\(\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx\) [1108]

Optimal result

Integrand size = 30, antiderivative size = 131 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {4 i a^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {2 a^2 \left (30 \sqrt {c-i d} d (i c+d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} \left (3 c^2-40 i c d-30 d^2+2 (3 c-5 i d) d \tan (e+f x)+3 d^2 \tan ^2(e+f x)\right )\right )}{15 d f} \] Input:

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

Output:

(-2*a^2*(30*Sqrt[c - I*d]*d*(I*c + d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqr 
t[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(3*c^2 - (40*I)*c*d - 30*d^2 + 2*(3 
*c - (5*I)*d)*d*Tan[e + f*x] + 3*d^2*Tan[e + f*x]^2)))/(15*d*f)
 

Rubi [A] (warning: unable to verify)

Time = 0.80 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4026, 3042, 4011, 3042, 4011, 3042, 4020, 25, 27, 73, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 \sqrt {2} a^4 (c-i d)^4 \int \frac {1}{\frac {4 i a^6 (c-i d)^6 \tan ^2(e+f x)}{d}-\frac {2 a^2 (i c+d)^3}{d}}d\sqrt {2 c+2 d \tan (e+f x)}}{d f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a^2 (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

(4*a^2*(c - I*d)^(3/2)*ArcTan[Sqrt[2]*a^2*(c - I*d)^(3/2)*Tan[e + f*x]])/f 
 + (4*a^2*(I*c + d)*Sqrt[c + d*Tan[e + f*x]])/f + (((4*I)/3)*a^2*(c + d*Ta 
n[e + f*x])^(3/2))/f - (2*a^2*(c + d*Tan[e + f*x])^(5/2))/(5*d*f)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 4011

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

rule 4020

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

rule 4026

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (110 ) = 220\).

Time = 0.40 (sec) , antiderivative size = 872, normalized size of antiderivative = 6.66

method	result	size
derivativedivides	\(\frac {2 a^{2} \left (-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i \sqrt {c +d \tan \left (f x +e \right )}\, d c +2 \sqrt {c +d \tan \left (f x +e \right )}\, d^{2}-2 d \left (\frac {\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}-\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )\right )}{f d}\)	\(872\)
default	\(\frac {2 a^{2} \left (-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i \sqrt {c +d \tan \left (f x +e \right )}\, d c +2 \sqrt {c +d \tan \left (f x +e \right )}\, d^{2}-2 d \left (\frac {\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}-\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )\right )}{f d}\)	\(872\)
parts	\(\text {Expression too large to display}\)	\(2493\)

Input:

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

Output:

2/f*a^2/d*(-1/5*(c+d*tan(f*x+e))^(5/2)+2/3*I*d*(c+d*tan(f*x+e))^(3/2)+2*I* 
(c+d*tan(f*x+e))^(1/2)*d*c+2*(c+d*tan(f*x+e))^(1/2)*d^2-2*d*(1/2/(2*(c^2+d 
^2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(1/2*(-I*c^2*(c^2+d^2)^(1/2)+I*d^2*(c 
^2+d^2)^(1/2)-I*c^3-I*c*d^2-2*c*d*(c^2+d^2)^(1/2)-c^2*d-d^3)*ln(d*tan(f*x+ 
e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2)) 
+2*(I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c* 
d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+ 
1/2*(-I*c^2*(c^2+d^2)^(1/2)+I*d^2*(c^2+d^2)^(1/2)-I*c^3-I*c*d^2-2*c*d*(c^2 
+d^2)^(1/2)-c^2*d-d^3)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2 
*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/ 
(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/2/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/(c^2+d^2 
)^(1/2)*(1/2*(I*c^2*(c^2+d^2)^(1/2)-I*d^2*(c^2+d^2)^(1/2)+I*c^3+I*c*d^2+2* 
c*d*(c^2+d^2)^(1/2)+c^2*d+d^3)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2 
*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(I*(2*(c^2+d^2)^(1/2)+2*c)^ 
(1/2)*c^3+I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1 
/2)*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-1/2*(I*c^2*(c^2+d^2)^(1/2)-I*d 
^2*(c^2+d^2)^(1/2)+I*c^3+I*c*d^2+2*c*d*(c^2+d^2)^(1/2)+c^2*d+d^3)*(2*(c^2+ 
d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f* 
x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*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. 646 vs. \(2 (105) = 210\).

Time = 0.15 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.93 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/15*(15*(d*f*e^(4*I*f*x + 4*I*e) + 2*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt( 
-(a^4*c^3 - 3*I*a^4*c^2*d - 3*a^4*c*d^2 + I*a^4*d^3)/f^2)*log(2*(-I*a^2*c^ 
2 - a^2*c*d + (f*e^(2*I*f*x + 2*I*e) + f)*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^4*c^3 - 3*I*a^4*c^2*d - 
 3*a^4*c*d^2 + I*a^4*d^3)/f^2) + (-I*a^2*c^2 - 2*a^2*c*d + I*a^2*d^2)*e^(2 
*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(-I*a^2*c - a^2*d)) - 15*(d*f*e^(4*I 
*f*x + 4*I*e) + 2*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^4*c^3 - 3*I*a^4* 
c^2*d - 3*a^4*c*d^2 + I*a^4*d^3)/f^2)*log(2*(-I*a^2*c^2 - a^2*c*d - (f*e^( 
2*I*f*x + 2*I*e) + f)*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^4*c^3 - 3*I*a^4*c^2*d - 3*a^4*c*d^2 + I*a^4 
*d^3)/f^2) + (-I*a^2*c^2 - 2*a^2*c*d + I*a^2*d^2)*e^(2*I*f*x + 2*I*e))*e^( 
-2*I*f*x - 2*I*e)/(-I*a^2*c - a^2*d)) - 2*(3*a^2*c^2 - 34*I*a^2*c*d - 23*a 
^2*d^2 + (3*a^2*c^2 - 46*I*a^2*c*d - 43*a^2*d^2)*e^(4*I*f*x + 4*I*e) + 2*( 
3*a^2*c^2 - 40*I*a^2*c*d - 27*a^2*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)))/(d*f*e^(4*I*f 
*x + 4*I*e) + 2*d*f*e^(2*I*f*x + 2*I*e) + d*f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (105) = 210\).

Time = 0.60 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.94 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {2 \, a^{2} {\left (\frac {30 \, \sqrt {2} {\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2} c \sqrt {-c + \sqrt {c^{2} + d^{2}}} - i \, \sqrt {2} \sqrt {-c + \sqrt {c^{2} + d^{2}}} d - \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {-c + \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-c + \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {3 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{4} - 10 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{5} - 30 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{5} - 30 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{6}}{d^{5}}\right )}}{15 \, f} \] Input:

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

Output:

-2/15*a^2*(30*sqrt(2)*(-I*c^2 - 2*c*d + I*d^2)*arctan(2*(sqrt(d*tan(f*x + 
e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(sqrt(2)*c*sqrt(-c + 
 sqrt(c^2 + d^2)) - I*sqrt(2)*sqrt(-c + sqrt(c^2 + d^2))*d - sqrt(2)*sqrt( 
c^2 + d^2)*sqrt(-c + sqrt(c^2 + d^2))))/(sqrt(-c + sqrt(c^2 + d^2))*(-I*d/ 
(c - sqrt(c^2 + d^2)) + 1)) + (3*(d*tan(f*x + e) + c)^(5/2)*d^4 - 10*I*(d* 
tan(f*x + e) + c)^(3/2)*d^5 - 30*I*sqrt(d*tan(f*x + e) + c)*c*d^5 - 30*sqr 
t(d*tan(f*x + e) + c)*d^6)/d^5)/f
 

Mupad [B] (verification not implemented)

Time = 7.55 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.50 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{3\,d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{3\,d\,f}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}-\left (c-d\,1{}\mathrm {i}\right )\,\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{d\,f}\right )\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}-\frac {2\,a^2\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}+\frac {\sqrt {4{}\mathrm {i}}\,a^2\,\mathrm {atan}\left (\frac {\sqrt {4{}\mathrm {i}}\,{\left (-d-c\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{2\,\left (c^2\,1{}\mathrm {i}+2\,c\,d-d^2\,1{}\mathrm {i}\right )}\right )\,{\left (-d-c\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{f} \] Input:

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

Output:

(4i^(1/2)*a^2*atan((4i^(1/2)*(- c*1i - d)^(3/2)*(c + d*tan(e + f*x))^(1/2) 
*1i)/(2*(2*c*d + c^2*1i - d^2*1i)))*(- c*1i - d)^(3/2)*2i)/f - (c - d*1i)* 
((2*a^2*(c - d*1i))/(d*f) - (2*a^2*(c + d*1i))/(d*f))*(c + d*tan(e + f*x)) 
^(1/2) - (2*a^2*(c + d*tan(e + f*x))^(5/2))/(5*d*f) - ((2*a^2*(c - d*1i))/ 
(3*d*f) - (2*a^2*(c + d*1i))/(3*d*f))*(c + d*tan(e + f*x))^(3/2)
 

Reduce [F]

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

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

Output:

a**2*(int(sqrt(tan(e + f*x)*d + c),x)*c - int(sqrt(tan(e + f*x)*d + c)*tan 
(e + f*x)**3,x)*d - int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*c + 2* 
int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*d*i + 2*int(sqrt(tan(e + f 
*x)*d + c)*tan(e + f*x),x)*c*i + int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x) 
,x)*d)