\(\int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1131]

Optimal result

Integrand size = 30, antiderivative size = 158 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {8 i a^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{3 (c-i d) d f (c+d \tan (e+f x))^{3/2}}+\frac {4 a^3 (i c-d) (c-4 i d)}{3 (c-i d)^2 d^2 f \sqrt {c+d \tan (e+f x)}} \] Output:

-8*I*a^3*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f+2/3 
*(c+I*d)*(a^3+I*a^3*tan(f*x+e))/(c-I*d)/d/f/(c+d*tan(f*x+e))^(3/2)+4/3*a^3 
*(I*c-d)*(c-4*I*d)/(c-I*d)^2/d^2/f/(c+d*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 1.90 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.78 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {2 a^3 \left (-\frac {12 i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2}}+\frac {(c+i d) \left (2 i c^2+9 c d-i d^2+3 d (i c+3 d) \tan (e+f x)\right )}{(c-i d)^2 d^2 (c+d \tan (e+f x))^{3/2}}\right )}{3 f} \] Input:

Integrate[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(2*a^3*(((-12*I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(c - I*d 
)^(5/2) + ((c + I*d)*((2*I)*c^2 + 9*c*d - I*d^2 + 3*d*(I*c + 3*d)*Tan[e + 
f*x]))/((c - I*d)^2*d^2*(c + d*Tan[e + f*x])^(3/2))))/(3*f)
 

Rubi [A] (warning: unable to verify)

Time = 0.88 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {3042, 4036, 25, 3042, 4074, 27, 3042, 4020, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4036

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4074

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \left (\frac {12 i a^6 d (c+i d)^3 \int \frac {1}{a^3 (c+i d)^2 (i c+d)-i a^9 (c+i d)^4 d^2 \tan ^2(e+f x)}d\sqrt {c+d \tan (e+f x)}}{f \left (c^2+d^2\right )}-\frac {2 a^3 (c+i d) (c-4 i d)}{d f (c-i d) \sqrt {c+d \tan (e+f x)}}\right )}{3 d (d+i c)}+\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{3 d f (c-i d) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (-\frac {12 a^3 d (c+i d) \text {arctanh}\left (\frac {a^3 d (c+i d) \tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d} \left (c^2+d^2\right )}-\frac {2 a^3 (c+i d) (c-4 i d)}{d f (c-i d) \sqrt {c+d \tan (e+f x)}}\right )}{3 d (d+i c)}+\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{3 d f (c-i d) (c+d \tan (e+f x))^{3/2}}\)

Input:

Int[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(2*(c + I*d)*(a^3 + I*a^3*Tan[e + f*x]))/(3*(c - I*d)*d*f*(c + d*Tan[e + f 
*x])^(3/2)) + (2*((-12*a^3*(c + I*d)*d*ArcTanh[(a^3*(c + I*d)*d*Tan[e + f* 
x])/Sqrt[c - I*d]])/(Sqrt[c - I*d]*(c^2 + d^2)*f) - (2*a^3*(c + I*d)*(c - 
(4*I)*d))/((c - I*d)*d*f*Sqrt[c + d*Tan[e + f*x]])))/(3*d*(I*c + d))
 

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 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 4036

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

rule 4074

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2757 vs. \(2 (135 ) = 270\).

Time = 0.50 (sec) , antiderivative size = 2758, normalized size of antiderivative = 17.46

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2758\)
default	\(\text {Expression too large to display}\)	\(2758\)
parts	\(\text {Expression too large to display}\)	\(8962\)

Input:

int((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

12*I/f*a^3/(c^2+d^2)^2/(c+d*tan(f*x+e))^(1/2)*c^2+2*I/f*a^3/(c^2+d^2)/(c+d 
*tan(f*x+e))^(3/2)*c-16/f*a^3*d/(c^2+d^2)^2/(c+d*tan(f*x+e))^(1/2)*c-4/f*a 
^3*d^3/(c^2+d^2)^(5/2)/(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))- 
4/f*a^3*d^3/(c^2+d^2)^(5/2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*t 
an(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))+2/f*a^3/d/(c^2+d^2)/(c+d*tan(f*x+e))^(3/2)*c^2-6*I/f*a^3*d^2/(c^2+d^2 
)^2/(c+d*tan(f*x+e))^(1/2)+12*I/f*a^3*d^2/(c^2+d^2)^(5/2)/(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))*c-6*I/f*a^3*d^2/(c^2+d^2)^(5/2)/(2*(c^2 
+d^2)^(1/2)+2*c)^(1/2)*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))*c+6*I/f*a^3*d^2/(c^2+d^2)^(5/2)/(2*(c 
^2+d^2)^(1/2)+2*c)^(1/2)*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))*c+12*I/f*a^3*d^2/(c^2+d^2)^(5/2)/(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))*c-2/3/f*a^3*d/(c^2+d^2)/( 
c+d*tan(f*x+e))^(3/2)+2*I/f*a^3/d^2/(c^2+d^2)^2/(c+d*tan(f*x+e))^(1/2)*c^4 
+6/f*a^3*d/(c^2+d^2)^(5/2)/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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))*c^2 
-6/f*a^3*d/(c^2+d^2)^(5/2)/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*ln(d*tan(f*x+e...
                                                                                    
                                                                                    
 

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. 963 vs. \(2 (126) = 252\).

Time = 0.16 (sec) , antiderivative size = 963, normalized size of antiderivative = 6.09 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas 
")
 

Output:

1/12*(3*sqrt(-64*I*a^6/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^2*d^3 + 5*I 
*c*d^4 + d^5)*f^2))*((c^4*d^2 - 4*I*c^3*d^3 - 6*c^2*d^4 + 4*I*c*d^5 + d^6) 
*f*e^(4*I*f*x + 4*I*e) + 2*(c^4*d^2 - 2*I*c^3*d^3 - 2*I*c*d^5 - d^6)*f*e^( 
2*I*f*x + 2*I*e) + (c^4*d^2 + 2*c^2*d^4 + d^6)*f)*log(1/4*(8*a^3*c + sqrt( 
-64*I*a^6/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^2*d^3 + 5*I*c*d^4 + d^5) 
*f^2))*((I*c^3 + 3*c^2*d - 3*I*c*d^2 - d^3)*f*e^(2*I*f*x + 2*I*e) + (I*c^3 
 + 3*c^2*d - 3*I*c*d^2 - d^3)*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)) + 8*(a^3*c - I*a^3*d)*e^(2*I*f*x + 2*I*e) 
)*e^(-2*I*f*x - 2*I*e)/a^3) - 3*sqrt(-64*I*a^6/((I*c^5 + 5*c^4*d - 10*I*c^ 
3*d^2 - 10*c^2*d^3 + 5*I*c*d^4 + d^5)*f^2))*((c^4*d^2 - 4*I*c^3*d^3 - 6*c^ 
2*d^4 + 4*I*c*d^5 + d^6)*f*e^(4*I*f*x + 4*I*e) + 2*(c^4*d^2 - 2*I*c^3*d^3 
- 2*I*c*d^5 - d^6)*f*e^(2*I*f*x + 2*I*e) + (c^4*d^2 + 2*c^2*d^4 + d^6)*f)* 
log(1/4*(8*a^3*c + sqrt(-64*I*a^6/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^ 
2*d^3 + 5*I*c*d^4 + d^5)*f^2))*((-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*f*e^( 
2*I*f*x + 2*I*e) + (-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*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)) + 8*(a^3*c - I* 
a^3*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a^3) - 16*(-I*a^3*c^3 - 2 
*a^3*c^2*d - 7*I*a^3*c*d^2 + 4*a^3*d^3 + (-I*a^3*c^3 - 5*a^3*c^2*d - I*a^3 
*c*d^2 - 5*a^3*d^3)*e^(4*I*f*x + 4*I*e) + (-2*I*a^3*c^3 - 7*a^3*c^2*d - 8* 
I*a^3*c*d^2 - a^3*d^3)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x ...
 

Sympy [F]

\[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=- i a^{3} \left (\int \frac {i}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx + \int \left (- \frac {3 \tan {\left (e + f x \right )}}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\right )\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (e + f x \right )}}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\right )\, dx\right ) \] Input:

integrate((a+I*a*tan(f*x+e))**3/(c+d*tan(f*x+e))**(5/2),x)
 

Output:

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

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima 
")
 

Output:

Timed out
 

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. 286 vs. \(2 (126) = 252\).

Time = 0.78 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.81 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {2 \, a^{3} {\left (\frac {12 \, \sqrt {2} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2} c \sqrt {-c + \sqrt {c^{2} + d^{2}}} - i \, \sqrt {2} \sqrt {-c + \sqrt {c^{2} + d^{2}}} d - \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {-c + \sqrt {c^{2} + d^{2}}}}\right )}{{\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} \sqrt {-c + \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {3 \, {\left (i \, d \tan \left (f x + e\right ) + i \, c\right )} c^{2} - i \, c^{3} + 6 \, {\left (d \tan \left (f x + e\right ) + c\right )} c d + c^{2} d + 9 \, {\left (i \, d \tan \left (f x + e\right ) + i \, c\right )} d^{2} - i \, c d^{2} + d^{3}}{{\left (c^{2} d^{2} - 2 i \, c d^{3} - d^{4}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\right )}}{3 \, f} \] Input:

integrate((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

2/3*a^3*(12*sqrt(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))))/((-I*c^2 - 2*c*d + I*d^2)*sqrt(-c + sqrt(c^2 + d^2))*(-I*d/(c 
 - sqrt(c^2 + d^2)) + 1)) + (3*(I*d*tan(f*x + e) + I*c)*c^2 - I*c^3 + 6*(d 
*tan(f*x + e) + c)*c*d + c^2*d + 9*(I*d*tan(f*x + e) + I*c)*d^2 - I*c*d^2 
+ d^3)/((c^2*d^2 - 2*I*c*d^3 - d^4)*(d*tan(f*x + e) + c)^(3/2)))/f
 

Mupad [B] (verification not implemented)

Time = 5.72 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.61 \[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {\frac {\left (a^3\,c^2+a^3\,c\,d\,2{}\mathrm {i}-a^3\,d^2\right )\,2{}\mathrm {i}}{3\,d^2\,f\,\left (c-d\,1{}\mathrm {i}\right )}-\frac {a^3\,\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (c^2-c\,d\,2{}\mathrm {i}+3\,d^2\right )\,2{}\mathrm {i}}{d^2\,f\,{\left (c-d\,1{}\mathrm {i}\right )}^2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {a^3\,\mathrm {atan}\left (\frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,\left (2\,c^8\,f^2+8\,c^6\,d^2\,f^2+12\,c^4\,d^4\,f^2+8\,c^2\,d^6\,f^2+2\,d^8\,f^2\right )}{2\,f\,{\left (-c+d\,1{}\mathrm {i}\right )}^{5/2}\,\left (f\,c^6+2{}\mathrm {i}\,f\,c^5\,d+f\,c^4\,d^2+4{}\mathrm {i}\,f\,c^3\,d^3-f\,c^2\,d^4+2{}\mathrm {i}\,f\,c\,d^5-f\,d^6\right )}\right )\,8{}\mathrm {i}}{f\,{\left (-c+d\,1{}\mathrm {i}\right )}^{5/2}} \] Input:

int((a + a*tan(e + f*x)*1i)^3/(c + d*tan(e + f*x))^(5/2),x)
 

Output:

(a^3*atan(((c + d*tan(e + f*x))^(1/2)*(2*c^8*f^2 + 2*d^8*f^2 + 8*c^2*d^6*f 
^2 + 12*c^4*d^4*f^2 + 8*c^6*d^2*f^2))/(2*f*(d*1i - c)^(5/2)*(c^6*f - d^6*f 
 - c^2*d^4*f + c^3*d^3*f*4i + c^4*d^2*f + c*d^5*f*2i + c^5*d*f*2i)))*8i)/( 
f*(d*1i - c)^(5/2)) - (((a^3*c^2 - a^3*d^2 + a^3*c*d*2i)*2i)/(3*d^2*f*(c - 
 d*1i)) - (a^3*(c + d*tan(e + f*x))*(c^2 - c*d*2i + 3*d^2)*2i)/(d^2*f*(c - 
 d*1i)^2))/(c + d*tan(e + f*x))^(3/2)
 

Reduce [F]

\[ \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:

int((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x)
 

Output:

(a**3*( - 2*sqrt(tan(e + f*x)*d + c) - 3*int((sqrt(tan(e + f*x)*d + c)*tan 
(e + f*x)**3)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + 
 f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*d**3*f*i - 6*int((sqrt(tan(e + f*x 
)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 
 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)*c*d**2*f*i - 3*int((sqrt( 
tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x 
)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*c**2*d*f*i - 12*int((sqrt(t 
an(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x) 
**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*d**3*f - 24* 
int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3*d**3 + 3*t 
an(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)*c*d* 
*2*f - 12*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3* 
d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*c**2*d* 
f + 9*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x))/(tan(e + f*x)**3*d**3 + 
3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)** 
2*d**3*f*i + 18*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x))/(tan(e + f*x)* 
*3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan( 
e + f*x)*c*d**2*f*i + 9*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x))/(tan(e 
 + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3) 
,x)*c**2*d*f*i))/(3*d*f*(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c*...