\(\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx\) [820]

Optimal result

Integrand size = 45, antiderivative size = 272 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=-\frac {5 a^{7/2} (4 i A+3 B) \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {5 a^3 (4 i A+3 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {5 a^2 (4 i A+3 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f} \]

[Out]

-5/4*a^(7/2)*(4*I*A+3*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))*c^(1/2)/f+5
/8*a^3*(4*I*A+3*B)*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)/f+5/24*a^2*(4*I*A+3*B)*(c-I*c*tan(f*x+e))
^(1/2)*(a+I*a*tan(f*x+e))^(3/2)/f+1/12*a*(4*I*A+3*B)*(c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(5/2)/f+1/4*B
*(c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(7/2)/f

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 272, 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, 81, 52, 65, 223, 209} \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=-\frac {5 a^{7/2} \sqrt {c} (3 B+4 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {5 a^3 (3 B+4 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {5 a^2 (3 B+4 i A) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (3 B+4 i A) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f} \]

[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)*((4*I)*A + 3*B)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e +
f*x]])])/(4*f) + (5*a^3*((4*I)*A + 3*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (5*a^2*
((4*I)*A + 3*B)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(24*f) + (a*((4*I)*A + 3*B)*(a + I*a*
Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]])/(12*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e +
f*x]])/(4*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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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)}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}+\frac {(a (4 A-3 i B) c) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = \frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}+\frac {\left (5 a^2 (4 A-3 i B) c\right ) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{12 f} \\ & = \frac {5 a^2 (4 i A+3 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}+\frac {\left (5 a^3 (4 A-3 i B) c\right ) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = \frac {5 a^3 (4 i A+3 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {5 a^2 (4 i A+3 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}+\frac {\left (5 a^4 (4 A-3 i B) c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = \frac {5 a^3 (4 i A+3 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {5 a^2 (4 i A+3 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}-\frac {\left (5 a^3 (4 i A+3 B) c\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 )}{4 f} \\ & = \frac {5 a^3 (4 i A+3 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {5 a^2 (4 i A+3 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}-\frac {\left (5 a^3 (4 i A+3 B) c\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 )}{4 f} \\ & = -\frac {5 a^{7/2} (4 i A+3 B) \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 )}{4 f}+\frac {5 a^3 (4 i A+3 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {5 a^2 (4 i A+3 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{24 f}+\frac {a (4 i A+3 B) (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{12 f}+\frac {B (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.96 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.81 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=\frac {a^{7/2} c (i+\tan (e+f x)) \left (-30 (4 A-3 i B) \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}+\sqrt {a} \sqrt {1-i \tan (e+f x)} (-i+\tan (e+f x)) \left (88 i A+72 B-9 (4 A-5 i B) \tan (e+f x)-8 i (A-3 i B) \tan ^2(e+f x)-6 i B \tan ^3(e+f x)\right )\right )}{24 f \sqrt {1-i \tan (e+f x)} \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]

[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)*c*(I + Tan[e + f*x])*(-30*(4*A - (3*I)*B)*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]]*(-I + Tan[e + f*x])*((88*I)*A + 72*B - 9*(4*A - (5*I)*
B)*Tan[e + f*x] - (8*I)*(A - (3*I)*B)*Tan[e + f*x]^2 - (6*I)*B*Tan[e + f*x]^3)))/(24*f*Sqrt[1 - I*Tan[e + f*x]
]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{3} \left (-6 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-8 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-45 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +45 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-24 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+60 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +88 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-36 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+72 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{24 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(349\)
default	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{3} \left (-6 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-8 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}-45 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +45 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-24 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+60 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +88 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-36 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+72 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{24 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(349\)
parts	\(-\frac {A \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{3} \left (2 i \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-22 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-15 a c \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right )+9 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\right )}{6 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}-\frac {B \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{3} \left (2 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+15 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -15 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+8 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-24 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{8 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}\)	\(403\)

[In]

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

[Out]

1/24/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*a^3*(-6*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(
1/2)*tan(f*x+e)^3-8*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2-45*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+45*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*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+60*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+88*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)-36*A*(a*c)^(1/2)*(a*c*(1+tan
(f*x+e)^2))^(1/2)*tan(f*x+e)+72*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2)/(a*c*(1+tan(f*x+e)^2))
^(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. 604 vs. \(2 (206) = 412\).

Time = 0.27 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.22 \[ \int (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 (16 \, A^{2} - 24 i \, A B - 9 \, B^{2}\right )} a^{7} c}{f^{2}}} {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (-4 i \, A - 3 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-4 i \, A - 3 \, 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 (16 \, A^{2} - 24 i \, A B - 9 \, B^{2}\right )} a^{7} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-4 i \, A - 3 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-4 i \, A - 3 \, B\right )} a^{3}}\right ) - 15 \, \sqrt {\frac {{\left (16 \, A^{2} - 24 i \, A B - 9 \, B^{2}\right )} a^{7} c}{f^{2}}} {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left ({\left (-4 i \, A - 3 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-4 i \, A - 3 \, 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 (16 \, A^{2} - 24 i \, A B - 9 \, B^{2}\right )} a^{7} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-4 i \, A - 3 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-4 i \, A - 3 \, B\right )} a^{3}}\right ) + 4 \, {\left (3 \, {\left (-44 i \, A - 49 \, B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 73 \, {\left (-4 i \, A - 3 \, B\right )} a^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 55 \, {\left (-4 i \, A - 3 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (-4 i \, A - 3 \, 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}}}{48 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e)),x, algorithm="fricas")

[Out]

-1/48*(15*sqrt((16*A^2 - 24*I*A*B - 9*B^2)*a^7*c/f^2)*(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e
^(2*I*f*x + 2*I*e) + f)*log(4*(2*((-4*I*A - 3*B)*a^3*e^(3*I*f*x + 3*I*e) + (-4*I*A - 3*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((16*A^2 - 24*I*A*B - 9*B^2)*a^7*c/
f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((-4*I*A - 3*B)*a^3*e^(2*I*f*x + 2*I*e) + (-4*I*A - 3*B)*a^3)) - 15*sqrt((16
*A^2 - 24*I*A*B - 9*B^2)*a^7*c/f^2)*(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e)
+ f)*log(4*(2*((-4*I*A - 3*B)*a^3*e^(3*I*f*x + 3*I*e) + (-4*I*A - 3*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((16*A^2 - 24*I*A*B - 9*B^2)*a^7*c/f^2)*(f*e^(2*I*f*x
+ 2*I*e) - f))/((-4*I*A - 3*B)*a^3*e^(2*I*f*x + 2*I*e) + (-4*I*A - 3*B)*a^3)) + 4*(3*(-44*I*A - 49*B)*a^3*e^(
7*I*f*x + 7*I*e) + 73*(-4*I*A - 3*B)*a^3*e^(5*I*f*x + 5*I*e) + 55*(-4*I*A - 3*B)*a^3*e^(3*I*f*x + 3*I*e) + 15*
(-4*I*A - 3*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)))/(f*e^
(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)

Sympy [F(-1)]

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

[In]

integrate((c-I*c*tan(f*x+e))**(1/2)*(a+I*a*tan(f*x+e))**(7/2)*(A+B*tan(f*x+e)),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. 1345 vs. \(2 (206) = 412\).

Time = 1.06 (sec) , antiderivative size = 1345, normalized size of antiderivative = 4.94 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e)),x, algorithm="maxima")

[Out]

96*(12*(44*A - 49*I*B)*a^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 292*(4*A - 3*I*B)*a^3*cos(5/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 220*(4*A - 3*I*B)*a^3*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) + 60*(4*A - 3*I*B)*a^3*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12*(44*I*A + 49*B)
*a^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 292*(4*I*A + 3*B)*a^3*sin(5/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) + 220*(4*I*A + 3*B)*a^3*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 60*(4*
I*A + 3*B)*a^3*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 30*((4*A - 3*I*B)*a^3*cos(8*f*x + 8*e) +
4*(4*A - 3*I*B)*a^3*cos(6*f*x + 6*e) + 6*(4*A - 3*I*B)*a^3*cos(4*f*x + 4*e) + 4*(4*A - 3*I*B)*a^3*cos(2*f*x +
2*e) - (-4*I*A - 3*B)*a^3*sin(8*f*x + 8*e) - 4*(-4*I*A - 3*B)*a^3*sin(6*f*x + 6*e) - 6*(-4*I*A - 3*B)*a^3*sin
(4*f*x + 4*e) - 4*(-4*I*A - 3*B)*a^3*sin(2*f*x + 2*e) + (4*A - 3*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*((4*A - 3*I*B)*a^3*c
os(8*f*x + 8*e) + 4*(4*A - 3*I*B)*a^3*cos(6*f*x + 6*e) + 6*(4*A - 3*I*B)*a^3*cos(4*f*x + 4*e) + 4*(4*A - 3*I*B
)*a^3*cos(2*f*x + 2*e) - (-4*I*A - 3*B)*a^3*sin(8*f*x + 8*e) - 4*(-4*I*A - 3*B)*a^3*sin(6*f*x + 6*e) - 6*(-4*I
*A - 3*B)*a^3*sin(4*f*x + 4*e) - 4*(-4*I*A - 3*B)*a^3*sin(2*f*x + 2*e) + (4*A - 3*I*B)*a^3)*arctan2(cos(1/2*ar
ctan2(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*((
-4*I*A - 3*B)*a^3*cos(8*f*x + 8*e) + 4*(-4*I*A - 3*B)*a^3*cos(6*f*x + 6*e) + 6*(-4*I*A - 3*B)*a^3*cos(4*f*x +
4*e) + 4*(-4*I*A - 3*B)*a^3*cos(2*f*x + 2*e) + (4*A - 3*I*B)*a^3*sin(8*f*x + 8*e) + 4*(4*A - 3*I*B)*a^3*sin(6*
f*x + 6*e) + 6*(4*A - 3*I*B)*a^3*sin(4*f*x + 4*e) + 4*(4*A - 3*I*B)*a^3*sin(2*f*x + 2*e) + (-4*I*A - 3*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*((4*I*A + 3*B)*a^3*cos(8*f*x + 8*e)
+ 4*(4*I*A + 3*B)*a^3*cos(6*f*x + 6*e) + 6*(4*I*A + 3*B)*a^3*cos(4*f*x + 4*e) + 4*(4*I*A + 3*B)*a^3*cos(2*f*x
+ 2*e) - (4*A - 3*I*B)*a^3*sin(8*f*x + 8*e) - 4*(4*A - 3*I*B)*a^3*sin(6*f*x + 6*e) - 6*(4*A - 3*I*B)*a^3*sin(4
*f*x + 4*e) - 4*(4*A - 3*I*B)*a^3*sin(2*f*x + 2*e) + (4*I*A + 3*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))*sqrt(a)*sqrt(c)/(f*(-4608*I*cos(8*f*x + 8*e) - 18432*I*cos(6*f*x + 6*e) - 27648
*I*cos(4*f*x + 4*e) - 18432*I*cos(2*f*x + 2*e) + 4608*sin(8*f*x + 8*e) + 18432*sin(6*f*x + 6*e) + 27648*sin(4*
f*x + 4*e) + 18432*sin(2*f*x + 2*e) - 4608*I))

Giac [F(-1)]

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

[In]

integrate((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[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)