3.1135 \(\int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=351 \[ \frac{d \left (9 i c^2 d+2 c^3+88 c d^2-45 i d^3\right )}{8 a^2 f (c-i d)^2 (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^{3/2}}+\frac{\left (2 i c^2-14 c d-47 i d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{8 a^2 f (c+i d)^{9/2}}+\frac{-9 d+2 i c}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{4 a^2 f (c-i d)^{5/2}}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \]

[Out]

((-I/4)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*(c - I*d)^(5/2)*f) + (((2*I)*c^2 - 14*c*d - (47*
I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(9/2)*f) + (d*(6*c^2 + (27*I)*c*d +
49*d^2))/(24*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*I)*c - 9*d)/(8*a^2*(c + I*d)^2*f*(1
+ I*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x]
)^(3/2)) + (d*(2*c^3 + (9*I)*c^2*d + 88*c*d^2 - (45*I)*d^3))/(8*a^2*(c - I*d)^2*(c + I*d)^4*f*Sqrt[c + d*Tan[e
+ f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.968385, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {3559, 3596, 3529, 3539, 3537, 63, 208} \[ \frac{d \left (9 i c^2 d+2 c^3+88 c d^2-45 i d^3\right )}{8 a^2 f (c-i d)^2 (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^{3/2}}+\frac{\left (2 i c^2-14 c d-47 i d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{8 a^2 f (c+i d)^{9/2}}+\frac{-9 d+2 i c}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{4 a^2 f (c-i d)^{5/2}}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-I/4)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*(c - I*d)^(5/2)*f) + (((2*I)*c^2 - 14*c*d - (47*
I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(9/2)*f) + (d*(6*c^2 + (27*I)*c*d +
49*d^2))/(24*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*I)*c - 9*d)/(8*a^2*(c + I*d)^2*f*(1
+ I*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x]
)^(3/2)) + (d*(2*c^3 + (9*I)*c^2*d + 88*c*d^2 - (45*I)*d^3))/(8*a^2*(c - I*d)^2*(c + I*d)^4*f*Sqrt[c + d*Tan[e
+ f*x]])

Rule 3559

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3596

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rule 3529

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

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 63

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 208

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

Rubi steps

\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a (4 i c-11 d)-\frac{7}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx}{4 a^2 (i c-d)}\\ &=\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a^2 \left (4 c^2+18 i c d-49 d^2\right )-\frac{5}{2} a^2 (2 c+9 i d) d \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx}{8 a^4 (c+i d)^2}\\ &=\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a^2 \left (4 c^3+18 i c^2 d-39 c d^2+45 i d^3\right )-\frac{1}{2} a^2 d \left (6 c^2+27 i c d+49 d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{8 a^4 (c+i d)^2 \left (c^2+d^2\right )}\\ &=\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{-\frac{1}{2} a^2 \left (4 c^4+18 i c^3 d-33 c^2 d^2+72 i c d^3+49 d^4\right )-\frac{1}{2} a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{8 a^4 (c+i d)^2 \left (c^2+d^2\right )^2}\\ &=\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}+\frac{\int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{8 a^2 (c-i d)^2}+\frac{\left (2 c^2+14 i c d-47 d^2\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{16 a^2 (c+i d)^4}\\ &=\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{8 a^2 (c-i d)^2 f}-\frac{\left (i \left (2 c^2+14 i c d-47 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{16 a^2 (c+i d)^4 f}\\ &=\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-\frac{i c}{d}+\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{4 a^2 (c-i d)^2 d f}-\frac{\left (2 c^2+14 i c d-47 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i c}{d}-\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{8 a^2 (c+i d)^4 d f}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{4 a^2 (c-i d)^{5/2} f}-\frac{\left (14 c d-i \left (2 c^2-47 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{8 a^2 (c+i d)^{9/2} f}+\frac{d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac{2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}

Mathematica [B]  time = 9.40147, size = 1004, normalized size = 2.86 \[ \frac{\sec ^2(e+f x) (\cos (2 e)+i \sin (2 e)) \left (\frac{2 \left (-45 i d^4+88 c d^3+9 i c^2 d^2+2 c^3 d\right ) \left (\frac{\tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right )}{2 \sqrt{-c-i d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right )}{2 \sqrt{i d-c}}\right ) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left (\tan ^2(e+f x)+1\right )}-\frac{i \left (4 c^4+18 i d c^3-33 d^2 c^2+72 i d^3 c+49 d^4\right ) \left (\frac{\tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right )}{\sqrt{-c-i d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right )}{\sqrt{i d-c}}\right ) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left (\tan ^2(e+f x)+1\right )}\right ) (\cos (f x)+i \sin (f x))^2}{16 (c-i d)^2 (c+i d)^4 f (i \tan (e+f x) a+a)^2}+\frac{\sec ^2(e+f x) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac{i (4 c+15 i d) \cos (2 f x)}{16 (c+i d)^4}+\frac{\left (9 i \cos (e) c^4-24 d \cos (e) c^3+9 i d \sin (e) c^3+75 i d^2 \cos (e) c^2-24 d^2 \sin (e) c^2+458 d^3 \cos (e) c+75 i d^3 \sin (e) c-192 i d^4 \cos (e)+10 d^4 \sin (e)\right ) \left (\frac{1}{48} \cos (2 e)+\frac{1}{48} i \sin (2 e)\right )}{(c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e))}+\frac{\cos (4 f x) \left (\frac{1}{16} i \cos (2 e)+\frac{1}{16} \sin (2 e)\right )}{(c+i d)^3}+\frac{(4 c+15 i d) \sin (2 f x)}{16 (c+i d)^4}+\frac{\left (\frac{1}{16} \cos (2 e)-\frac{1}{16} i \sin (2 e)\right ) \sin (4 f x)}{(c+i d)^3}-\frac{4 \left (\frac{3}{2} \cos (2 e-f x) d^5-\frac{3}{2} \cos (2 e+f x) d^5+\frac{3}{2} i \sin (2 e-f x) d^5-\frac{3}{2} i \sin (2 e+f x) d^5+\frac{7}{2} i c \cos (2 e-f x) d^4-\frac{7}{2} i c \cos (2 e+f x) d^4-\frac{7}{2} c \sin (2 e-f x) d^4+\frac{7}{2} c \sin (2 e+f x) d^4\right )}{3 (c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} \cos (2 e) d^5+\frac{2}{3} i \sin (2 e) d^5}{(c-i d)^2 (c+i d)^4 (c \cos (e+f x)+d \sin (e+f x))^2}\right ) (\cos (f x)+i \sin (f x))^2}{f (i \tan (e+f x) a+a)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((I/16)*(4*c +
(15*I)*d)*Cos[2*f*x])/(c + I*d)^4 + (((9*I)*c^4*Cos[e] - 24*c^3*d*Cos[e] + (75*I)*c^2*d^2*Cos[e] + 458*c*d^3*
Cos[e] - (192*I)*d^4*Cos[e] + (9*I)*c^3*d*Sin[e] - 24*c^2*d^2*Sin[e] + (75*I)*c*d^3*Sin[e] + 10*d^4*Sin[e])*(C
os[2*e]/48 + (I/48)*Sin[2*e]))/((c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + (Cos[4*f*x]*((I/16)*Cos[2*e]
+ Sin[2*e]/16))/(c + I*d)^3 + ((4*c + (15*I)*d)*Sin[2*f*x])/(16*(c + I*d)^4) + ((Cos[2*e]/16 - (I/16)*Sin[2*e]
)*Sin[4*f*x])/(c + I*d)^3 + ((2*d^5*Cos[2*e])/3 + ((2*I)/3)*d^5*Sin[2*e])/((c - I*d)^2*(c + I*d)^4*(c*Cos[e +
f*x] + d*Sin[e + f*x])^2) - (4*(((7*I)/2)*c*d^4*Cos[2*e - f*x] + (3*d^5*Cos[2*e - f*x])/2 - ((7*I)/2)*c*d^4*Co
s[2*e + f*x] - (3*d^5*Cos[2*e + f*x])/2 - (7*c*d^4*Sin[2*e - f*x])/2 + ((3*I)/2)*d^5*Sin[2*e - f*x] + (7*c*d^4
*Sin[2*e + f*x])/2 - ((3*I)/2)*d^5*Sin[2*e + f*x]))/(3*(c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])*(c*Cos[e
+ f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]^2*(Cos[2*e] + I*Sin[2*e])*(Cos[f*x] +
I*Sin[f*x])^2*(((-I)*(4*c^4 + (18*I)*c^3*d - 33*c^2*d^2 + (72*I)*c*d^3 + 49*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f
*x]]/Sqrt[-c - I*d]]/Sqrt[-c - I*d] - ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/Sqrt[-c + I*d])*Sec[e +
f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2)) + (2*(2*c^3*d + (9*I)*c^2*
d^2 + 88*c*d^3 - (45*I)*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/(2*Sqrt[-c - I*d]) + ArcTan[Sqrt
[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/(2*Sqrt[-c + I*d]))*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] +
d*Sin[e + f*x])*(1 + Tan[e + f*x]^2))))/(16*(c - I*d)^2*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])^2)

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Maple [B]  time = 0.09, size = 2312, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x)

[Out]

5/2/f/a^2*d^3/(I*d-c)^(5/2)/(c+I*d)^5*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2-5/4/f/a^2*d/(I*d-c)^(5/
2)/(c+I*d)^5*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^4+8/f/a^2*d^3/(I*d-c)^2/(c+I*d)^5/(c+d*tan(f*x+e))
^(1/2)*c^2+2/3/f/a^2*d^3/(c+I*d)^4/(I*d-c)^2/(c+d*tan(f*x+e))^(3/2)*c^2+1/4*I/f/a^2/(I*d-c)^(5/2)/(c+I*d)^5*ar
ctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^5-23/8*I/f/a^2*d^4/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^
2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^4+11/8*I/f/a^2*d^6/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I
*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^2+15/8*I/f/a^2*d^6/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+
c^2)*(c+d*tan(f*x+e))^(3/2)*c+57/8*I/f/a^2*d^2/(I*d-c)^2/(c+I*d)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c
+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^5+4*I/f/a^2*d^4/(I*d-c)^2/(c+I*d)^5/(c+d*tan(f*x+e))^(1/2)*c-5/2*I/f/a^
2*d^2/(I*d-c)^(5/2)/(c+I*d)^5*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^3+5/4*I/f/a^2*d^4/(I*d-c)^(5/2)/(
c+I*d)^5*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c-13/8/f/a^2*d^7/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))
^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)-47/8/f/a^2*d^7/(I*d-c)^2/(c+I*d)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1
/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))-9/8/f/a^2*d^3/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^
2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c^4+7/2/f/a^2*d^3/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*
d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^5-15/8/f/a^2*d^3/(I*d-c)^2/(c+I*d)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan(
(c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^4+1/4/f/a^2*d/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d
+c^2)*(c+d*tan(f*x+e))^(3/2)*c^6-3/f/a^2*d^5/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d
*tan(f*x+e))^(3/2)*c^2-1/4/f/a^2*d/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e
))^(1/2)*c^7+31/4/f/a^2*d^5/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2
)*c^3+4/f/a^2*d^7/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c+15/8*I
/f/a^2*d^8/(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)+2/3/f/a^2*d^5/(
c+I*d)^4/(I*d-c)^2/(c+d*tan(f*x+e))^(3/2)-1/4/f/a^2*d^5/(I*d-c)^(5/2)/(c+I*d)^5*arctan((c+d*tan(f*x+e))^(1/2)/
(I*d-c)^(1/2))+4/f/a^2*d^5/(I*d-c)^2/(c+I*d)^5/(c+d*tan(f*x+e))^(1/2)+61/8*I/f/a^2*d^6/(I*d-c)^2/(c+I*d)^5/(-d
^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c+15*I/f/a^2*d^4/(I*d-c)^2/(c+I*d
)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3+15/4*I/f/a^2*d^4/(I*d-
c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c^3-1/4*I/f/a^2/(I*d-c)^2/(c+I*
d)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^7+2/f/a^2*d/(I*d-c)^2/(
c+I*d)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^6+15/8*I/f/a^2*d^2/
(I*d-c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c^5-19/8*I/f/a^2*d^2/(I*d-
c)^2/(c+I*d)^5/(-I*d+d*tan(f*x+e))^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^6-39/4/f/a^2*d^5/(I*d-c)^2/(c
+I*d)^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 62.2461, size = 7791, normalized size = 22.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

((12*(a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) + (24*a^2*c^8 +
48*I*a^2*c^7*d + 48*a^2*c^6*d^2 + 144*I*a^2*c^5*d^3 + 144*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 + 48*I*a^2*c*d^7 - 2
4*a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (12*a^2*c^8 + 48*I*a^2*c^7*d - 48*a^2*c^6*d^2 + 48*I*a^2*c^5*d^3 - 120*a^2*
c^4*d^4 - 48*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 - 48*I*a^2*c*d^7 + 12*a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(-I/((16
*I*a^4*c^5 + 80*a^4*c^4*d - 160*I*a^4*c^3*d^2 - 160*a^4*c^2*d^3 + 80*I*a^4*c*d^4 + 16*a^4*d^5)*f^2))*log((((8*
I*a^2*c^3 + 24*a^2*c^2*d - 24*I*a^2*c*d^2 - 8*a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (8*I*a^2*c^3 + 24*a^2*c^2*d - 2
4*I*a^2*c*d^2 - 8*a^2*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-
I/((16*I*a^4*c^5 + 80*a^4*c^4*d - 160*I*a^4*c^3*d^2 - 160*a^4*c^2*d^3 + 80*I*a^4*c*d^4 + 16*a^4*d^5)*f^2)) + 2
*(c - I*d)*e^(2*I*f*x + 2*I*e) + 2*c)*e^(-2*I*f*x - 2*I*e)) - (12*(a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4
*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) + (24*a^2*c^8 + 48*I*a^2*c^7*d + 48*a^2*c^6*d^2 + 144*I*a^2*c^5*
d^3 + 144*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 + 48*I*a^2*c*d^7 - 24*a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (12*a^2*c^8 +
48*I*a^2*c^7*d - 48*a^2*c^6*d^2 + 48*I*a^2*c^5*d^3 - 120*a^2*c^4*d^4 - 48*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 - 48*
I*a^2*c*d^7 + 12*a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(-I/((16*I*a^4*c^5 + 80*a^4*c^4*d - 160*I*a^4*c^3*d^2 - 1
60*a^4*c^2*d^3 + 80*I*a^4*c*d^4 + 16*a^4*d^5)*f^2))*log((((-8*I*a^2*c^3 - 24*a^2*c^2*d + 24*I*a^2*c*d^2 + 8*a^
2*d^3)*f*e^(2*I*f*x + 2*I*e) + (-8*I*a^2*c^3 - 24*a^2*c^2*d + 24*I*a^2*c*d^2 + 8*a^2*d^3)*f)*sqrt(((c - I*d)*e
^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-I/((16*I*a^4*c^5 + 80*a^4*c^4*d - 160*I*a^4*c^3
*d^2 - 160*a^4*c^2*d^3 + 80*I*a^4*c*d^4 + 16*a^4*d^5)*f^2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) + 2*c)*e^(-2*I*f
*x - 2*I*e)) - (12*(a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) +
(24*a^2*c^8 + 48*I*a^2*c^7*d + 48*a^2*c^6*d^2 + 144*I*a^2*c^5*d^3 + 144*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 + 48*I
*a^2*c*d^7 - 24*a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (12*a^2*c^8 + 48*I*a^2*c^7*d - 48*a^2*c^6*d^2 + 48*I*a^2*c^5*
d^3 - 120*a^2*c^4*d^4 - 48*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 - 48*I*a^2*c*d^7 + 12*a^2*d^8)*f*e^(4*I*f*x + 4*I*e)
)*sqrt((4*I*c^4 - 56*c^3*d - 384*I*c^2*d^2 + 1316*c*d^3 + 2209*I*d^4)/((-64*I*a^4*c^9 + 576*a^4*c^8*d + 2304*I
*a^4*c^7*d^2 - 5376*a^4*c^6*d^3 - 8064*I*a^4*c^5*d^4 + 8064*a^4*c^4*d^5 + 5376*I*a^4*c^3*d^6 - 2304*a^4*c^2*d^
7 - 576*I*a^4*c*d^8 + 64*a^4*d^9)*f^2))*log((2*c^3 + 16*I*c^2*d - 61*c*d^2 - 47*I*d^3 + ((8*I*a^2*c^5 - 40*a^2
*c^4*d - 80*I*a^2*c^3*d^2 + 80*a^2*c^2*d^3 + 40*I*a^2*c*d^4 - 8*a^2*d^5)*f*e^(2*I*f*x + 2*I*e) + (8*I*a^2*c^5
- 40*a^2*c^4*d - 80*I*a^2*c^3*d^2 + 80*a^2*c^2*d^3 + 40*I*a^2*c*d^4 - 8*a^2*d^5)*f)*sqrt(((c - I*d)*e^(2*I*f*x
+ 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((4*I*c^4 - 56*c^3*d - 384*I*c^2*d^2 + 1316*c*d^3 + 2209*I
*d^4)/((-64*I*a^4*c^9 + 576*a^4*c^8*d + 2304*I*a^4*c^7*d^2 - 5376*a^4*c^6*d^3 - 8064*I*a^4*c^5*d^4 + 8064*a^4*
c^4*d^5 + 5376*I*a^4*c^3*d^6 - 2304*a^4*c^2*d^7 - 576*I*a^4*c*d^8 + 64*a^4*d^9)*f^2)) + (2*c^3 + 14*I*c^2*d -
47*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-8*I*a^2*c^5 + 40*a^2*c^4*d + 80*I*a^2*c^3*d^2 - 80*a^2*
c^2*d^3 - 40*I*a^2*c*d^4 + 8*a^2*d^5)*f)) + (12*(a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2
*d^8)*f*e^(8*I*f*x + 8*I*e) + (24*a^2*c^8 + 48*I*a^2*c^7*d + 48*a^2*c^6*d^2 + 144*I*a^2*c^5*d^3 + 144*I*a^2*c^
3*d^5 - 48*a^2*c^2*d^6 + 48*I*a^2*c*d^7 - 24*a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (12*a^2*c^8 + 48*I*a^2*c^7*d - 4
8*a^2*c^6*d^2 + 48*I*a^2*c^5*d^3 - 120*a^2*c^4*d^4 - 48*I*a^2*c^3*d^5 - 48*a^2*c^2*d^6 - 48*I*a^2*c*d^7 + 12*a
^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt((4*I*c^4 - 56*c^3*d - 384*I*c^2*d^2 + 1316*c*d^3 + 2209*I*d^4)/((-64*I*a^4
*c^9 + 576*a^4*c^8*d + 2304*I*a^4*c^7*d^2 - 5376*a^4*c^6*d^3 - 8064*I*a^4*c^5*d^4 + 8064*a^4*c^4*d^5 + 5376*I*
a^4*c^3*d^6 - 2304*a^4*c^2*d^7 - 576*I*a^4*c*d^8 + 64*a^4*d^9)*f^2))*log((2*c^3 + 16*I*c^2*d - 61*c*d^2 - 47*I
*d^3 + ((-8*I*a^2*c^5 + 40*a^2*c^4*d + 80*I*a^2*c^3*d^2 - 80*a^2*c^2*d^3 - 40*I*a^2*c*d^4 + 8*a^2*d^5)*f*e^(2*
I*f*x + 2*I*e) + (-8*I*a^2*c^5 + 40*a^2*c^4*d + 80*I*a^2*c^3*d^2 - 80*a^2*c^2*d^3 - 40*I*a^2*c*d^4 + 8*a^2*d^5
)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((4*I*c^4 - 56*c^3*d - 384*
I*c^2*d^2 + 1316*c*d^3 + 2209*I*d^4)/((-64*I*a^4*c^9 + 576*a^4*c^8*d + 2304*I*a^4*c^7*d^2 - 5376*a^4*c^6*d^3 -
8064*I*a^4*c^5*d^4 + 8064*a^4*c^4*d^5 + 5376*I*a^4*c^3*d^6 - 2304*a^4*c^2*d^7 - 576*I*a^4*c*d^8 + 64*a^4*d^9)
*f^2)) + (2*c^3 + 14*I*c^2*d - 47*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-8*I*a^2*c^5 + 40*a^2*c^4
*d + 80*I*a^2*c^3*d^2 - 80*a^2*c^2*d^3 - 40*I*a^2*c*d^4 + 8*a^2*d^5)*f)) + (3*I*c^5 - 3*c^4*d + 6*I*c^3*d^2 -
6*c^2*d^3 + 3*I*c*d^4 - 3*d^5 + (9*I*c^5 - 6*c^4*d + 114*I*c^3*d^2 + 632*c^2*d^3 - 735*I*c*d^4 - 202*d^5)*e^(8
*I*f*x + 8*I*e) + (30*I*c^5 - 45*c^4*d + 276*I*c^3*d^2 + 1090*c^2*d^3 - 402*I*c*d^4 + 103*d^5)*e^(6*I*f*x + 6*
I*e) + (36*I*c^5 - 75*c^4*d + 192*I*c^3*d^2 + 386*c^2*d^3 + 348*I*c*d^4 + 269*d^5)*e^(4*I*f*x + 4*I*e) + (18*I
*c^5 - 39*c^4*d + 36*I*c^3*d^2 - 78*c^2*d^3 + 18*I*c*d^4 - 39*d^5)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I
*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))/(48*(a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*
d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) + (96*a^2*c^8 + 192*I*a^2*c^7*d + 192*a^2*c^6*d^2 + 576*I*a^2*c^5*d^3 + 5
76*I*a^2*c^3*d^5 - 192*a^2*c^2*d^6 + 192*I*a^2*c*d^7 - 96*a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (48*a^2*c^8 + 192*I
*a^2*c^7*d - 192*a^2*c^6*d^2 + 192*I*a^2*c^5*d^3 - 480*a^2*c^4*d^4 - 192*I*a^2*c^3*d^5 - 192*a^2*c^2*d^6 - 192
*I*a^2*c*d^7 + 48*a^2*d^8)*f*e^(4*I*f*x + 4*I*e))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))**2/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.7767, size = 960, normalized size = 2.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-2*d^3*(2*(2*I*c^2 - 14*c*d - 47*I*d^2)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x
+ e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c
+ 8*sqrt(c^2 + d^2))))/((8*a^2*c^4*d^3*f + 32*I*a^2*c^3*d^4*f - 48*a^2*c^2*d^5*f - 32*I*a^2*c*d^6*f + 8*a^2*d^
7*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) - (12*(d*tan(f*x + e) + c)*c + c^2 + (-6*
I*d*tan(f*x + e) - 6*I*c)*d + d^2)/((3*a^2*c^6*f + 6*I*a^2*c^5*d*f + 3*a^2*c^4*d^2*f + 12*I*a^2*c^3*d^3*f - 3*
a^2*c^2*d^4*f + 6*I*a^2*c*d^5*f - 3*a^2*d^6*f)*(d*tan(f*x + e) + c)^(3/2)) + 2*arctan(4*(sqrt(d*tan(f*x + e) +
c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) - I*sqrt(-8*c + 8*sqrt(c^2
+ d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/((4*I*a^2*c^2*d^3*f + 8*a^2*c*d^4*f - 4*I*a^2*d^
5*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - (2*(d*tan(f*x + e) + c)^(3/2)*c - 2*sq
rt(d*tan(f*x + e) + c)*c^2 + 13*I*(d*tan(f*x + e) + c)^(3/2)*d - 17*I*sqrt(d*tan(f*x + e) + c)*c*d + 15*sqrt(d
*tan(f*x + e) + c)*d^2)/((16*a^2*c^4*d^2*f + 64*I*a^2*c^3*d^3*f - 96*a^2*c^2*d^4*f - 64*I*a^2*c*d^5*f + 16*a^2
*d^6*f)*(d*tan(f*x + e) - I*d)^2))