\(\int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx\) [821]

Optimal result

Integrand size = 45, antiderivative size = 283 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {5 a^{7/2} (3 i A+4 B) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {5 a^3 (3 i A+4 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c f}-\frac {5 a^2 (3 i A+4 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c f}-\frac {a (3 i A+4 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c f} \]

[Out]

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*tan(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3669, 79, 52, 65, 223, 209} \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {5 a^{7/2} (4 B+3 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {5 a^3 (4 B+3 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 c f}-\frac {5 a^2 (4 B+3 i A) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 c f}-\frac {a (4 B+3 i A) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 c f}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{f \sqrt {c-i c \tan (e+f x)}} \]

[In]

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

[Out]

(5*a^(7/2)*((3*I)*A + 4*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/
(Sqrt[c]*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (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) - (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) - (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)

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{5/2} (A+B x)}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {(a (3 A-4 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{f \sqrt {c-i c \tan (e+f x)}}-\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}-\frac {\left (5 a^2 (3 A-4 i B)\right ) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{3 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^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}-\frac {\left (5 a^3 (3 A-4 i B)\right ) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 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}-\frac {\left (5 a^4 (3 A-4 i B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 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}+\frac {\left (5 a^3 (3 i A+4 B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{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}+\frac {\left (5 a^3 (3 i A+4 B)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{f} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.51 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.65 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {a^{7/2} \left (\frac {30 (3 A-4 i B) \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) (i+\tan (e+f x))}{\sqrt {1-i \tan (e+f x)}}+\frac {\sqrt {a} (-i+\tan (e+f x)) \left (72 A-94 i B+(-21 i A-34 B) \tan (e+f x)+(3 A-10 i B) \tan ^2(e+f x)+2 B \tan ^3(e+f x)\right )}{\sqrt {a+i a \tan (e+f x)}}\right )}{6 f \sqrt {c-i c \tan (e+f x)}} \]

[In]

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

[Out]

(a^(7/2)*((30*(3*A - (4*I)*B)*ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt[2]*Sqrt[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]])

Maple [B] (verified)

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}}\)	\(688\)

[In]

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,method=_RETURNVERBOSE)

[Out]

-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+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*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+ta
n(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+tan(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)

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. 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 )}} \]

[In]

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")

[Out]

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)

Sympy [F(-1)]

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

[In]

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)

[Out]

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. 1329 vs. \(2 (219) = 438\).

Time = 0.71 (sec) , antiderivative size = 1329, normalized size of antiderivative = 4.70 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

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")

[Out]

-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), cos(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), co
s(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*arct
an2(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 + 2*e) + (3*A - 4*I*B)*a^3*sin(6*f*x + 6*e) + 3*(3*A - 4*I*B)*a^
3*sin(4*f*x + 4*e) + 3*(3*A - 4*I*B)*a^3*sin(2*f*x + 2*e) + (-3*I*A - 4*B)*a^3)*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*sin(1/2*arctan2(sin(
2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 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 + 2*e) - (3*A - 4*I*B)*a^3*sin(6*f*x + 6*e) - 3*(3*A - 4*I*B)*a^3*sin(
4*f*x + 4*e) - 3*(3*A - 4*I*B)*a^3*sin(2*f*x + 2*e) + (3*I*A + 4*B)*a^3)*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*sin(1/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) + 1) + 12*(8*(I*A + B)*a^3*cos(6*f*x + 6*e) + 24*(I*A + B)*a^3*cos(4*f*x + 4*e) + 24
*(I*A + B)*a^3*cos(2*f*x + 2*e) - 8*(A - I*B)*a^3*sin(6*f*x + 6*e) - 24*(A - I*B)*a^3*sin(4*f*x + 4*e) - 24*(A
- I*B)*a^3*sin(2*f*x + 2*e) + 5*(3*I*A + 4*B)*a^3)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt
(a)*sqrt(c)/((-72*I*c*cos(6*f*x + 6*e) - 216*I*c*cos(4*f*x + 4*e) - 216*I*c*cos(2*f*x + 2*e) + 72*c*sin(6*f*x
+ 6*e) + 216*c*sin(4*f*x + 4*e) + 216*c*sin(2*f*x + 2*e) - 72*I*c)*f)

Giac [F]

\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]

[In]

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")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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