\(\int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx\) [1151]

Optimal result

Integrand size = 32, antiderivative size = 257 \[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} \left (15 c^2-10 i c d-7 d^2\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{4 f}-\frac {i \sqrt {2} \sqrt {a} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {(7 c-i d) d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f} \] Output:

-1/4*(-1)^(1/4)*a^(1/2)*d^(1/2)*(15*c^2-10*I*c*d-7*d^2)*arctanh((-1)^(3/4) 
*d^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c+d*tan(f*x+e))^(1/2))/f-I*2^(1 
/2)*a^(1/2)*(c-I*d)^(5/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/( 
c-I*d)^(1/2)/(a+I*a*tan(f*x+e))^(1/2))/f+1/4*(7*c-I*d)*d*(a+I*a*tan(f*x+e) 
)^(1/2)*(c+d*tan(f*x+e))^(1/2)/f+1/2*d*(a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f 
*x+e))^(3/2)/f
 

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.27 \[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\frac {4 i \sqrt {2} (c-i d)^2 \sqrt {-a (c-i d)} \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )+(7 c-i d) d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}+2 d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}+\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {i a (c+i d)} \sqrt {d} \left (15 c^2-10 i c d-7 d^2\right ) \text {arcsinh}\left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a} \sqrt {i a (c+i d)}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{\sqrt {i a} \sqrt {c+d \tan (e+f x)}}}{4 f} \] Input:

Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

((4*I)*Sqrt[2]*(c - I*d)^2*Sqrt[-(a*(c - I*d))]*ArcTan[(Sqrt[-(a*(c - I*d) 
)]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])] + (7* 
c - I*d)*d*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + 2*d*Sqrt[ 
a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2) + ((-1)^(1/4)*Sqrt[a]*Sqr 
t[I*a*(c + I*d)]*Sqrt[d]*(15*c^2 - (10*I)*c*d - 7*d^2)*ArcSinh[((-1)^(1/4) 
*Sqrt[a]*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[I*a]*Sqrt[I*a*(c + I*d) 
])]*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(Sqrt[I*a]*Sqrt[c + d*Tan[e + f* 
x]]))/(4*f)
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 4043, 27, 3042, 4080, 27, 3042, 4084, 3042, 4027, 221, 4082, 66, 221}

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.

\(\displaystyle \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}dx\)

\(\Big \downarrow \) 4043

\(\displaystyle \frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}-\frac {\int -\frac {1}{2} \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (a \left (4 c^2-i d c-3 d^2\right )+a (7 c-i d) d \tan (e+f x)\right )dx}{2 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (a \left (4 c^2-i d c-3 d^2\right )+a (7 c-i d) d \tan (e+f x)\right )dx}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (a \left (4 c^2-i d c-3 d^2\right )+a (7 c-i d) d \tan (e+f x)\right )dx}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 4080

\(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (8 c^3-9 i d c^2-14 d^2 c+i d^3\right ) a^2+d \left (15 c^2-10 i d c-7 d^2\right ) \tan (e+f x) a^2\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (8 c^3-9 i d c^2-14 d^2 c+i d^3\right ) a^2+d \left (15 c^2-10 i d c-7 d^2\right ) \tan (e+f x) a^2\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (8 c^3-9 i d c^2-14 d^2 c+i d^3\right ) a^2+d \left (15 c^2-10 i d c-7 d^2\right ) \tan (e+f x) a^2\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 4084

\(\displaystyle \frac {\frac {8 a^2 (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+a d \left (15 i c^2+10 c d-7 i d^2\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 a^2 (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+a d \left (15 i c^2+10 c d-7 i d^2\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 4027

\(\displaystyle \frac {\frac {a d \left (15 i c^2+10 c d-7 i d^2\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {16 i a^4 (c-i d)^3 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {a d \left (15 i c^2+10 c d-7 i d^2\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {8 i \sqrt {2} a^{5/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 4082

\(\displaystyle \frac {\frac {\frac {a^3 d \left (15 i c^2+10 c d-7 i d^2\right ) \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f}-\frac {8 i \sqrt {2} a^{5/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {\frac {2 a^3 d \left (15 i c^2+10 c d-7 i d^2\right ) \int \frac {1}{i a-\frac {d (i \tan (e+f x) a+a)}{c+d \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {8 i \sqrt {2} a^{5/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 (-1)^{3/4} a^{5/2} \sqrt {d} \left (15 i c^2+10 c d-7 i d^2\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {8 i \sqrt {2} a^{5/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a d (7 c-i d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 a}+\frac {d \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

Input:

Int[Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(d*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*f) + (((2*(-1 
)^(3/4)*a^(5/2)*Sqrt[d]*((15*I)*c^2 + 10*c*d - (7*I)*d^2)*ArcTanh[((-1)^(3 
/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]]) 
])/f - ((8*I)*Sqrt[2]*a^(5/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqr 
t[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f)/(2* 
a) + (a*(7*c - I*d)*d*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) 
/f)/(4*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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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 4043

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])
 

rule 4080

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*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(f*(m + n))), x] + Simp[ 
1/(a*(m + n))   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)*Sim 
p[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))*T 
an[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c 
 - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 0]
 

rule 4082

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]
 

rule 4084

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2129 vs. \(2 (202 ) = 404\).

Time = 0.44 (sec) , antiderivative size = 2130, normalized size of antiderivative = 8.29

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2130\)
default	\(\text {Expression too large to display}\)	\(2130\)

Input:

int((a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOS 
E)
 

Output:

-1/16/f*(a*(1+I*tan(f*x+e)))^(1/2)*(c+d*tan(f*x+e))^(1/2)*(6*I*2^(1/2)*(-a 
*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)* 
d^3*tan(f*x+e)-16*I*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*t 
an(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e) 
))^(1/2))/(I+tan(f*x+e)))*a*c^3*d*tan(f*x+e)-16*I*(I*a*d)^(1/2)*ln((3*a*c+ 
I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c 
+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*a*c*d^3*tan(f*x+e) 
+4*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan 
(f*x+e)))^(1/2)*c*d^2*tan(f*x+e)^2-8*I*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x 
+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+ 
e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*a*c^4+18*I*2^(1/2)*(-a*(I*d-c 
))^(1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d*t 
an(f*x+e)+8*I*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x 
+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/ 
2))/(I+tan(f*x+e)))*a*d^4+15*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d* 
tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/ 
2)+a*d)/(I*a*d)^(1/2))*a*c^3*d+16*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/ 
2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^2-15*2^(1/2)*(-a*(I*d-c 
))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f 
*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c^3*d*tan(f*x+e)-3*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. 1117 vs. \(2 (193) = 386\).

Time = 0.13 (sec) , antiderivative size = 1117, normalized size of antiderivative = 4.35 \[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="fr 
icas")
 

Output:

-1/8*(4*sqrt(2)*(f*e^(2*I*f*x + 2*I*e) + f)*sqrt(-(a*c^5 - 5*I*a*c^4*d - 1 
0*a*c^3*d^2 + 10*I*a*c^2*d^3 + 5*a*c*d^4 - I*a*d^5)/f^2)*log(-(I*sqrt(2)*f 
*sqrt(-(a*c^5 - 5*I*a*c^4*d - 10*a*c^3*d^2 + 10*I*a*c^2*d^3 + 5*a*c*d^4 - 
I*a*d^5)/f^2)*e^(I*f*x + I*e) - sqrt(2)*(c^2 - 2*I*c*d - d^2 + (c^2 - 2*I* 
c*d - d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + 
I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f 
*x - I*e)/(c^2 - 2*I*c*d - d^2)) - 4*sqrt(2)*(f*e^(2*I*f*x + 2*I*e) + f)*s 
qrt(-(a*c^5 - 5*I*a*c^4*d - 10*a*c^3*d^2 + 10*I*a*c^2*d^3 + 5*a*c*d^4 - I* 
a*d^5)/f^2)*log(-(-I*sqrt(2)*f*sqrt(-(a*c^5 - 5*I*a*c^4*d - 10*a*c^3*d^2 + 
 10*I*a*c^2*d^3 + 5*a*c*d^4 - I*a*d^5)/f^2)*e^(I*f*x + I*e) - sqrt(2)*(c^2 
 - 2*I*c*d - d^2 + (c^2 - 2*I*c*d - d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I 
*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2 
*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(c^2 - 2*I*c*d - d^2)) - 2*sqrt(2) 
*(3*(3*c*d - I*d^2)*e^(3*I*f*x + 3*I*e) + (9*c*d + I*d^2)*e^(I*f*x + I*e)) 
*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)) 
*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) + (f*e^(2*I*f*x + 2*I*e) + f)*sqrt((225 
*I*a*c^4*d + 300*a*c^3*d^2 - 310*I*a*c^2*d^3 - 140*a*c*d^4 + 49*I*a*d^5)/f 
^2)*log((sqrt(2)*(15*c^2 - 10*I*c*d - 7*d^2 + (15*c^2 - 10*I*c*d - 7*d^2)* 
e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2* 
I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) + 2*I*f*sqrt((22...
 

Sympy [F]

\[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int \sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a+I*a*tan(f*x+e))**(1/2)*(c+d*tan(f*x+e))**(5/2),x)
 

Output:

Integral(sqrt(I*a*(tan(e + f*x) - I))*(c + d*tan(e + f*x))**(5/2), x)
 

Maxima [F]

\[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int { \sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(I*a*tan(f*x + e) + a)*(d*tan(f*x + e) + c)^(5/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="gi 
ac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int \sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:

int((a + a*tan(e + f*x)*1i)^(1/2)*(c + d*tan(e + f*x))^(5/2),x)
 

Output:

int((a + a*tan(e + f*x)*1i)^(1/2)*(c + d*tan(e + f*x))^(5/2), x)
 

Reduce [F]

\[ \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\sqrt {a}\, \left (\left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c d +\left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}d x \right ) c^{2}\right ) \] Input:

int((a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x)
 

Output:

sqrt(a)*(int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x 
)**2,x)*d**2 + 2*int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan 
(e + f*x),x)*c*d + int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c),x 
)*c**2)