3.1136 \(\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=446 \[ \frac{d \left (-26 c^2 d^2+11 i c^3 d+2 c^4+253 i c d^3+150 d^4\right )}{16 a^3 f (c-i d)^2 (c+i d)^5 \sqrt{c+d \tan (e+f x)}}+\frac{d \left (33 i c^2 d+6 c^3-83 c d^2+154 i d^3\right )}{48 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{\left (-16 c^2 d+2 i c^3-61 i c d^2+152 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 f (c+i d)^{11/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 f (c-i d)^{5/2}}+\frac{-4 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \]

[Out]

((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(5/2)*f) + (((2*I)*c^3 - 16*c^2*d - (6
1*I)*c*d^2 + 152*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(11/2)*f) + (d*(6*c^3
+ (33*I)*c^2*d - 83*c*d^2 + (154*I)*d^3))/(48*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])^(3/2)) - 1/(6*
(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)) + (I*c - 4*d)/(8*a*(c + I*d)^2*f*(a + I*a*Tan
[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)) + (2*c^2 + (11*I)*c*d - 30*d^2)/(16*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e
+ f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(2*c^4 + (11*I)*c^3*d - 26*c^2*d^2 + (253*I)*c*d^3 + 150*d^4))/(16*a^
3*(c - I*d)^2*(c + I*d)^5*f*Sqrt[c + d*Tan[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.48194, antiderivative size = 446, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{d \left (-26 c^2 d^2+11 i c^3 d+2 c^4+253 i c d^3+150 d^4\right )}{16 a^3 f (c-i d)^2 (c+i d)^5 \sqrt{c+d \tan (e+f x)}}+\frac{d \left (33 i c^2 d+6 c^3-83 c d^2+154 i d^3\right )}{48 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{\left (-16 c^2 d+2 i c^3-61 i c d^2+152 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 f (c+i d)^{11/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 f (c-i d)^{5/2}}+\frac{-4 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(5/2)*f) + (((2*I)*c^3 - 16*c^2*d - (6
1*I)*c*d^2 + 152*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(11/2)*f) + (d*(6*c^3
+ (33*I)*c^2*d - 83*c*d^2 + (154*I)*d^3))/(48*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])^(3/2)) - 1/(6*
(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)) + (I*c - 4*d)/(8*a*(c + I*d)^2*f*(a + I*a*Tan
[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)) + (2*c^2 + (11*I)*c*d - 30*d^2)/(16*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e
+ f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(2*c^4 + (11*I)*c^3*d - 26*c^2*d^2 + (253*I)*c*d^3 + 150*d^4))/(16*a^
3*(c - I*d)^2*(c + I*d)^5*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))^3 (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{3}{2} a (2 i c-5 d)-\frac{9}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx}{6 a^2 (i c-d)}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{3}{2} a^2 \left (4 c^2+15 i c d-32 d^2\right )-\frac{21}{2} a^2 (c+4 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{\frac{3}{2} a^3 \left (4 i c^3-22 c^2 d-67 i c d^2+154 d^3\right )+\frac{15}{2} a^3 d \left (2 i c^2-11 c d-30 i d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac{d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{\frac{3}{2} a^3 \left (4 i c^4-22 c^3 d-57 i c^2 d^2+99 c d^3-150 i d^4\right )+\frac{3}{2} a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )}\\ &=\frac{d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 f \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{\frac{3}{2} a^3 \left (4 i c^5-22 c^4 d-51 i c^3 d^2+66 c^2 d^3-233 i c d^4-154 d^5\right )-\frac{3}{2} a^3 d \left (11 c^3 d-i \left (2 c^4-26 c^2 d^2+253 i c d^3+150 d^4\right )\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )^2}\\ &=\frac{d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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}{16 a^3 (c-i d)^2}+\frac{\left (2 c^3+16 i c^2 d-61 c d^2-152 i d^3\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)^5}\\ &=\frac{d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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 )}{16 a^3 (c-i d)^2 f}-\frac{\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 (c+i d)^5 f}\\ &=\frac{d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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 )}{8 a^3 (c-i d)^2 d f}+\frac{\left (i \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right )\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 )}{16 a^3 (c+i d)^5 d f}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 (c-i d)^{5/2} f}+\frac{\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 (c+i d)^{11/2} f}+\frac{d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac{i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac{2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}

Mathematica [B]  time = 10.2644, size = 1160, normalized size = 2.6 \[ \frac{\sec ^3(e+f x) (\cos (3 e)+i \sin (3 e)) \left (\frac{2 \left (150 d^5+253 i c d^4-26 c^2 d^3+11 i c^3 d^2+2 c^4 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^5+22 i d c^4-51 d^2 c^3-66 i d^3 c^2-233 d^4 c+154 i d^5\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))^3}{32 (c-i d)^2 (c+i d)^5 f (i \tan (e+f x) a+a)^3}+\frac{\sec ^3(e+f x) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac{\left (18 c^2+103 i d c-208 d^2\right ) \cos (2 f x) \left (\frac{1}{96} i \cos (e)-\frac{\sin (e)}{96}\right )}{(c+i d)^5}+\frac{(9 c+26 i d) \cos (4 f x) \left (\frac{1}{96} i \cos (e)+\frac{\sin (e)}{96}\right )}{(c+i d)^4}+\frac{\left (11 i \cos (e) c^5-50 d \cos (e) c^4+11 i d \sin (e) c^4-51 i d^2 \cos (e) c^3-50 d^2 \sin (e) c^3-296 d^3 \cos (e) c^2-51 i d^3 \sin (e) c^2+1208 i d^4 \cos (e) c-296 d^4 \sin (e) c+576 d^5 \cos (e)+120 i d^5 \sin (e)\right ) \left (\frac{1}{96} \cos (3 e)+\frac{1}{96} i \sin (3 e)\right )}{(c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e))}+\frac{\cos (6 f x) \left (\frac{1}{48} i \cos (3 e)+\frac{1}{48} \sin (3 e)\right )}{(c+i d)^3}+\frac{\left (18 c^2+103 i d c-208 d^2\right ) \left (\frac{\cos (e)}{96}+\frac{1}{96} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^5}+\frac{(9 c+26 i d) \left (\frac{\cos (e)}{96}-\frac{1}{96} i \sin (e)\right ) \sin (4 f x)}{(c+i d)^4}+\frac{\left (\frac{1}{48} \cos (3 e)-\frac{1}{48} i \sin (3 e)\right ) \sin (6 f x)}{(c+i d)^3}+\frac{2 \left (-\frac{9}{2} i \cos (3 e-f x) d^6+\frac{9}{2} i \cos (3 e+f x) d^6+\frac{9}{2} \sin (3 e-f x) d^6-\frac{9}{2} \sin (3 e+f x) d^6+\frac{17}{2} c \cos (3 e-f x) d^5-\frac{17}{2} c \cos (3 e+f x) d^5+\frac{17}{2} i c \sin (3 e-f x) d^5-\frac{17}{2} i c \sin (3 e+f x) d^5\right )}{3 (c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} i d^6 \cos (3 e)-\frac{2}{3} d^6 \sin (3 e)}{(c-i d)^2 (c+i d)^5 (c \cos (e+f x)+d \sin (e+f x))^2}\right ) (\cos (f x)+i \sin (f x))^3}{f (i \tan (e+f x) a+a)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((18*c^2 + (10
3*I)*c*d - 208*d^2)*Cos[2*f*x]*((I/96)*Cos[e] - Sin[e]/96))/(c + I*d)^5 + ((9*c + (26*I)*d)*Cos[4*f*x]*((I/96)
*Cos[e] + Sin[e]/96))/(c + I*d)^4 + (((11*I)*c^5*Cos[e] - 50*c^4*d*Cos[e] - (51*I)*c^3*d^2*Cos[e] - 296*c^2*d^
3*Cos[e] + (1208*I)*c*d^4*Cos[e] + 576*d^5*Cos[e] + (11*I)*c^4*d*Sin[e] - 50*c^3*d^2*Sin[e] - (51*I)*c^2*d^3*S
in[e] - 296*c*d^4*Sin[e] + (120*I)*d^5*Sin[e])*(Cos[3*e]/96 + (I/96)*Sin[3*e]))/((c - I*d)^2*(c + I*d)^5*(c*Co
s[e] + d*Sin[e])) + (Cos[6*f*x]*((I/48)*Cos[3*e] + Sin[3*e]/48))/(c + I*d)^3 + ((18*c^2 + (103*I)*c*d - 208*d^
2)*(Cos[e]/96 + (I/96)*Sin[e])*Sin[2*f*x])/(c + I*d)^5 + ((9*c + (26*I)*d)*(Cos[e]/96 - (I/96)*Sin[e])*Sin[4*f
*x])/(c + I*d)^4 + ((Cos[3*e]/48 - (I/48)*Sin[3*e])*Sin[6*f*x])/(c + I*d)^3 + (((2*I)/3)*d^6*Cos[3*e] - (2*d^6
*Sin[3*e])/3)/((c - I*d)^2*(c + I*d)^5*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + (2*((17*c*d^5*Cos[3*e - f*x])/2
- ((9*I)/2)*d^6*Cos[3*e - f*x] - (17*c*d^5*Cos[3*e + f*x])/2 + ((9*I)/2)*d^6*Cos[3*e + f*x] + ((17*I)/2)*c*d^5
*Sin[3*e - f*x] + (9*d^6*Sin[3*e - f*x])/2 - ((17*I)/2)*c*d^5*Sin[3*e + f*x] - (9*d^6*Sin[3*e + f*x])/2))/(3*(
c - I*d)^2*(c + I*d)^5*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^3)
+ (Sec[e + f*x]^3*(Cos[3*e] + I*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(((-I)*(4*c^5 + (22*I)*c^4*d - 51*c^3*d^2
- (66*I)*c^2*d^3 - 233*c*d^4 + (154*I)*d^5)*(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^4*d + (11*I)*c^3*d^2 - 26*c^2*d^3 + (253*I)*c*d^4 + 1
50*d^5)*(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))))/(32*(c - I*d)^2*(c + I*d)^5*f*(a + I*a*Tan[e + f*x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.105, size = 4108, normalized size = 9.2 \begin{align*} \text{output 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))^3/(c+d*tan(f*x+e))^(5/2),x)

[Out]

15/8*I/f/a^3*d^4/(I*d-c)^(5/2)/(c+I*d)^6*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2+2/3*I/f/a^3*d^4/(I*d
-c)^2/(c+I*d)^6/(c+d*tan(f*x+e))^(3/2)*c^3+19/2/f/a^3*d^9/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(
-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))-35/8/f/a^3*d^11/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*
x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)+27/8/f/a^3*d^9/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f
*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)+10*I/f/a^3*d^4/(I*d-c)^2/(c+I*d)^6/(c+d*tan(f*x
+e))^(1/2)*c^2+2/3*I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^6/(c+d*tan(f*x+e))^(3/2)*c-15/8*I/f/a^3*d^2/(I*d-c)^(5/2)/(c+
I*d)^6*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^4-2/3/f/a^3*d^7/(I*d-c)^2/(c+I*d)^6/(c+d*tan(f*x+e))^(3/
2)-1/8*I/f/a^3/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)
/(-I*d-c)^(1/2))*c^9+83/6*I/f/a^3*d^8/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)
*(c+d*tan(f*x+e))^(3/2)*c^2+25/16*I/f/a^3*d^2/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^
2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^9+143/3*I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^
2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^4-123/16*I/f/a^3*d^8/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d
^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c-11/4*I/f/a^3*d^2/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/
(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^8-229/8*I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*
x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^5+51/4*I/f/a^3*d^8/(I*d-c)^2/(c+I*d)^6/(-I*d+d
*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^3+297/16*I/f/a^3*d^10/(I*d-c)^2/(c+I*d)
^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c+47/2*I/f/a^3*d^4/(I*d-c)^2/(c
+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^6-227/16*I/f/a^3*d^6/(I*
d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^3+19/16*I/f/a^3
*d^2/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^7-85/16
*I/f/a^3*d^4/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c
^5-177/16*I/f/a^3*d^4/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e)
)^(1/2)/(-I*d-c)^(1/2))*c^5-635/16*I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/
2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3-365/16*I/f/a^3*d^8/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*
I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c-85/4*I/f/a^3*d^4/(I*d-c)^2/(c+I*d)
^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^7+83/6/f/a^3*d^3/(I*d-c)^2/(c
+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^7+11/3/f/a^3*d^5/(I*d-c)
^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^5-211/6/f/a^3*d^7/(
I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^3+7/8/f/a^3*d
^7/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1
/2))*c^2+12/f/a^3*d^5/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e)
)^(1/2)*c^6+95/2/f/a^3*d^7/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f
*x+e))^(1/2)*c^4-99/4/f/a^3*d^9/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*
tan(f*x+e))^(3/2)*c+1/8/f/a^3*d/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*
tan(f*x+e))^(1/2)*c^10-73/8/f/a^3*d^3/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)
*(c+d*tan(f*x+e))^(1/2)*c^8+175/8/f/a^3*d^9/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*
d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^2+1/8/f/a^3*d/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^
2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^8-125/8/f/a^3*d^3/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c
)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^6-51/2/f/a^3*d^5/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3
*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^4-91/12*I/f/a^3*d^10/(I*d-c)^2/(c
+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)-41/8/f/a^3*d^3/(I*d-c)^2/(
c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^6-29/4/f/a^3*d^5/(I*d-c
)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^4+11/8/f/a^3*d^7/(
I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^2-1/4/f/a^3*d
/(I*d-c)^2/(c+I*d)^6/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^9+5/4/f/a^3
*d/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1
/2))*c^8-2/3/f/a^3*d^5/(I*d-c)^2/(c+I*d)^6/(c+d*tan(f*x+e))^(3/2)*c^2-4/f/a^3*d^5/(I*d-c)^2/(c+I*d)^6/(c+d*tan
(f*x+e))^(1/2)*c+5/2/f/a^3*d^3/(I*d-c)^(5/2)/(c+I*d)^6*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^3-3/4/f/
a^3*d^5/(I*d-c)^(5/2)/(c+I*d)^6*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c-3/4/f/a^3*d/(I*d-c)^(5/2)/(c+I*
d)^6*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^5+1/8*I/f/a^3/(I*d-c)^(5/2)/(c+I*d)^6*arctan((c+d*tan(f*x+
e))^(1/2)/(I*d-c)^(1/2))*c^6+6*I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^6/(c+d*tan(f*x+e))^(1/2)-1/8*I/f/a^3*d^6/(I*d-c)^
(5/2)/(c+I*d)^6*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))+91/16*I/f/a^3*d^2/(I*d-c)^2/(c+I*d)^6/(-I*d^3-3*c
*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^7

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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))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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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))**3/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 2.01707, size = 1449, normalized size = 3.25 \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))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-2*d^4*(2*(-2*I*c^3 + 16*c^2*d + 61*I*c*d^2 - 152*d^3)*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))))/((16*a^3*c^5*d^4*f + 80*I*a^3*c^4*d^5*f - 160*a^3*c^3*d^6*f - 160*I*a^3
*c^2*d^7*f + 80*a^3*c*d^8*f + 16*I*a^3*d^9*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1))
- 2*I*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^3*c^2
*d^4*f - 16*I*a^3*c*d^5*f - 8*a^3*d^6*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - (6
*I*(d*tan(f*x + e) + c)^4*c^4 - 12*I*(d*tan(f*x + e) + c)^3*c^5 + 6*I*(d*tan(f*x + e) + c)^2*c^6 - 33*(d*tan(f
*x + e) + c)^4*c^3*d + 84*(d*tan(f*x + e) + c)^3*c^4*d - 51*(d*tan(f*x + e) + c)^2*c^5*d - 78*I*(d*tan(f*x + e
) + c)^4*c^2*d^2 + 256*I*(d*tan(f*x + e) + c)^3*c^3*d^2 - 198*I*(d*tan(f*x + e) + c)^2*c^4*d^2 - 759*(d*tan(f*
x + e) + c)^4*c*d^3 + 1856*(d*tan(f*x + e) + c)^3*c^2*d^3 - 1446*(d*tan(f*x + e) + c)^2*c^3*d^3 + 384*(d*tan(f
*x + e) + c)*c^4*d^3 + 32*c^5*d^3 + 450*I*(d*tan(f*x + e) + c)^4*d^4 + 844*I*(d*tan(f*x + e) + c)^3*c*d^4 - 23
34*I*(d*tan(f*x + e) + c)^2*c^2*d^4 + (960*I*d*tan(f*x + e) + 960*I*c)*c^3*d^4 + 96*I*c^4*d^4 + 1196*(d*tan(f*
x + e) + c)^3*d^5 - 243*(d*tan(f*x + e) + c)^2*c*d^5 - 576*(d*tan(f*x + e) + c)*c^2*d^5 - 64*c^3*d^5 - 978*I*(
d*tan(f*x + e) + c)^2*d^6 + (192*I*d*tan(f*x + e) + 192*I*c)*c*d^6 + 64*I*c^2*d^6 - 192*(d*tan(f*x + e) + c)*d
^7 - 96*c*d^7 - 32*I*d^8)/((96*a^3*c^7*d^3*f + 288*I*a^3*c^6*d^4*f - 96*a^3*c^5*d^5*f + 480*I*a^3*c^4*d^6*f -
480*a^3*c^3*d^7*f + 96*I*a^3*c^2*d^8*f - 288*a^3*c*d^9*f - 96*I*a^3*d^10*f)*(-I*(d*tan(f*x + e) + c)^(3/2) + I
*sqrt(d*tan(f*x + e) + c)*c - sqrt(d*tan(f*x + e) + c)*d)^3))