\(\int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx\) [1138]

Optimal result

Integrand size = 32, antiderivative size = 250 \[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx=-\frac {\sqrt [4]{-1} a^{3/2} (i c+3 d) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}} \] Output:

-(-1)^(1/4)*a^(3/2)*(I*c+3*d)*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))/d^(1/2)/f-2*I*2^(1/2)*a^(3/2)*(c-I 
*d)^(1/2)*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+a^2*(c+I*d)*(c+d*tan(f*x+e))^(1/2)/d/f/(a+I*a*tan 
(f*x+e))^(1/2)-a^2*(c+d*tan(f*x+e))^(3/2)/d/f/(a+I*a*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 3.07 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.94 \[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx=\frac {a \left (2 i \sqrt {2} \sqrt {-a (c-i d)} \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )+i \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}+\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {c+i d} (c-3 i d) \arcsin \left (\frac {\sqrt [4]{-1} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+i d}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{\sqrt {d} \sqrt {c+d \tan (e+f x)}}\right )}{f} \] Input:

Integrate[(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {3042, 4039, 27, 3042, 4078, 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.

\(\displaystyle \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}dx\)

\(\Big \downarrow \) 4039

\(\displaystyle \frac {a \int -\frac {(a (i c-5 d)+a (c-3 i d) \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{2 \sqrt {i \tan (e+f x) a+a}}dx}{d}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \int \frac {(a (i c-5 d)+a (c-3 i d) \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}dx}{2 d}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(a (i c-5 d)+a (c-3 i d) \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}dx}{2 d}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4078

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4084

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4027

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

Int[(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]],x]
 

Output:

-((a^2*(c + d*Tan[e + f*x])^(3/2))/(d*f*Sqrt[a + I*a*Tan[e + f*x]])) - (a* 
(-(((-2*(-1)^(3/4)*a^(5/2)*(c - (3*I)*d)*Sqrt[d]*ArcTanh[((-1)^(3/4)*Sqrt[ 
d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/f - (( 
4*I)*Sqrt[2]*a^(5/2)*Sqrt[c - I*d]*d*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*T 
an[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f)/a^2) - (2*a* 
(c + I*d)*Sqrt[c + d*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])))/(2*d)
 

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 4039

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + 
 d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[a/(d*(m + n - 1)) 
Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + 
a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x 
] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 
 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] 
 && (IntegerQ[m] || IntegersQ[2*m, 2*n])
 

rule 4078

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))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(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 - 1)*Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a 
*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] 
&& NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 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. 865 vs. \(2 (201 ) = 402\).

Time = 0.42 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.46

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {c +d \tan \left (f x +e \right )}\, a \left (2 i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) a c +2 i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) a d +4 i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a c +3 i \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a d \sqrt {2}\, \sqrt {-a \left (i d -c \right )}+2 i \sqrt {i a d}\, \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}-2 \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) a c +2 \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {i a d}\, a d -\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a c +4 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a d \right ) \sqrt {2}}{4 f \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}\, \sqrt {-a \left (i d -c \right )}}\)	\(866\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {c +d \tan \left (f x +e \right )}\, a \left (2 i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) a c +2 i \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) a d +4 i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a c +3 i \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a d \sqrt {2}\, \sqrt {-a \left (i d -c \right )}+2 i \sqrt {i a d}\, \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}-2 \sqrt {i a d}\, \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) a c +2 \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {i a d}\, a d -\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a c +4 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a d \tan \left (f x +e \right )+i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}+a d}{2 \sqrt {i a d}}\right ) a d \right ) \sqrt {2}}{4 f \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i a d}\, \sqrt {-a \left (i d -c \right )}}\)	\(866\)

Input:

int((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/4/f*(a*(1+I*tan(f*x+e)))^(1/2)*(c+d*tan(f*x+e))^(1/2)*a*(2*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+2 
*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*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*I*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*d*2^(1/2)*(-a*(I 
*d-c))^(1/2)+2*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+ 
e))*(1+I*tan(f*x+e)))^(1/2)-2*(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+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*ta 
n(f*x+e)))^(1/2))/(I+tan(f*x+e)))*(I*a*d)^(1/2)*a*d-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+4*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*d)/(a*(c+d*tan(f*x+e))*(1+I*ta 
n(f*x+e)))^(1/2)/(I*a*d)^(1/2)*2^(1/2)/(-a*(I*d-c))^(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. 745 vs. \(2 (192) = 384\).

Time = 0.11 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.98 \[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx =\text {Too large to display} \] Input:

integrate((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2),x, algorithm="fr 
icas")
 

Output:

1/2*(2*I*sqrt(2)*a*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) - 2*s 
qrt(2)*f*sqrt(-(a^3*c - I*a^3*d)/f^2)*log(-(I*sqrt(2)*f*sqrt(-(a^3*c - I*a 
^3*d)/f^2)*e^(I*f*x + I*e) - sqrt(2)*(a*e^(2*I*f*x + 2*I*e) + a)*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)/a) + 2*sqrt(2)*f*sqrt(-(a^3*c - 
 I*a^3*d)/f^2)*log(-(-I*sqrt(2)*f*sqrt(-(a^3*c - I*a^3*d)/f^2)*e^(I*f*x + 
I*e) - sqrt(2)*(a*e^(2*I*f*x + 2*I*e) + a)*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)/a) - f*sqrt((-I*a^3*c^2 - 6*a^3*c*d + 9*I*a^3*d^2)/(d 
*f^2))*log((2*I*d*f*sqrt((-I*a^3*c^2 - 6*a^3*c*d + 9*I*a^3*d^2)/(d*f^2))*e 
^(I*f*x + I*e) + sqrt(2)*(I*a*c + 3*a*d + (I*a*c + 3*a*d)*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)/(I*a*c + 3*a*d)) 
+ f*sqrt((-I*a^3*c^2 - 6*a^3*c*d + 9*I*a^3*d^2)/(d*f^2))*log((-2*I*d*f*sqr 
t((-I*a^3*c^2 - 6*a^3*c*d + 9*I*a^3*d^2)/(d*f^2))*e^(I*f*x + I*e) + sqrt(2 
)*(I*a*c + 3*a*d + (I*a*c + 3*a*d)*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)/(I*a*c + 3*a*d)))/f
 

Sympy [F]

\[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt {c + d \tan {\left (e + f x \right )}}\, dx \] Input:

integrate((a+I*a*tan(f*x+e))**(3/2)*(c+d*tan(f*x+e))**(1/2),x)
 

Output:

Integral((I*a*(tan(e + f*x) - I))**(3/2)*sqrt(c + d*tan(e + f*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2),x, algorithm="ma 
xima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(3*d-c>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

Exception generated. \[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \] Input:

int((a + a*tan(e + f*x)*1i)^(3/2)*(c + d*tan(e + f*x))^(1/2),x)
 

Output:

int((a + a*tan(e + f*x)*1i)^(3/2)*(c + d*tan(e + f*x))^(1/2), x)
 

Reduce [F]

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

int((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2),x)
 

Output:

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