\(\int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx\) [844]

Optimal result

Integrand size = 45, antiderivative size = 284 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\frac {2 (i A-6 B) c^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{a^{5/2} f}+\frac {(i A-6 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a^3 f}+\frac {2 (i A-6 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a^2 f \sqrt {a+i a \tan (e+f x)}}-\frac {2 (i A-6 B) c (c-i c \tan (e+f x))^{5/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \] Output:

2*(I*A-6*B)*c^(7/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c 
*tan(f*x+e))^(1/2))/a^(5/2)/f+(I*A-6*B)*c^3*(a+I*a*tan(f*x+e))^(1/2)*(c-I* 
c*tan(f*x+e))^(1/2)/a^3/f+2/3*(I*A-6*B)*c^2*(c-I*c*tan(f*x+e))^(3/2)/a^2/f 
/(a+I*a*tan(f*x+e))^(1/2)-2/15*(I*A-6*B)*c*(c-I*c*tan(f*x+e))^(5/2)/a/f/(a 
+I*a*tan(f*x+e))^(3/2)+1/5*(I*A-B)*(c-I*c*tan(f*x+e))^(7/2)/f/(a+I*a*tan(f 
*x+e))^(5/2)
 

Mathematica [A] (verified)

Time = 14.91 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.72 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\frac {2 \sqrt {2} c^2 e^{-4 i (e+f x)} \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{3/2} \left (i A \left (3-2 e^{2 i (e+f x)}+10 e^{4 i (e+f x)}+15 e^{6 i (e+f x)}\right )-3 B \left (1-4 e^{2 i (e+f x)}+20 e^{4 i (e+f x)}+30 e^{6 i (e+f x)}\right )+15 i (A+6 i B) e^{5 i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) \arctan \left (e^{i (e+f x)}\right )\right )}{15 a^2 f \sqrt {a+i a \tan (e+f x)}} \] Input:

Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan 
[e + f*x])^(5/2),x]
 

Output:

(2*Sqrt[2]*c^2*(c/(1 + E^((2*I)*(e + f*x))))^(3/2)*(I*A*(3 - 2*E^((2*I)*(e 
 + f*x)) + 10*E^((4*I)*(e + f*x)) + 15*E^((6*I)*(e + f*x))) - 3*B*(1 - 4*E 
^((2*I)*(e + f*x)) + 20*E^((4*I)*(e + f*x)) + 30*E^((6*I)*(e + f*x))) + (1 
5*I)*(A + (6*I)*B)*E^((5*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))*ArcTan[E^ 
(I*(e + f*x))]))/(15*a^2*E^((4*I)*(e + f*x))*f*Sqrt[a + I*a*Tan[e + f*x]])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {3042, 4071, 87, 57, 57, 60, 45, 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.

\(\displaystyle \int \frac {(c-i c \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(a+i a \tan (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c-i c \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(a+i a \tan (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4071

\(\displaystyle \frac {a c \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(i \tan (e+f x) a+a)^{7/2}}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}-\frac {(A+6 i B) \int \frac {(c-i c \tan (e+f x))^{5/2}}{(i \tan (e+f x) a+a)^{5/2}}d\tan (e+f x)}{5 a}\right )}{f}\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}-\frac {(A+6 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{3 a (a+i a \tan (e+f x))^{3/2}}-\frac {5 c \int \frac {(c-i c \tan (e+f x))^{3/2}}{(i \tan (e+f x) a+a)^{3/2}}d\tan (e+f x)}{3 a}\right )}{5 a}\right )}{f}\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}-\frac {(A+6 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{3 a (a+i a \tan (e+f x))^{3/2}}-\frac {5 c \left (\frac {2 i (c-i c \tan (e+f x))^{3/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {3 c \int \frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}d\tan (e+f x)}{a}\right )}{3 a}\right )}{5 a}\right )}{f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}-\frac {(A+6 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{3 a (a+i a \tan (e+f x))^{3/2}}-\frac {5 c \left (\frac {2 i (c-i c \tan (e+f x))^{3/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {3 c \left (c \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )}{f}\)

\(\Big \downarrow \) 45

\(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}-\frac {(A+6 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{3 a (a+i a \tan (e+f x))^{3/2}}-\frac {5 c \left (\frac {2 i (c-i c \tan (e+f x))^{3/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {3 c \left (2 c \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}-\frac {(A+6 i B) \left (\frac {2 i (c-i c \tan (e+f x))^{5/2}}{3 a (a+i a \tan (e+f x))^{3/2}}-\frac {5 c \left (\frac {2 i (c-i c \tan (e+f x))^{3/2}}{a \sqrt {a+i a \tan (e+f x)}}-\frac {3 c \left (-\frac {2 i \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a}}-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )}{f}\)

Input:

Int[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f 
*x])^(5/2),x]
 

Output:

(a*c*(((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(5*a*c*(a + I*a*Tan[e + f*x 
])^(5/2)) - ((A + (6*I)*B)*((((2*I)/3)*(c - I*c*Tan[e + f*x])^(5/2))/(a*(a 
 + I*a*Tan[e + f*x])^(3/2)) - (5*c*(((2*I)*(c - I*c*Tan[e + f*x])^(3/2))/( 
a*Sqrt[a + I*a*Tan[e + f*x]]) - (3*c*(((-2*I)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt 
[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/Sqrt[a] - ( 
I*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/a))/a))/(3*a)))/( 
5*a)))/f
 

Defintions of rubi rules used

rule 45

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] && EqQ[b*c + a*d, 0] &&  !GtQ[c, 0]
 

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 4071

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si 
mp[a*(c/f)   Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x 
, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c 
+ a*d, 0] && EqQ[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. 834 vs. \(2 (235 ) = 470\).

Time = 0.81 (sec) , antiderivative size = 835, normalized size of antiderivative = 2.94

method	result	size
derivativedivides	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (246 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+60 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}-15 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{4}-90 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{4}-474 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-360 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}-15 B \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{4}-90 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -94 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+90 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+46 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{3}+26 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+540 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+360 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+564 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-60 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-15 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -74 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-141 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{15 f \,a^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \left (i-\tan \left (f x +e \right )\right )^{4}}\)	\(835\)
default	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (246 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+60 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}-15 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{4}-90 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{4}-474 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-360 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{3}-15 B \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{4}-90 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -94 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+90 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+46 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{3}+26 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+540 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )^{2}+360 B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+564 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-60 i A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-15 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -74 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-141 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{15 f \,a^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \left (i-\tan \left (f x +e \right )\right )^{4}}\)	\(835\)
parts	\(\text {Expression too large to display}\)	\(895\)

Input:

int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(5/2),x,m 
ethod=_RETURNVERBOSE)
 

Output:

1/15/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*c^3/a^3*(246 
*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^3+60*I*A*ln((a*c* 
tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan( 
f*x+e)^3-15*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)) 
/(a*c)^(1/2))*a*c*tan(f*x+e)^4-90*I*B*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c* 
(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^4-474*I*B*(a*c)^(1/2) 
*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-360*B*ln((a*c*tan(f*x+e)+(a*c)^(1 
/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^3-15*B*(a*c* 
(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^4-90*I*B*ln((a*c*tan(f*x+e) 
+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c-94*I*A*(a*c)^( 
1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+90*A*ln((a*c*tan(f*x+e)+(a* 
c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^2+46*A* 
(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^3+26*I*A*(a*c)^(1/2)*( 
a*c*(1+tan(f*x+e)^2))^(1/2)+540*I*B*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1 
+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^2+360*B*ln((a*c*tan(f*x 
+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)+ 
564*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2-60*I*A*ln((a*c 
*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c*tan 
(f*x+e)-15*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/ 
(a*c)^(1/2))*a*c-74*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+...
 

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. 533 vs. \(2 (218) = 436\).

Time = 0.11 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.88 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=-\frac {{\left (15 \, a^{3} \sqrt {\frac {{\left (A^{2} + 12 i \, A B - 36 \, B^{2}\right )} c^{7}}{a^{5} f^{2}}} f e^{\left (5 i \, f x + 5 i \, e\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (i \, A - 6 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (i \, A - 6 \, B\right )} c^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} - a^{3} f\right )} \sqrt {\frac {{\left (A^{2} + 12 i \, A B - 36 \, B^{2}\right )} c^{7}}{a^{5} f^{2}}}\right )}}{{\left (-i \, A + 6 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, A + 6 \, B\right )} c^{3}}\right ) - 15 \, a^{3} \sqrt {\frac {{\left (A^{2} + 12 i \, A B - 36 \, B^{2}\right )} c^{7}}{a^{5} f^{2}}} f e^{\left (5 i \, f x + 5 i \, e\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (i \, A - 6 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (i \, A - 6 \, B\right )} c^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} - a^{3} f\right )} \sqrt {\frac {{\left (A^{2} + 12 i \, A B - 36 \, B^{2}\right )} c^{7}}{a^{5} f^{2}}}\right )}}{{\left (-i \, A + 6 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, A + 6 \, B\right )} c^{3}}\right ) + 4 \, {\left (15 \, {\left (-i \, A + 6 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, {\left (-i \, A + 6 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (i \, A - 6 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 \, {\left (-i \, A + B\right )} c^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-5 i \, f x - 5 i \, e\right )}}{30 \, a^{3} f} \] Input:

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(5/ 
2),x, algorithm="fricas")
 

Output:

-1/30*(15*a^3*sqrt((A^2 + 12*I*A*B - 36*B^2)*c^7/(a^5*f^2))*f*e^(5*I*f*x + 
 5*I*e)*log(-4*(2*((I*A - 6*B)*c^3*e^(3*I*f*x + 3*I*e) + (I*A - 6*B)*c^3*e 
^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I 
*e) + 1)) + (a^3*f*e^(2*I*f*x + 2*I*e) - a^3*f)*sqrt((A^2 + 12*I*A*B - 36* 
B^2)*c^7/(a^5*f^2)))/((-I*A + 6*B)*c^3*e^(2*I*f*x + 2*I*e) + (-I*A + 6*B)* 
c^3)) - 15*a^3*sqrt((A^2 + 12*I*A*B - 36*B^2)*c^7/(a^5*f^2))*f*e^(5*I*f*x 
+ 5*I*e)*log(-4*(2*((I*A - 6*B)*c^3*e^(3*I*f*x + 3*I*e) + (I*A - 6*B)*c^3* 
e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2* 
I*e) + 1)) - (a^3*f*e^(2*I*f*x + 2*I*e) - a^3*f)*sqrt((A^2 + 12*I*A*B - 36 
*B^2)*c^7/(a^5*f^2)))/((-I*A + 6*B)*c^3*e^(2*I*f*x + 2*I*e) + (-I*A + 6*B) 
*c^3)) + 4*(15*(-I*A + 6*B)*c^3*e^(6*I*f*x + 6*I*e) + 10*(-I*A + 6*B)*c^3* 
e^(4*I*f*x + 4*I*e) + 2*(I*A - 6*B)*c^3*e^(2*I*f*x + 2*I*e) + 3*(-I*A + B) 
*c^3)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))) 
*e^(-5*I*f*x - 5*I*e)/(a^3*f)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(7/2)/(a+I*a*tan(f*x+e))**( 
5/2),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. 1025 vs. \(2 (218) = 436\).

Time = 0.36 (sec) , antiderivative size = 1025, normalized size of antiderivative = 3.61 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(5/ 
2),x, algorithm="maxima")
 

Output:

15*(60*(A + 6*I*B)*c^3*cos(6*f*x + 6*e) + 40*(A + 6*I*B)*c^3*cos(4*f*x + 4 
*e) - 8*(A + 6*I*B)*c^3*cos(2*f*x + 2*e) + 60*(I*A - 6*B)*c^3*sin(6*f*x + 
6*e) + 40*(I*A - 6*B)*c^3*sin(4*f*x + 4*e) + 8*(-I*A + 6*B)*c^3*sin(2*f*x 
+ 2*e) + 12*(A + I*B)*c^3 + 30*((A + 6*I*B)*c^3*cos(7/2*arctan2(sin(2*f*x 
+ 2*e), cos(2*f*x + 2*e))) + (A + 6*I*B)*c^3*cos(5/2*arctan2(sin(2*f*x + 2 
*e), cos(2*f*x + 2*e))) + (I*A - 6*B)*c^3*sin(7/2*arctan2(sin(2*f*x + 2*e) 
, cos(2*f*x + 2*e))) + (I*A - 6*B)*c^3*sin(5/2*arctan2(sin(2*f*x + 2*e), c 
os(2*f*x + 2*e))))*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2 
*e))), sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 30*((A 
+ 6*I*B)*c^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (A + 6 
*I*B)*c^3*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (I*A - 6* 
B)*c^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (I*A - 6*B)* 
c^3*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*arctan2(cos(1/2* 
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x + 
 2*e), cos(2*f*x + 2*e))) + 1) + 15*((I*A - 6*B)*c^3*cos(7/2*arctan2(sin(2 
*f*x + 2*e), cos(2*f*x + 2*e))) + (I*A - 6*B)*c^3*cos(5/2*arctan2(sin(2*f* 
x + 2*e), cos(2*f*x + 2*e))) - (A + 6*I*B)*c^3*sin(7/2*arctan2(sin(2*f*x + 
 2*e), cos(2*f*x + 2*e))) - (A + 6*I*B)*c^3*sin(5/2*arctan2(sin(2*f*x + 2* 
e), cos(2*f*x + 2*e))))*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 
2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*s...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(5/ 
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 TypeError: Bad Argument TypeError: Bad Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:

int(((A + B*tan(e + f*x))*(c - c*tan(e + f*x)*1i)^(7/2))/(a + a*tan(e + f* 
x)*1i)^(5/2),x)
 

Output:

int(((A + B*tan(e + f*x))*(c - c*tan(e + f*x)*1i)^(7/2))/(a + a*tan(e + f* 
x)*1i)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:

int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2)/(a+I*a*tan(f*x+e))^(5/2),x)
 

Output:

(sqrt(c)*c**3*( - int(sqrt( - tan(e + f*x)*i + 1)/(sqrt(tan(e + f*x)*i + 1 
)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt(tan(e 
 + f*x)*i + 1)),x)*a - int((sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**4)/( 
sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*tan( 
e + f*x)*i - sqrt(tan(e + f*x)*i + 1)),x)*b*i - int((sqrt( - tan(e + f*x)* 
i + 1)*tan(e + f*x)**3)/(sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2 - 2*sqrt 
(tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt(tan(e + f*x)*i + 1)),x)*a*i + 3 
*int((sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**3)/(sqrt(tan(e + f*x)*i + 
1)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt(tan( 
e + f*x)*i + 1)),x)*b + 3*int((sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x)**2 
)/(sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*t 
an(e + f*x)*i - sqrt(tan(e + f*x)*i + 1)),x)*a + 3*int((sqrt( - tan(e + f* 
x)*i + 1)*tan(e + f*x)**2)/(sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2 - 2*s 
qrt(tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt(tan(e + f*x)*i + 1)),x)*b*i 
+ 3*int((sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/(sqrt(tan(e + f*x)*i + 
1)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt(tan( 
e + f*x)*i + 1)),x)*a*i - int((sqrt( - tan(e + f*x)*i + 1)*tan(e + f*x))/( 
sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*tan( 
e + f*x)*i - sqrt(tan(e + f*x)*i + 1)),x)*b))/(sqrt(a)*a**2)