Integrand size = 45, antiderivative size = 228 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {3 (2 i A-3 B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a} f}+\frac {3 (2 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a f}+\frac {(2 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}} \]
3*(2*I*A-3*B)*c^(5/2)*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))/f/a^(1/2)+3/2*(2*I*A-3*B)*c^2*(a+I*a*tan(f*x+e))^(1/ 2)*(c-I*c*tan(f*x+e))^(1/2)/a/f+1/2*(2*I*A-3*B)*c*(a+I*a*tan(f*x+e))^(1/2) *(c-I*c*tan(f*x+e))^(3/2)/a/f+(I*A-B)*(c-I*c*tan(f*x+e))^(5/2)/f/(a+I*a*ta n(f*x+e))^(1/2)
Time = 7.74 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {c^2 \sqrt {c-i c \tan (e+f x)} \left (-((2 A+3 i B) (i+\tan (e+f x)))-B (i+\tan (e+f x))^2+6 (2 i A-3 B) \left (1+\frac {\arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {1-i \tan (e+f x)}}\right )\right )}{2 f \sqrt {a+i a \tan (e+f x)}} \]
(c^2*Sqrt[c - I*c*Tan[e + f*x]]*(-((2*A + (3*I)*B)*(I + Tan[e + f*x])) - B *(I + Tan[e + f*x])^2 + 6*((2*I)*A - 3*B)*(1 + (ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[1 - I *Tan[e + f*x]]))))/(2*f*Sqrt[a + I*a*Tan[e + f*x]])
Time = 0.43 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3042, 4071, 87, 60, 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.
| \(\displaystyle \int \frac {(c-i c \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{\sqrt {a+i a \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-i c \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{\sqrt {a+i a \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(i \tan (e+f x) a+a)^{3/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{a c \sqrt {a+i a \tan (e+f x)}}-\frac {(2 A+3 i B) \int \frac {(c-i c \tan (e+f x))^{3/2}}{\sqrt {i \tan (e+f x) a+a}}d\tan (e+f x)}{a}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{a c \sqrt {a+i a \tan (e+f x)}}-\frac {(2 A+3 i B) \left (\frac {3}{2} c \int \frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}d\tan (e+f x)-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{a c \sqrt {a+i a \tan (e+f x)}}-\frac {(2 A+3 i B) \left (\frac {3}{2} c \left (c \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{f}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{a c \sqrt {a+i a \tan (e+f x)}}-\frac {(2 A+3 i B) \left (\frac {3}{2} c \left (2 c \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a c \left (\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{a c \sqrt {a+i a \tan (e+f x)}}-\frac {(2 A+3 i B) \left (\frac {3}{2} c \left (-\frac {2 i \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {a}}-\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{a}\right )-\frac {i \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 a}\right )}{a}\right )}{f}\) |
(a*c*(((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(a*c*Sqrt[a + I*a*Tan[e + f *x]]) - ((2*A + (3*I)*B)*(((-1/2*I)*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Ta n[e + f*x])^(3/2))/a + (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))/2))/a))/f
3.9.30.3.1 Defintions of rubi rules used
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]
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]
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]))))
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (188 ) = 376\).
Time = 0.39 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.48
| method | result | size |
| derivativedivides | \(\frac {i \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^{2} \left (6 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 )^{2}+18 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 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-9 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}+B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-6 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 -12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+12 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 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}-14 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+9 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 +19 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-10 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{2 f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(566\) |
| default | \(\frac {i \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^{2} \left (6 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 )^{2}+18 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 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-9 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}+B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-6 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 -12 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+12 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 A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{2}-14 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+9 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 +19 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-10 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{2 f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(566\) |
| parts | \(\frac {i A \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^{2} \left (3 i \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 ) \tan \left (f x +e \right )^{2} a c -3 i \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 +6 \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 ) \tan \left (f x +e \right ) a c -6 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-5 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}-\frac {B \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^{2} \left (9 i \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 ) \tan \left (f x +e \right )^{2} a c -i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-9 i \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 +18 \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 ) \tan \left (f x +e \right ) a c -19 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+4 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-14 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{2 f a \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \left (i-\tan \left (f x +e \right )\right )^{2} \sqrt {a c}}\) | \(633\) |
int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(1/2),x,m ethod=_RETURNVERBOSE)
1/2*I/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*c^2/a*(6*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)^2+18*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*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x +e)^2))^(1/2)*tan(f*x+e)^2-9*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+B*(a*c)^(1/2)*(a*c*(1+tan( f*x+e)^2))^(1/2)*tan(f*x+e)^3-6*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-12*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x +e)^2))^(1/2)*tan(f*x+e)+12*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*A*(a*c*(1+tan(f*x+e)^2))^(1 /2)*(a*c)^(1/2)*tan(f*x+e)^2-14*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/ 2)+9*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+19*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-10*A*( a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c*(1+tan(f*x+e)^2))^(1/2)/(I-t an(f*x+e))^2/(a*c)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (176) = 352\).
Time = 0.28 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.40 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=-\frac {3 \, \sqrt {\frac {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (3 i \, f x + 3 i \, e\right )} + a f e^{\left (i \, f x + i \, e\right )}\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2} 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}} + \sqrt {\frac {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} - a f\right )}\right )}}{{\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2}}\right ) - 3 \, \sqrt {\frac {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (3 i \, f x + 3 i \, e\right )} + a f e^{\left (i \, f x + i \, e\right )}\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2} 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}} - \sqrt {\frac {{\left (4 \, A^{2} + 12 i \, A B - 9 \, B^{2}\right )} c^{5}}{a f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} - a f\right )}\right )}}{{\left (2 i \, A - 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2 i \, A - 3 \, B\right )} c^{2}}\right ) + 4 \, {\left (3 \, {\left (-2 i \, A + 3 \, B\right )} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, {\left (-2 i \, A + 3 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, {\left (-i \, A + B\right )} c^{2}\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}}}{4 \, {\left (a f e^{\left (3 i \, f x + 3 i \, e\right )} + a f e^{\left (i \, f x + i \, e\right )}\right )}} \]
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(1/ 2),x, algorithm="fricas")
-1/4*(3*sqrt((4*A^2 + 12*I*A*B - 9*B^2)*c^5/(a*f^2))*(a*f*e^(3*I*f*x + 3*I *e) + a*f*e^(I*f*x + I*e))*log(4*(2*((2*I*A - 3*B)*c^2*e^(3*I*f*x + 3*I*e) + (2*I*A - 3*B)*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sq rt(c/(e^(2*I*f*x + 2*I*e) + 1)) + sqrt((4*A^2 + 12*I*A*B - 9*B^2)*c^5/(a*f ^2))*(a*f*e^(2*I*f*x + 2*I*e) - a*f))/((2*I*A - 3*B)*c^2*e^(2*I*f*x + 2*I* e) + (2*I*A - 3*B)*c^2)) - 3*sqrt((4*A^2 + 12*I*A*B - 9*B^2)*c^5/(a*f^2))* (a*f*e^(3*I*f*x + 3*I*e) + a*f*e^(I*f*x + I*e))*log(4*(2*((2*I*A - 3*B)*c^ 2*e^(3*I*f*x + 3*I*e) + (2*I*A - 3*B)*c^2*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)) - sqrt((4*A^2 + 12*I* A*B - 9*B^2)*c^5/(a*f^2))*(a*f*e^(2*I*f*x + 2*I*e) - a*f))/((2*I*A - 3*B)* c^2*e^(2*I*f*x + 2*I*e) + (2*I*A - 3*B)*c^2)) + 4*(3*(-2*I*A + 3*B)*c^2*e^ (4*I*f*x + 4*I*e) + 5*(-2*I*A + 3*B)*c^2*e^(2*I*f*x + 2*I*e) + 4*(-I*A + B )*c^2)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) )/(a*f*e^(3*I*f*x + 3*I*e) + a*f*e^(I*f*x + I*e))
\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )}\right )}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )}}\, dx \]
Integral((-I*c*(tan(e + f*x) + I))**(5/2)*(A + B*tan(e + f*x))/sqrt(I*a*(t an(e + f*x) - I)), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1319 vs. \(2 (176) = 352\).
Time = 0.66 (sec) , antiderivative size = 1319, normalized size of antiderivative = 5.79 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Too large to display} \]
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(1/ 2),x, algorithm="maxima")
4*(12*(2*A + 3*I*B)*c^2*cos(4*f*x + 4*e) + 20*(2*A + 3*I*B)*c^2*cos(2*f*x + 2*e) + 12*(2*I*A - 3*B)*c^2*sin(4*f*x + 4*e) + 20*(2*I*A - 3*B)*c^2*sin( 2*f*x + 2*e) + 16*(A + I*B)*c^2 + 6*((2*A + 3*I*B)*c^2*cos(5/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*A + 3*I*B)*c^2*cos(3/2*arctan2(si n(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*A + 3*I*B)*c^2*cos(1/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 3*B)*c^2*sin(5/2*arctan2(sin( 2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*I*A - 3*B)*c^2*sin(3/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 3*B)*c^2*sin(1/2*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) + 6*((2*A + 3*I*B)*c^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*A + 3*I*B)*c^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*A + 3*I*B)*c^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 3*B)*c^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*I*A - 3*B)*c^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 3*B)*c^2*sin(1/2*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) + 3*((2*I*A - 3*B)* c^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*I*A - 3*B) *c^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2*I*A - 3*...
\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {i \, a \tan \left (f x + e\right ) + a}} \,d x } \]
integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(1/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]