)*arctan((cot(d*x +c)-csc(d*x+c))^(1/2))*2^(1/2)-4*cos(d*x+c)*arctan((cot(d*x+c)-csc(d*x+c)) ^(1/2)*2^(1/2)+1)-4*cos(d*x+c)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2 )-1)-2*cos(d*x+c)*ln((-(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-csc(d*x+c)+co t(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 )))*(a*(1+I*tan(d*x+c)))^(1/2)*cos(d*x+c)/(I*sin(d*x+c)+cos(d*x+c)-1)/cot( d*x+c)^(3/2)/(cot(d*x+c)-csc(d*x+c))^(1/2)/(cos(d*x+c)+1)*2^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (133) = 266\).
Time = 0.27 (sec) , antiderivative size = 610, normalized size of antiderivative = 3.49 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {4 \, \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 (i \, e^{\left (3 i \, d x + 3 i \, c\right )} - i \, e^{\left (i \, d x + i \, c\right )}\right )} + {\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 ) - {\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 ) - {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {i \, a}{d^{2}}} \log \left (-16 \, {\left (2 \, \sqrt {2} {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\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 \, a}{d^{2}}} + 3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {i \, a}{d^{2}}} \log \left (16 \, {\left (2 \, \sqrt {2} {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\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 \, a}{d^{2}}} - 3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right )}{4 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-1/4*(4*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))*(I*e^(3*I*d*x + 3*I*c) - I*e^(I*d*x + I*c)) + (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)*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)*sqrt(-I*a/d^2)*log(-16*(2*sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) - a* d*e^(I*d*x + I*c))*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(-I*a/d^2) + 3*a^2*e^(2*I*d*x + 2*I*c) - a^2)*e^(-2*I*d*x - 2*I*c)) + (d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-I *a/d^2)*log(16*(2*sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))* 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(-I*a/d^2) - 3*a^2*e^(2*I*d*x + 2*I*c) + a^2)*e^(- 2*I*d*x - 2*I*c)))/(d*e^(2*I*d*x + 2*I*c) + d)
\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]