Integrand size = 45, antiderivative size = 283 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {5 a^{7/2} (3 i A+4 B) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {5 a^3 (3 i A+4 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c f}-\frac {5 a^2 (3 i A+4 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c f}-\frac {a (3 i A+4 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c f} \]
5*a^(7/2)*(3*I*A+4*B)*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/c^(1/2)-5/2*a^3*(3*I*A+4*B)*(a+I*a*tan(f*x+e))^(1/ 2)*(c-I*c*tan(f*x+e))^(1/2)/c/f-5/6*a^2*(3*I*A+4*B)*(c-I*c*tan(f*x+e))^(1/ 2)*(a+I*a*tan(f*x+e))^(3/2)/c/f-1/3*a*(3*I*A+4*B)*(c-I*c*tan(f*x+e))^(1/2) *(a+I*a*tan(f*x+e))^(5/2)/c/f-(I*A+B)*(a+I*a*tan(f*x+e))^(7/2)/f/(c-I*c*ta n(f*x+e))^(1/2)
Time = 8.51 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.65 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {a^{7/2} \left (\frac {30 (3 A-4 i B) \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) (i+\tan (e+f x))}{\sqrt {1-i \tan (e+f x)}}+\frac {\sqrt {a} (-i+\tan (e+f x)) \left (72 A-94 i B+(-21 i A-34 B) \tan (e+f x)+(3 A-10 i B) \tan ^2(e+f x)+2 B \tan ^3(e+f x)\right )}{\sqrt {a+i a \tan (e+f x)}}\right )}{6 f \sqrt {c-i c \tan (e+f x)}} \]
(a^(7/2)*((30*(3*A - (4*I)*B)*ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt[2]*S qrt[a])]*(I + Tan[e + f*x]))/Sqrt[1 - I*Tan[e + f*x]] + (Sqrt[a]*(-I + Tan [e + f*x])*(72*A - (94*I)*B + ((-21*I)*A - 34*B)*Tan[e + f*x] + (3*A - (10 *I)*B)*Tan[e + f*x]^2 + 2*B*Tan[e + f*x]^3))/Sqrt[a + I*a*Tan[e + f*x]]))/ (6*f*Sqrt[c - I*c*Tan[e + f*x]])
Time = 0.44 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.95, 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, 60, 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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (-\frac {(3 A-4 i B) \int \frac {(i \tan (e+f x) a+a)^{5/2}}{\sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (-\frac {(3 A-4 i B) \left (\frac {5}{3} a \int \frac {(i \tan (e+f x) a+a)^{3/2}}{\sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)+\frac {i (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (-\frac {(3 A-4 i B) \left (\frac {5}{3} a \left (\frac {3}{2} a \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)+\frac {i (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 c}\right )+\frac {i (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a c \left (-\frac {(3 A-4 i B) \left (\frac {5}{3} a \left (\frac {3}{2} a \left (a \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)+\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}\right )+\frac {i (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 c}\right )+\frac {i (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {a c \left (-\frac {(3 A-4 i B) \left (\frac {5}{3} a \left (\frac {3}{2} a \left (2 a \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}+\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}\right )+\frac {i (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 c}\right )+\frac {i (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a c \left (-\frac {(3 A-4 i B) \left (\frac {5}{3} a \left (\frac {3}{2} a \left (\frac {i \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c}-\frac {2 i \sqrt {a} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c}}\right )+\frac {i (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 c}\right )+\frac {i (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c}\right )}{c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{a c \sqrt {c-i c \tan (e+f x)}}\right )}{f}\) |
(a*c*(-(((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(a*c*Sqrt[c - I*c*Tan[e + f*x]])) - ((3*A - (4*I)*B)*(((I/3)*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]])/c + (5*a*(((I/2)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/c + (3*a*(((-2*I)*Sqrt[a]*ArcTan[(Sqrt[c]*Sqrt[a + I*a* Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/Sqrt[c] + (I*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/c))/2))/3))/c))/f
3.9.21.3.1 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. 626 vs. \(2 (234 ) = 468\).
Time = 0.40 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.22
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (-60 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}+8 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}-2 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+90 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 )+18 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}+45 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}-3 A \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 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 +128 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+120 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 )+24 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-72 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-45 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 -93 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-94 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f c \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}}\) | \(627\) |
| default | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (-60 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}+8 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}-2 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+90 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 )+18 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}+45 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}-3 A \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 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 +128 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+120 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 )+24 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-72 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-45 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 -93 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-94 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{6 f c \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}}\) | \(627\) |
| parts | \(-\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (30 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 ) a c +6 i \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+15 \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 -\tan \left (f x +e \right )^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-24 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-15 a c \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 )-31 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\right )}{2 f c \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 )^{2}}+\frac {B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (30 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 -4 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+\tan \left (f x +e \right )^{4} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-30 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 -64 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-60 \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 -12 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+47 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{3 f c \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 )^{2}}\) | \(688\) |
int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/2),x,m ethod=_RETURNVERBOSE)
-1/6/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^3/c*(-60*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+8*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x +e)^3-2*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^4+90*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)+18*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+4 5*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)^2-3*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+ e)^3+60*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+128*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e )+120*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)+24*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f *x+e)^2-72*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)-45*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-93*A*(a*c )^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-94*B*(a*c)^(1/2)*(a*c*(1+t an(f*x+e)^2))^(1/2))/(a*c*(1+tan(f*x+e)^2))^(1/2)/(I+tan(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. 590 vs. \(2 (219) = 438\).
Time = 0.27 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.08 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {15 \, \sqrt {\frac {{\left (9 \, A^{2} - 24 i \, A B - 16 \, B^{2}\right )} a^{7}}{c f^{2}}} {\left (c f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + c f\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (-3 i \, A - 4 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-3 i \, A - 4 \, B\right )} a^{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}} + \sqrt {\frac {{\left (9 \, A^{2} - 24 i \, A B - 16 \, B^{2}\right )} a^{7}}{c f^{2}}} {\left (c f e^{\left (2 i \, f x + 2 i \, e\right )} - c f\right )}\right )}}{{\left (-3 i \, A - 4 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A - 4 \, B\right )} a^{3}}\right ) - 15 \, \sqrt {\frac {{\left (9 \, A^{2} - 24 i \, A B - 16 \, B^{2}\right )} a^{7}}{c f^{2}}} {\left (c f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + c f\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (-3 i \, A - 4 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-3 i \, A - 4 \, B\right )} a^{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}} - \sqrt {\frac {{\left (9 \, A^{2} - 24 i \, A B - 16 \, B^{2}\right )} a^{7}}{c f^{2}}} {\left (c f e^{\left (2 i \, f x + 2 i \, e\right )} - c f\right )}\right )}}{{\left (-3 i \, A - 4 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A - 4 \, B\right )} a^{3}}\right ) - 4 \, {\left (24 \, {\left (i \, A + B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 33 \, {\left (3 i \, A + 4 \, B\right )} a^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 40 \, {\left (3 i \, A + 4 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (3 i \, A + 4 \, B\right )} a^{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}}}{12 \, {\left (c f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + c f\right )}} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/ 2),x, algorithm="fricas")
1/12*(15*sqrt((9*A^2 - 24*I*A*B - 16*B^2)*a^7/(c*f^2))*(c*f*e^(4*I*f*x + 4 *I*e) + 2*c*f*e^(2*I*f*x + 2*I*e) + c*f)*log(4*(2*((-3*I*A - 4*B)*a^3*e^(3 *I*f*x + 3*I*e) + (-3*I*A - 4*B)*a^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)) + sqrt((9*A^2 - 24*I*A*B - 16*B^2)*a^7/(c*f^2))*(c*f*e^(2*I*f*x + 2*I*e) - c*f))/((-3*I*A - 4*B)*a^3 *e^(2*I*f*x + 2*I*e) + (-3*I*A - 4*B)*a^3)) - 15*sqrt((9*A^2 - 24*I*A*B - 16*B^2)*a^7/(c*f^2))*(c*f*e^(4*I*f*x + 4*I*e) + 2*c*f*e^(2*I*f*x + 2*I*e) + c*f)*log(4*(2*((-3*I*A - 4*B)*a^3*e^(3*I*f*x + 3*I*e) + (-3*I*A - 4*B)*a ^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)) - sqrt((9*A^2 - 24*I*A*B - 16*B^2)*a^7/(c*f^2))*(c*f*e^(2*I* f*x + 2*I*e) - c*f))/((-3*I*A - 4*B)*a^3*e^(2*I*f*x + 2*I*e) + (-3*I*A - 4 *B)*a^3)) - 4*(24*(I*A + B)*a^3*e^(7*I*f*x + 7*I*e) + 33*(3*I*A + 4*B)*a^3 *e^(5*I*f*x + 5*I*e) + 40*(3*I*A + 4*B)*a^3*e^(3*I*f*x + 3*I*e) + 15*(3*I* A + 4*B)*a^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)))/(c*f*e^(4*I*f*x + 4*I*e) + 2*c*f*e^(2*I*f*x + 2*I *e) + c*f)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1329 vs. \(2 (219) = 438\).
Time = 0.71 (sec) , antiderivative size = 1329, normalized size of antiderivative = 4.70 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\text {Too large to display} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/ 2),x, algorithm="maxima")
-6*(12*(9*A - 20*I*B)*a^3*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2* e))) + 32*(6*A - 11*I*B)*a^3*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12*(9*I*A + 20*B)*a^3*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f* x + 2*e))) + 32*(6*I*A + 11*B)*a^3*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2 *f*x + 2*e))) - 30*((3*A - 4*I*B)*a^3*cos(6*f*x + 6*e) + 3*(3*A - 4*I*B)*a ^3*cos(4*f*x + 4*e) + 3*(3*A - 4*I*B)*a^3*cos(2*f*x + 2*e) - (-3*I*A - 4*B )*a^3*sin(6*f*x + 6*e) - 3*(-3*I*A - 4*B)*a^3*sin(4*f*x + 4*e) - 3*(-3*I*A - 4*B)*a^3*sin(2*f*x + 2*e) + (3*A - 4*I*B)*a^3)*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), co s(2*f*x + 2*e))) + 1) - 30*((3*A - 4*I*B)*a^3*cos(6*f*x + 6*e) + 3*(3*A - 4*I*B)*a^3*cos(4*f*x + 4*e) + 3*(3*A - 4*I*B)*a^3*cos(2*f*x + 2*e) - (-3*I *A - 4*B)*a^3*sin(6*f*x + 6*e) - 3*(-3*I*A - 4*B)*a^3*sin(4*f*x + 4*e) - 3 *(-3*I*A - 4*B)*a^3*sin(2*f*x + 2*e) + (3*A - 4*I*B)*a^3)*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) + 12*(8*(A - I*B)*a^3*cos(6*f*x + 6*e) + 24 *(A - I*B)*a^3*cos(4*f*x + 4*e) + 24*(A - I*B)*a^3*cos(2*f*x + 2*e) + 8*(I *A + B)*a^3*sin(6*f*x + 6*e) + 24*(I*A + B)*a^3*sin(4*f*x + 4*e) + 24*(I*A + B)*a^3*sin(2*f*x + 2*e) + 5*(3*A - 4*I*B)*a^3)*cos(1/2*arctan2(sin(2*f* x + 2*e), cos(2*f*x + 2*e))) + 15*((-3*I*A - 4*B)*a^3*cos(6*f*x + 6*e) + 3 *(-3*I*A - 4*B)*a^3*cos(4*f*x + 4*e) + 3*(-3*I*A - 4*B)*a^3*cos(2*f*x +...
\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]