\(\int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) [751]

Optimal result

Integrand size = 28, antiderivative size = 174 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {(1+i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {26 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{15 d}-\frac {2 i \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{15 d}-\frac {2 \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d} \] Output:

(-1-I)*a^(1/2)*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+26/15*cot(d*x+c)^(1/2)*(a+I*a*ta 
n(d*x+c))^(1/2)/d-2/15*I*cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d-2/5*c 
ot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {\sqrt {\cot (c+d x)} \left (15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}+2 \left (-13+i \cot (c+d x)+3 \cot ^2(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}\right )}{15 d} \] Input:

Integrate[Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]],x]
 

Output:

-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
 

Rubi [A] (verified)

Time = 1.11 (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

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {-\frac {\frac {(15+15 i) a^{5/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {26 a^2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\)

Input:

Int[Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]],x]
 

Output:

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))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4027

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]
 

rule 4044

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])
 

rule 4081

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]
 

rule 4729

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]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (139 ) = 278\).

Time = 5.94 (sec) , antiderivative size = 645, normalized size of antiderivative = 3.71

method	result	size
default	\(-\frac {\sec \left (d x +c \right )^{3} \left (i \sin \left (d x +c \right )^{3} \left (-15 \cos \left (d x +c \right )-15\right ) \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}\, \sin \left (d x +c \right )-\sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sqrt {-2 \csc \left (d x +c \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, \sin \left (d x +c \right )+\sin \left (d x +c \right )+\cos \left (d x +c \right )-1}\right )+\sin \left (d x +c \right )^{3} \left (15 \cos \left (d x +c \right )+15\right ) \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}\, \sin \left (d x +c \right )+\sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sqrt {-2 \csc \left (d x +c \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, \sin \left (d x +c \right )-\sin \left (d x +c \right )-\cos \left (d x +c \right )+1}\right )+i \sin \left (d x +c \right )^{3} \left (-30 \cos \left (d x +c \right )-30\right ) \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 )+\sin \left (d x +c \right )^{3} \left (30 \cos \left (d x +c \right )+30\right ) \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 )+i \sin \left (d x +c \right )^{3} \left (-30 \cos \left (d x +c \right )-30\right ) \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 )+\sin \left (d x +c \right )^{3} \left (30 \cos \left (d x +c \right )+30\right ) \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 )+i \sin \left (d x +c \right ) \left (34 \cos \left (d x +c \right )^{3}+32 \cos \left (d x +c \right )^{2}-28 \cos \left (d x +c \right )-26\right ) \sqrt {2}+\left (-34 \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )+26\right ) \sqrt {2}\, \sin \left (d x +c \right )^{2}\right ) \cot \left (d x +c \right )^{\frac {7}{2}} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {2}}{30 d \left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right )}\)	\(645\)

Input:

int(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30/d*sec(d*x+c)^3*(I*sin(d*x+c)^3*(-15*cos(d*x+c)-15)*(cot(d*x+c)-csc(d 
*x+c))^(1/2)*ln(-((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-sin(d*x 
+c)-cos(d*x+c)+1)/((-2*csc(d*x+c)*sin(1/2*d*x+1/2*c)^2)^(1/2)*2^(1/2)*sin( 
d*x+c)+sin(d*x+c)+cos(d*x+c)-1))+sin(d*x+c)^3*(15*cos(d*x+c)+15)*(cot(d*x+ 
c)-csc(d*x+c))^(1/2)*ln(-((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c) 
+sin(d*x+c)+cos(d*x+c)-1)/((-2*csc(d*x+c)*sin(1/2*d*x+1/2*c)^2)^(1/2)*2^(1 
/2)*sin(d*x+c)-sin(d*x+c)-cos(d*x+c)+1))+I*sin(d*x+c)^3*(-30*cos(d*x+c)-30 
)*(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)+sin(d*x+c)^3*(30*cos(d*x+c)+30)*(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)+I*sin(d*x+c)^3*(-30*cos(d*x+c)-3 
0)*(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)+sin(d*x+c)^3*(30*cos(d*x+c)+30)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arcta 
n((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1)+I*sin(d*x+c)*(34*cos(d*x+c)^3+3 
2*cos(d*x+c)^2-28*cos(d*x+c)-26)*2^(1/2)+(-34*cos(d*x+c)^2-2*cos(d*x+c)+26 
)*2^(1/2)*sin(d*x+c)^2)*cot(d*x+c)^(7/2)*(a*(1+I*tan(d*x+c)))^(1/2)/(I*cos 
(d*x+c)+I-sin(d*x+c))*2^(1/2)
 

Fricas [B] (verification not implemented)

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 = 0.12 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.18 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {8 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (17 \, e^{\left (5 i \, d x + 5 i \, c\right )} - 20 \, e^{\left (3 i \, d x + 3 i \, c\right )} + 15 \, e^{\left (i \, d x + i \, c\right )}\right )} - 15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {8 i \, a}{d^{2}}} \log \left (-{\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} - 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{60 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

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)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**(7/2)*(a+I*a*tan(d*x+c))**(1/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

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 = 0.28 (sec) , antiderivative size = 1134, normalized size of antiderivative = 6.52 \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

-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/...
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:

int(cot(c + d*x)^(7/2)*(a + a*tan(c + d*x)*1i)^(1/2),x)
 

Output:

int(cot(c + d*x)^(7/2)*(a + a*tan(c + d*x)*1i)^(1/2), x)
 

Reduce [F]

\[ \int \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3}d x \right ) \] Input:

int(cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x)
 

Output:

sqrt(a)*int(sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d*x))*cot(c + d*x)**3,x)