\(\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx\) [817]

Optimal result

Integrand size = 45, antiderivative size = 267 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=-\frac {5 i a^{7/2} A c^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {5 a^3 A c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f} \]

[Out]

-5/8*I*a^(7/2)*A*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))/f+5/16*a^3*
A*c^3*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)*tan(f*x+e)/f+5/24*a^2*A*c^2*tan(f*x+e)*(a+I*a*tan(f*x+
e))^(3/2)*(c-I*c*tan(f*x+e))^(3/2)/f+1/6*a*A*c*tan(f*x+e)*(a+I*a*tan(f*x+e))^(5/2)*(c-I*c*tan(f*x+e))^(5/2)/f+
1/7*B*(a+I*a*tan(f*x+e))^(7/2)*(c-I*c*tan(f*x+e))^(7/2)/f

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 267, 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, 38, 65, 223, 209} \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=-\frac {5 i a^{7/2} A c^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {5 a^3 A c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f} \]

[In]

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

[Out]

(((-5*I)/8)*a^(7/2)*A*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])
])/f + (5*a^3*A*c^3*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(16*f) + (5*a^2*A*c^2*
Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) + (a*A*c*Tan[e + f*x]*(a + I*a*
Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(6*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x
])^(7/2))/(7*f)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

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 (a+i a x)^{5/2} (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}+\frac {(a A c) \text {Subst}\left (\int (a+i a x)^{5/2} (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}+\frac {\left (5 a^2 A c^2\right ) \text {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{6 f} \\ & = \frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}+\frac {\left (5 a^3 A c^3\right ) \text {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = \frac {5 a^3 A c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}+\frac {\left (5 a^4 A c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{16 f} \\ & = \frac {5 a^3 A c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {\left (5 i a^3 A c^4\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 )}{8 f} \\ & = \frac {5 a^3 A c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {\left (5 i a^3 A c^4\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 )}{8 f} \\ & = -\frac {5 i a^{7/2} A 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 )}{8 f}+\frac {5 a^3 A c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.56 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.60 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {a^{7/2} c^4 \left (-\frac {6720 A \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) (i+\tan (e+f x))}{\sqrt {1-i \tan (e+f x)}}+\frac {\sqrt {a} \sec ^8(e+f x) (1536 B+1981 A \sin (2 (e+f x))+700 A \sin (4 (e+f x))+105 A \sin (6 (e+f x)))}{\sqrt {a+i a \tan (e+f x)}}\right )}{10752 f \sqrt {c-i c \tan (e+f x)}} \]

[In]

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

[Out]

(a^(7/2)*c^4*((-6720*A*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]*Sec[e + f*x]^8*(1536*B + 1981*A*Sin[2*(e + f*x)] + 700*A*Sin[4*(e + f*x)] + 105*A*Sin[6*
(e + f*x)]))/Sqrt[a + I*a*Tan[e + f*x]]))/(10752*f*Sqrt[c - I*c*Tan[e + f*x]])

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.01

method	result	size
parts	\(\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{3} \left (8 \tan \left (f x +e \right )^{5} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+26 \tan \left (f x +e \right )^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+15 a c \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right )+33 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\right )}{48 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}+\frac {B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (\tan \left (f x +e \right )^{4}+2 \tan \left (f x +e \right )^{2}+1\right )}{7 f}\)	\(269\)
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{3} \left (48 B \tan \left (f x +e \right )^{6} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+56 A \tan \left (f x +e \right )^{5} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+144 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+182 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+144 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+105 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +231 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+48 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{336 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(314\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{3} \left (48 B \tan \left (f x +e \right )^{6} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+56 A \tan \left (f x +e \right )^{5} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+144 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+182 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+144 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+105 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +231 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+48 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{336 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(314\)

[In]

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

[Out]

1/48*A/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^3*c^3*(8*tan(f*x+e)^5*(a*c*(1+tan(f*x+e)^2))
^(1/2)*(a*c)^(1/2)+26*tan(f*x+e)^3*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+15*a*c*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))+33*tan(f*x+e)*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c*(1
+tan(f*x+e)^2))^(1/2)/(a*c)^(1/2)+1/7*B/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^3*c^3*(1+ta
n(f*x+e)^2)*(tan(f*x+e)^4+2*tan(f*x+e)^2+1)

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. 693 vs. \(2 (207) = 414\).

Time = 0.27 (sec) , antiderivative size = 693, normalized size of antiderivative = 2.60 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {105 \, \sqrt {\frac {A^{2} a^{7} c^{7}}{f^{2}}} {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (A a^{3} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + A a^{3} 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}} - \sqrt {\frac {A^{2} a^{7} c^{7}}{f^{2}}} {\left (i \, f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, f\right )}\right )}}{A a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + A a^{3} c^{3}}\right ) - 105 \, \sqrt {\frac {A^{2} a^{7} c^{7}}{f^{2}}} {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (A a^{3} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + A a^{3} 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}} - \sqrt {\frac {A^{2} a^{7} c^{7}}{f^{2}}} {\left (-i \, f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, f\right )}\right )}}{A a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + A a^{3} c^{3}}\right ) + 4 \, {\left (-105 i \, A a^{3} c^{3} e^{\left (13 i \, f x + 13 i \, e\right )} - 700 i \, A a^{3} c^{3} e^{\left (11 i \, f x + 11 i \, e\right )} - 1981 i \, A a^{3} c^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 3072 \, B a^{3} c^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 1981 i \, A a^{3} c^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 700 i \, A a^{3} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 105 i \, A a^{3} 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}}}{672 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + 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))^(7/2),x, algorithm="fricas")

[Out]

1/672*(105*sqrt(A^2*a^7*c^7/f^2)*(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*f*x + 8*I*
e) + 20*f*e^(6*I*f*x + 6*I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x + 2*I*e) + f)*log(4*(2*(A*a^3*c^3*e^
(3*I*f*x + 3*I*e) + A*a^3*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)) - sqrt(A^2*a^7*c^7/f^2)*(I*f*e^(2*I*f*x + 2*I*e) - I*f))/(A*a^3*c^3*e^(2*I*f*x + 2*I*e) + A*a^3*c^3)) - 1
05*sqrt(A^2*a^7*c^7/f^2)*(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*f*x + 8*I*e) + 20*
f*e^(6*I*f*x + 6*I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x + 2*I*e) + f)*log(4*(2*(A*a^3*c^3*e^(3*I*f*x
+ 3*I*e) + A*a^3*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)) - s
qrt(A^2*a^7*c^7/f^2)*(-I*f*e^(2*I*f*x + 2*I*e) + I*f))/(A*a^3*c^3*e^(2*I*f*x + 2*I*e) + A*a^3*c^3)) + 4*(-105*
I*A*a^3*c^3*e^(13*I*f*x + 13*I*e) - 700*I*A*a^3*c^3*e^(11*I*f*x + 11*I*e) - 1981*I*A*a^3*c^3*e^(9*I*f*x + 9*I*
e) + 3072*B*a^3*c^3*e^(7*I*f*x + 7*I*e) + 1981*I*A*a^3*c^3*e^(5*I*f*x + 5*I*e) + 700*I*A*a^3*c^3*e^(3*I*f*x +
3*I*e) + 105*I*A*a^3*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)))
/(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*f*x + 8*I*e) + 20*f*e^(6*I*f*x + 6*I*e) +
15*f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x + 2*I*e) + f)

Sympy [F(-1)]

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

[In]

integrate((a+I*a*tan(f*x+e))**(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(7/2),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. 1901 vs. \(2 (207) = 414\).

Time = 2.74 (sec) , antiderivative size = 1901, normalized size of antiderivative = 7.12 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(420*A*a^3*c^3*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2800*A*a^3*c^3*cos(11/2*arctan2(sin(2*
f*x + 2*e), cos(2*f*x + 2*e))) + 7924*A*a^3*c^3*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12288*I
*B*a^3*c^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 7924*A*a^3*c^3*cos(5/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e))) - 2800*A*a^3*c^3*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 420*A*a^3*c^3*
cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 420*I*A*a^3*c^3*sin(13/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) + 2800*I*A*a^3*c^3*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 7924*I*A*a^3*c^3*sin
(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12288*B*a^3*c^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) - 7924*I*A*a^3*c^3*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2800*I*A*a^3*c^3*sin(3/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 420*I*A*a^3*c^3*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) + 210*(A*a^3*c^3*cos(14*f*x + 14*e) + 7*A*a^3*c^3*cos(12*f*x + 12*e) + 21*A*a^3*c^3*cos(10*f*x + 10*e) +
35*A*a^3*c^3*cos(8*f*x + 8*e) + 35*A*a^3*c^3*cos(6*f*x + 6*e) + 21*A*a^3*c^3*cos(4*f*x + 4*e) + 7*A*a^3*c^3*c
os(2*f*x + 2*e) + I*A*a^3*c^3*sin(14*f*x + 14*e) + 7*I*A*a^3*c^3*sin(12*f*x + 12*e) + 21*I*A*a^3*c^3*sin(10*f*
x + 10*e) + 35*I*A*a^3*c^3*sin(8*f*x + 8*e) + 35*I*A*a^3*c^3*sin(6*f*x + 6*e) + 21*I*A*a^3*c^3*sin(4*f*x + 4*e
) + 7*I*A*a^3*c^3*sin(2*f*x + 2*e) + A*a^3*c^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) + 210*(A*a^3*c^3*cos(14*f*x + 14*e) + 7*A*a^3*c^3*co
s(12*f*x + 12*e) + 21*A*a^3*c^3*cos(10*f*x + 10*e) + 35*A*a^3*c^3*cos(8*f*x + 8*e) + 35*A*a^3*c^3*cos(6*f*x +
6*e) + 21*A*a^3*c^3*cos(4*f*x + 4*e) + 7*A*a^3*c^3*cos(2*f*x + 2*e) + I*A*a^3*c^3*sin(14*f*x + 14*e) + 7*I*A*a
^3*c^3*sin(12*f*x + 12*e) + 21*I*A*a^3*c^3*sin(10*f*x + 10*e) + 35*I*A*a^3*c^3*sin(8*f*x + 8*e) + 35*I*A*a^3*c
^3*sin(6*f*x + 6*e) + 21*I*A*a^3*c^3*sin(4*f*x + 4*e) + 7*I*A*a^3*c^3*sin(2*f*x + 2*e) + A*a^3*c^3)*arctan2(co
s(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)
+ 105*(I*A*a^3*c^3*cos(14*f*x + 14*e) + 7*I*A*a^3*c^3*cos(12*f*x + 12*e) + 21*I*A*a^3*c^3*cos(10*f*x + 10*e)
+ 35*I*A*a^3*c^3*cos(8*f*x + 8*e) + 35*I*A*a^3*c^3*cos(6*f*x + 6*e) + 21*I*A*a^3*c^3*cos(4*f*x + 4*e) + 7*I*A*
a^3*c^3*cos(2*f*x + 2*e) - A*a^3*c^3*sin(14*f*x + 14*e) - 7*A*a^3*c^3*sin(12*f*x + 12*e) - 21*A*a^3*c^3*sin(10
*f*x + 10*e) - 35*A*a^3*c^3*sin(8*f*x + 8*e) - 35*A*a^3*c^3*sin(6*f*x + 6*e) - 21*A*a^3*c^3*sin(4*f*x + 4*e) -
7*A*a^3*c^3*sin(2*f*x + 2*e) + I*A*a^3*c^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) + 105*(-I*A*a^3*c^3*cos(14*f*x + 14*e) - 7*I*A*a^3*c^3*cos(12*f*x + 12*e) - 21*I*A*a^3*c^3*cos(10*f*x + 10*
e) - 35*I*A*a^3*c^3*cos(8*f*x + 8*e) - 35*I*A*a^3*c^3*cos(6*f*x + 6*e) - 21*I*A*a^3*c^3*cos(4*f*x + 4*e) - 7*I
*A*a^3*c^3*cos(2*f*x + 2*e) + A*a^3*c^3*sin(14*f*x + 14*e) + 7*A*a^3*c^3*sin(12*f*x + 12*e) + 21*A*a^3*c^3*sin
(10*f*x + 10*e) + 35*A*a^3*c^3*sin(8*f*x + 8*e) + 35*A*a^3*c^3*sin(6*f*x + 6*e) + 21*A*a^3*c^3*sin(4*f*x + 4*e
) + 7*A*a^3*c^3*sin(2*f*x + 2*e) - I*A*a^3*c^3)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + s
in(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*(-672*I*cos(14*f*x + 14*e) - 4704*I*cos(12*f*x + 12*e) - 14112*I*cos(10*f*x + 10*e)
- 23520*I*cos(8*f*x + 8*e) - 23520*I*cos(6*f*x + 6*e) - 14112*I*cos(4*f*x + 4*e) - 4704*I*cos(2*f*x + 2*e) + 6
72*sin(14*f*x + 14*e) + 4704*sin(12*f*x + 12*e) + 14112*sin(10*f*x + 10*e) + 23520*sin(8*f*x + 8*e) + 23520*si
n(6*f*x + 6*e) + 14112*sin(4*f*x + 4*e) + 4704*sin(2*f*x + 2*e) - 672*I))

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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