Integrand size = 28, antiderivative size = 180 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 \sqrt [4]{-1} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {a} d}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {a} d}-\frac {1}{d \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}} \]
-2*(-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))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d/a^(1/2)+(-1/2+1/2*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)/d/a^(1/2)-1/d/cot(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2)
Time = 1.10 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-4 (-1)^{3/4} a \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {1+i \tan (c+d x)} \sqrt {\tan (c+d x)}-2 a \tan (c+d x)+i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{2 a d \sqrt {a+i a \tan (c+d x)}} \]
(Sqrt[Cot[c + d*x]]*(-4*(-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]] - 2*a*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]]]*Sq rt[I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]))/(2*a*d*Sqrt[a + I*a*Tan[ c + d*x]])
Time = 0.97 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.92, 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, 4041, 27, 3042, 4084, 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.
\(\displaystyle \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cot (c+d x)^{3/2} \sqrt {a+i a \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4729 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\tan (c+d x)^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx\) |
\(\Big \downarrow \) 4041 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\int -\frac {\sqrt {i \tan (c+d x) a+a} (a-2 i a \tan (c+d x))}{2 \sqrt {\tan (c+d x)}}dx}{a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} (a-2 i a \tan (c+d x))}{\sqrt {\tan (c+d x)}}dx}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} (a-2 i a \tan (c+d x))}{\sqrt {\tan (c+d x)}}dx}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 4084 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 4027 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {2 i a^3 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}+2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 4082 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {2 a^2 \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\frac {4 a^2 \int \frac {1}{1-\frac {i a \tan (c+d x)}{i \tan (c+d x) a+a}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {4 \sqrt [4]{-1} a^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {(1-i) a^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}}{2 a^2}-\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-4*(-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 - ((1 - I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[ c + d*x]]])/d)/(2*a^2) - Sqrt[Tan[c + d*x]]/(d*Sqrt[a + I*a*Tan[c + d*x]]) )
3.8.72.3.1 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_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m* ((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Ta n[e + f*x], 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] && GtQ[n, 1] && (In tegerQ[m] || 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[((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*b + a*B)/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] - Simp[B/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[ e + f*x]), 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] && NeQ[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. 562 vs. \(2 (141 ) = 282\).
Time = 37.69 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.13
method | result | size |
default | \(\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-i \sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-i\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+i \sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+1\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+i \sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+i\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )-i \sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-1\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+2 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\right )-\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-\sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-i\right )+\sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+1\right )+\sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}+i\right )-\sqrt {2}\, \ln \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}-1\right )-2 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\right ) \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right ) \left (\csc \left (d x +c \right ) \left (1-\cos \left (d x +c \right )\right )^{2}-\sin \left (d x +c \right )\right )}{d \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}}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \left (1-\cos \left (d x +c \right )\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 {3}{2}}}\) | \(563\) |
(-1/2-1/2*I)/d*(-I*2^(1/2)*ln((cot(d*x+c)-csc(d*x+c))^(1/2)-I)*(csc(d*x+c) -cot(d*x+c))+I*2^(1/2)*ln((cot(d*x+c)-csc(d*x+c))^(1/2)+1)*(csc(d*x+c)-cot (d*x+c))+I*2^(1/2)*ln((cot(d*x+c)-csc(d*x+c))^(1/2)+I)*(csc(d*x+c)-cot(d*x +c))-I*2^(1/2)*ln((cot(d*x+c)-csc(d*x+c))^(1/2)-1)*(csc(d*x+c)-cot(d*x+c)) +I*(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+2*I*arctan((1/2+1/2*I)*(cot(d*x+c )-csc(d*x+c))^(1/2)*2^(1/2))-(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-2^(1/2) *ln((cot(d*x+c)-csc(d*x+c))^(1/2)-I)+2^(1/2)*ln((cot(d*x+c)-csc(d*x+c))^(1 /2)+1)+2^(1/2)*ln((cot(d*x+c)-csc(d*x+c))^(1/2)+I)-2^(1/2)*ln((cot(d*x+c)- csc(d*x+c))^(1/2)-1)-2*arctan((1/2+1/2*I)*(cot(d*x+c)-csc(d*x+c))^(1/2)*2^ (1/2))*(csc(d*x+c)-cot(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)/(cot(d*x+ c)-csc(d*x+c))^(1/2)/(1-cos(d*x+c))/(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos( d*x+c))^2-sin(d*x+c)))^(3/2)*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+c))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (134) = 268\).
Time = 0.28 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.44 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {{\left (a d \sqrt {-\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-2 \, {\left (\sqrt {2} {\left (i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a 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 {2 i}{a d^{2}}} - 2 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - a d \sqrt {-\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-2 \, {\left (\sqrt {2} {\left (-i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a 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 {2 i}{a d^{2}}} - 2 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + a d \sqrt {-\frac {4 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-16 \, {\left (\sqrt {2} {\left (a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} - a^{2} 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 {4 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 ) - a d \sqrt {-\frac {4 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (16 \, {\left (\sqrt {2} {\left (a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} - a^{2} 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 {4 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 ) - 2 \, \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 (2 i \, d x + 2 i \, c\right )} + i\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a d} \]
1/4*(a*d*sqrt(-2*I/(a*d^2))*e^(I*d*x + I*c)*log(-2*(sqrt(2)*(I*a*d*e^(2*I* d*x + 2*I*c) - I*a*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(-2*I/(a*d^2)) - 2*I*a*e^(I* d*x + I*c))*e^(-I*d*x - I*c)) - a*d*sqrt(-2*I/(a*d^2))*e^(I*d*x + I*c)*log (-2*(sqrt(2)*(-I*a*d*e^(2*I*d*x + 2*I*c) + I*a*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 (-2*I/(a*d^2)) - 2*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + a*d*sqrt(-4*I/ (a*d^2))*e^(I*d*x + I*c)*log(-16*(sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) - a^2 *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(-4*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)) - a*d*sqrt(-4*I/(a*d^2))*e^(I*d* x + I*c)*log(16*(sqrt(2)*(a^2*d*e^(3*I*d*x + 3*I*c) - a^2*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(-4*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)) - 2*sqrt(2)*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))*(-I*e^(2*I*d*x + 2*I*c) + I))*e^(-I*d*x - I*c)/(a*d)
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]