∫ (a+b tan(e+fx))2(A+B tan(e+fx)+C tan2(e+fx))___________________________________

3.123 \(\int \frac {(a+b \tan (e+f x))^2 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=358 \[ -\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{3 d^3 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}-\frac {(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}+\frac {2 b^2 \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 f \left (c^2+d^2\right )} \]

[Out]

-(a-I*b)^2*(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f-(a+I*b)^2*(B-I*(A-C))*arc

tanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(5/2)/f+2/3*(-a*d+b*c)*(b*(4*c^4*C-B*c^3*d-2*c^2*(A-5*C)*d^

2-7*B*c*d^3+4*A*d^4)+3*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/d^3/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2)+2/3*b^2*(4*c^

2*C-B*c*d+(A+3*C)*d^2)*(c+d*tan(f*x+e))^(1/2)/d^3/(c^2+d^2)/f-2/3*(A*d^2-B*c*d+C*c^2)*(a+b*tan(f*x+e))^2/d/(c^

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

________________________________________________________________________________________

Rubi [A] time = 1.55, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3645, 3635, 3630, 3539, 3537, 63, 208} \[ \frac {2 (b c-a d) \left (3 a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right )\right )}{3 d^3 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}-\frac {(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}+\frac {2 b^2 \left (d^2 (A+3 C)-B c d+4 c^2 C\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 f \left (c^2+d^2\right )} \]

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

-(((a - I*b)^2*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) - ((a + I

*b)^2*(B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(c^2*C - B*c*d

+ A*d^2)*(a + b*Tan[e + f*x])^2)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*c - a*d)*(b*(4*c^4*C

- B*c^3*d - 2*c^2*(A - 5*C)*d^2 - 7*B*c*d^3 + 4*A*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/(3*d^3*(c^2

+ d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b^2*(4*c^2*C - B*c*d + (A + 3*C)*d^2)*Sqrt[c + d*Tan[e + f*x]])/(3*

d^3*(c^2 + d^2)*f)

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]

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 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 3630

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (

f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])

^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]

&& !LeQ[m, -1]

Rule 3635

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e

_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(c^2*C - B*c*d + A*d^2)*

(c + d*Tan[e + f*x])^(n + 1))/(d^2*f*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x

])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d +

a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &&

NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

Rule 3645

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t

an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*d^2 + c*(c*C - B*d))*(a + b*T

an[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I

nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c

*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1

) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -

a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {(a+b \tan (e+f x)) \left (\frac {1}{2} \left (2 A d \left (\frac {3 a c}{2}+2 b d\right )+2 \left (2 b c-\frac {3 a d}{2}\right ) (c C-B d)\right )+\frac {3}{2} d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+\frac {1}{2} b \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 d \left (c^2+d^2\right )}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} \left (b^2 \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )-3 a^2 d^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+6 a b d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-\frac {3}{2} d^2 \left (2 a b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)+\frac {1}{2} b^2 \left (c^2+d^2\right ) \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {2 \int \frac {-\frac {3}{2} d^2 \left (a^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-2 a b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-\frac {3}{2} d^2 \left (2 a b \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {\left ((a-i b)^2 (A-i B-C)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)^2}+\frac {\left ((a+i b)^2 (A+i B-C)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c+i d)^2}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {\left ((a-i b)^2 (i A+B-i C)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (c-i d)^2 f}-\frac {\left (i (a+i b)^2 (A+i B-C)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (c+i d)^2 f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}-\frac {\left ((a-i b)^2 (A-i B-C)\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 )}{(c-i d)^2 d f}-\frac {\left ((a+i b)^2 (A+i B-C)\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 )}{(c+i d)^2 d f}\\ &=-\frac {(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 (b c-a d) \left (b \left (4 c^4 C-B c^3 d-2 c^2 (A-5 C) d^2-7 B c d^3+4 A d^4\right )+3 a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 \left (4 c^2 C-B c d+(A+3 C) d^2\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] time = 6.56, size = 502, normalized size = 1.40 \[ \frac {2 C (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (-\frac {(4 a C d+b B d-4 b c C) (a+b \tan (e+f x))}{d f (c+d \tan (e+f x))^{3/2}}-\frac {-\frac {2 \left (8 a^2 C d^2+a b B d^2-16 a b c C d-A b^2 d^2-2 b^2 B c d+8 b^2 c^2 C+b^2 C d^2\right )}{3 d (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (\frac {\left (\frac {3}{2} c d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )+\frac {3}{2} d^4 \left (-\left (a^2 (A-C)\right )+2 a b B+b^2 (A-C)\right )\right ) \left (\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{3 (-d+i c) (c+d \tan (e+f x))^{3/2}}-\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{3 (d+i c) (c+d \tan (e+f x))^{3/2}}\right )}{d}-\frac {3}{2} d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{(-d+i c) \sqrt {c+d \tan (e+f x)}}-\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{(d+i c) \sqrt {c+d \tan (e+f x)}}\right )\right )}{3 d}}{2 d f}\right )}{d} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

(2*C*(a + b*Tan[e + f*x])^2)/(d*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(-(((-4*b*c*C + b*B*d + 4*a*C*d)*(a + b*Tan

[e + f*x]))/(d*f*(c + d*Tan[e + f*x])^(3/2))) - ((-2*(8*b^2*c^2*C - 2*b^2*B*c*d - 16*a*b*c*C*d - A*b^2*d^2 + a

*b*B*d^2 + 8*a^2*C*d^2 + b^2*C*d^2))/(3*d*(c + d*Tan[e + f*x])^(3/2)) + (2*((((3*c*(a^2*B - b^2*B + 2*a*b*(A -

C))*d^3)/2 + (3*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^4)/2)*(-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*T

an[e + f*x])/(c - I*d)]/((I*c + d)*(c + d*Tan[e + f*x])^(3/2)) + Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e

+ f*x])/(c + I*d)]/(3*(I*c - d)*(c + d*Tan[e + f*x])^(3/2))))/d - (3*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(-(H

ypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*Sqrt[c + d*Tan[e + f*x]])) + Hyperge

ometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)]/((I*c - d)*Sqrt[c + d*Tan[e + f*x]])))/2))/(3*d))/(2*

d*f)))/d

________________________________________________________________________________________

fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B] time = 0.52, size = 61833, normalized size = 172.72 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [B] time = 116.90, size = 88684, normalized size = 247.72 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*tan(e + f*x))^2*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(5/2),x)

[Out]

atan((((c + d*tan(e + f*x))^(1/2)*(96*A^2*a^2*b^2*d^18*f^3 - 16*A^2*b^4*d^18*f^3 - 16*A^2*a^4*d^18*f^3 + 320*A

^2*a^4*c^4*d^14*f^3 + 1024*A^2*a^4*c^6*d^12*f^3 + 1440*A^2*a^4*c^8*d^10*f^3 + 1024*A^2*a^4*c^10*d^8*f^3 + 320*

A^2*a^4*c^12*d^6*f^3 - 16*A^2*a^4*c^16*d^2*f^3 + 320*A^2*b^4*c^4*d^14*f^3 + 1024*A^2*b^4*c^6*d^12*f^3 + 1440*A

^2*b^4*c^8*d^10*f^3 + 1024*A^2*b^4*c^10*d^8*f^3 + 320*A^2*b^4*c^12*d^6*f^3 - 16*A^2*b^4*c^16*d^2*f^3 - 256*A^2

*a*b^3*c*d^17*f^3 + 256*A^2*a^3*b*c*d^17*f^3 - 1280*A^2*a*b^3*c^3*d^15*f^3 - 2304*A^2*a*b^3*c^5*d^13*f^3 - 128

0*A^2*a*b^3*c^7*d^11*f^3 + 1280*A^2*a*b^3*c^9*d^9*f^3 + 2304*A^2*a*b^3*c^11*d^7*f^3 + 1280*A^2*a*b^3*c^13*d^5*

f^3 + 256*A^2*a*b^3*c^15*d^3*f^3 + 1280*A^2*a^3*b*c^3*d^15*f^3 + 2304*A^2*a^3*b*c^5*d^13*f^3 + 1280*A^2*a^3*b*

c^7*d^11*f^3 - 1280*A^2*a^3*b*c^9*d^9*f^3 - 2304*A^2*a^3*b*c^11*d^7*f^3 - 1280*A^2*a^3*b*c^13*d^5*f^3 - 256*A^

2*a^3*b*c^15*d^3*f^3 - 1920*A^2*a^2*b^2*c^4*d^14*f^3 - 6144*A^2*a^2*b^2*c^6*d^12*f^3 - 8640*A^2*a^2*b^2*c^8*d^

10*f^3 - 6144*A^2*a^2*b^2*c^10*d^8*f^3 - 1920*A^2*a^2*b^2*c^12*d^6*f^3 + 96*A^2*a^2*b^2*c^16*d^2*f^3) + ((((8*

A^2*a^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2

- 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c

^4*d*f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2

*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b

^2)*(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) -

4*A^2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*

f^2 + 16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3

*c^4*d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^

2*d^3*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^

4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(32*A*b^2*d^21*f^4 - 32*A*a^2*d^21*f^4 - (c + d*tan(e + f*x))^(1/2)*((((8*A^2*a

^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32

*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*

f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*

f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(

16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^

2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 +

16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*

d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^2*d^3

*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4

+ 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^

9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*

f^5 + 64*c^21*d^2*f^5) - 160*A*a^2*c^2*d^19*f^4 - 128*A*a^2*c^4*d^17*f^4 + 896*A*a^2*c^6*d^15*f^4 + 3136*A*a^2

*c^8*d^13*f^4 + 4928*A*a^2*c^10*d^11*f^4 + 4480*A*a^2*c^12*d^9*f^4 + 2432*A*a^2*c^14*d^7*f^4 + 736*A*a^2*c^16*

d^5*f^4 + 96*A*a^2*c^18*d^3*f^4 + 160*A*b^2*c^2*d^19*f^4 + 128*A*b^2*c^4*d^17*f^4 - 896*A*b^2*c^6*d^15*f^4 - 3

136*A*b^2*c^8*d^13*f^4 - 4928*A*b^2*c^10*d^11*f^4 - 4480*A*b^2*c^12*d^9*f^4 - 2432*A*b^2*c^14*d^7*f^4 - 736*A*

b^2*c^16*d^5*f^4 - 96*A*b^2*c^18*d^3*f^4 + 192*A*a*b*c*d^20*f^4 + 1472*A*a*b*c^3*d^18*f^4 + 4864*A*a*b*c^5*d^1

6*f^4 + 8960*A*a*b*c^7*d^14*f^4 + 9856*A*a*b*c^9*d^12*f^4 + 6272*A*a*b*c^11*d^10*f^4 + 1792*A*a*b*c^13*d^8*f^4

- 256*A*a*b*c^15*d^6*f^4 - 320*A*a*b*c^17*d^4*f^4 - 64*A*a*b*c^19*d^2*f^4))*((((8*A^2*a^4*c^5*f^2 + 8*A^2*b^4

*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32*A^2*a*b^3*d^5*f^2 + 3

2*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*f^2 + 160*A^2*a^3*b*c^

4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*f^2 + 480*A^2*a^2*b^2*

c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(16*c^10*f^4 + 16*d^10*

f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^2*a^4*c^5*f^2 - 4*A^2*

b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 + 16*A^2*a*b^3*d^5*f^2

- 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*d*f^2 - 80*A^2*a^3*b*c

^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^2*d^3*f^2 - 240*A^2*a^2*b^2

*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1

/2)*1i + ((c + d*tan(e + f*x))^(1/2)*(96*A^2*a^2*b^2*d^18*f^3 - 16*A^2*b^4*d^18*f^3 - 16*A^2*a^4*d^18*f^3 + 32

0*A^2*a^4*c^4*d^14*f^3 + 1024*A^2*a^4*c^6*d^12*f^3 + 1440*A^2*a^4*c^8*d^10*f^3 + 1024*A^2*a^4*c^10*d^8*f^3 + 3

20*A^2*a^4*c^12*d^6*f^3 - 16*A^2*a^4*c^16*d^2*f^3 + 320*A^2*b^4*c^4*d^14*f^3 + 1024*A^2*b^4*c^6*d^12*f^3 + 144

0*A^2*b^4*c^8*d^10*f^3 + 1024*A^2*b^4*c^10*d^8*f^3 + 320*A^2*b^4*c^12*d^6*f^3 - 16*A^2*b^4*c^16*d^2*f^3 - 256*

A^2*a*b^3*c*d^17*f^3 + 256*A^2*a^3*b*c*d^17*f^3 - 1280*A^2*a*b^3*c^3*d^15*f^3 - 2304*A^2*a*b^3*c^5*d^13*f^3 -

1280*A^2*a*b^3*c^7*d^11*f^3 + 1280*A^2*a*b^3*c^9*d^9*f^3 + 2304*A^2*a*b^3*c^11*d^7*f^3 + 1280*A^2*a*b^3*c^13*d

^5*f^3 + 256*A^2*a*b^3*c^15*d^3*f^3 + 1280*A^2*a^3*b*c^3*d^15*f^3 + 2304*A^2*a^3*b*c^5*d^13*f^3 + 1280*A^2*a^3

*b*c^7*d^11*f^3 - 1280*A^2*a^3*b*c^9*d^9*f^3 - 2304*A^2*a^3*b*c^11*d^7*f^3 - 1280*A^2*a^3*b*c^13*d^5*f^3 - 256

*A^2*a^3*b*c^15*d^3*f^3 - 1920*A^2*a^2*b^2*c^4*d^14*f^3 - 6144*A^2*a^2*b^2*c^6*d^12*f^3 - 8640*A^2*a^2*b^2*c^8

*d^10*f^3 - 6144*A^2*a^2*b^2*c^10*d^8*f^3 - 1920*A^2*a^2*b^2*c^12*d^6*f^3 + 96*A^2*a^2*b^2*c^16*d^2*f^3) - (((

(8*A^2*a^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*

f^2 - 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^

3*c^4*d*f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*

c^2*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^

6*b^2)*(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2

) - 4*A^2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d

^2*f^2 + 16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*

b^3*c^4*d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b

*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6

*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A

^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5

*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A

^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/

4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^

8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 2

4*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 + 16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*

d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*

A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(

16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*

f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13

440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*A*a^2*d^2

1*f^4 + 32*A*b^2*d^21*f^4 - 160*A*a^2*c^2*d^19*f^4 - 128*A*a^2*c^4*d^17*f^4 + 896*A*a^2*c^6*d^15*f^4 + 3136*A*

a^2*c^8*d^13*f^4 + 4928*A*a^2*c^10*d^11*f^4 + 4480*A*a^2*c^12*d^9*f^4 + 2432*A*a^2*c^14*d^7*f^4 + 736*A*a^2*c^

16*d^5*f^4 + 96*A*a^2*c^18*d^3*f^4 + 160*A*b^2*c^2*d^19*f^4 + 128*A*b^2*c^4*d^17*f^4 - 896*A*b^2*c^6*d^15*f^4

- 3136*A*b^2*c^8*d^13*f^4 - 4928*A*b^2*c^10*d^11*f^4 - 4480*A*b^2*c^12*d^9*f^4 - 2432*A*b^2*c^14*d^7*f^4 - 736

*A*b^2*c^16*d^5*f^4 - 96*A*b^2*c^18*d^3*f^4 + 192*A*a*b*c*d^20*f^4 + 1472*A*a*b*c^3*d^18*f^4 + 4864*A*a*b*c^5*

d^16*f^4 + 8960*A*a*b*c^7*d^14*f^4 + 9856*A*a*b*c^9*d^12*f^4 + 6272*A*a*b*c^11*d^10*f^4 + 1792*A*a*b*c^13*d^8*

f^4 - 256*A*a*b*c^15*d^6*f^4 - 320*A*a*b*c^17*d^4*f^4 - 64*A*a*b*c^19*d^2*f^4))*((((8*A^2*a^4*c^5*f^2 + 8*A^2*

b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32*A^2*a*b^3*d^5*f^2

+ 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*f^2 + 160*A^2*a^3*b

*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*f^2 + 480*A^2*a^2*b

^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(16*c^10*f^4 + 16*d^

10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^2*a^4*c^5*f^2 - 4*A

^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 + 16*A^2*a*b^3*d^5*f

^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*d*f^2 - 80*A^2*a^3*

b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^2*d^3*f^2 - 240*A^2*a^2*

b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))

^(1/2)*1i)/(((c + d*tan(e + f*x))^(1/2)*(96*A^2*a^2*b^2*d^18*f^3 - 16*A^2*b^4*d^18*f^3 - 16*A^2*a^4*d^18*f^3 +

320*A^2*a^4*c^4*d^14*f^3 + 1024*A^2*a^4*c^6*d^12*f^3 + 1440*A^2*a^4*c^8*d^10*f^3 + 1024*A^2*a^4*c^10*d^8*f^3

+ 320*A^2*a^4*c^12*d^6*f^3 - 16*A^2*a^4*c^16*d^2*f^3 + 320*A^2*b^4*c^4*d^14*f^3 + 1024*A^2*b^4*c^6*d^12*f^3 +

1440*A^2*b^4*c^8*d^10*f^3 + 1024*A^2*b^4*c^10*d^8*f^3 + 320*A^2*b^4*c^12*d^6*f^3 - 16*A^2*b^4*c^16*d^2*f^3 - 2

56*A^2*a*b^3*c*d^17*f^3 + 256*A^2*a^3*b*c*d^17*f^3 - 1280*A^2*a*b^3*c^3*d^15*f^3 - 2304*A^2*a*b^3*c^5*d^13*f^3

- 1280*A^2*a*b^3*c^7*d^11*f^3 + 1280*A^2*a*b^3*c^9*d^9*f^3 + 2304*A^2*a*b^3*c^11*d^7*f^3 + 1280*A^2*a*b^3*c^1

3*d^5*f^3 + 256*A^2*a*b^3*c^15*d^3*f^3 + 1280*A^2*a^3*b*c^3*d^15*f^3 + 2304*A^2*a^3*b*c^5*d^13*f^3 + 1280*A^2*

a^3*b*c^7*d^11*f^3 - 1280*A^2*a^3*b*c^9*d^9*f^3 - 2304*A^2*a^3*b*c^11*d^7*f^3 - 1280*A^2*a^3*b*c^13*d^5*f^3 -

256*A^2*a^3*b*c^15*d^3*f^3 - 1920*A^2*a^2*b^2*c^4*d^14*f^3 - 6144*A^2*a^2*b^2*c^6*d^12*f^3 - 8640*A^2*a^2*b^2*

c^8*d^10*f^3 - 6144*A^2*a^2*b^2*c^10*d^8*f^3 - 1920*A^2*a^2*b^2*c^12*d^6*f^3 + 96*A^2*a^2*b^2*c^16*d^2*f^3) +

((((8*A^2*a^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d

^2*f^2 - 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a

*b^3*c^4*d*f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3

*b*c^2*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4

*a^6*b^2)*(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(

1/2) - 4*A^2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^

3*d^2*f^2 + 16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2

*a*b^3*c^4*d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^

3*b*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*

c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(32*A*b^2*d^21*f^4 - 32*A*a^2*d^21*f^4 - (c + d*tan(e + f*x))^(1/2)*((((8

*A^2*a^4*c^5*f^2 + 8*A^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^

2 - 32*A^2*a*b^3*d^5*f^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*

c^4*d*f^2 + 160*A^2*a^3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^

2*d^3*f^2 + 480*A^2*a^2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*

b^2)*(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)

- 4*A^2*a^4*c^5*f^2 - 4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2

*f^2 + 16*A^2*a*b^3*d^5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^

3*c^4*d*f^2 - 80*A^2*a^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c

^2*d^3*f^2 - 240*A^2*a^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d

^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13

440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^1

9*d^4*f^5 + 64*c^21*d^2*f^5) - 160*A*a^2*c^2*d^19*f^4 - 128*A*a^2*c^4*d^17*f^4 + 896*A*a^2*c^6*d^15*f^4 + 3136

*A*a^2*c^8*d^13*f^4 + 4928*A*a^2*c^10*d^11*f^4 + 4480*A*a^2*c^12*d^9*f^4 + 2432*A*a^2*c^14*d^7*f^4 + 736*A*a^2

*c^16*d^5*f^4 + 96*A*a^2*c^18*d^3*f^4 + 160*A*b^2*c^2*d^19*f^4 + 128*A*b^2*c^4*d^17*f^4 - 896*A*b^2*c^6*d^15*f

^4 - 3136*A*b^2*c^8*d^13*f^4 - 4928*A*b^2*c^10*d^11*f^4 - 4480*A*b^2*c^12*d^9*f^4 - 2432*A*b^2*c^14*d^7*f^4 -

736*A*b^2*c^16*d^5*f^4 - 96*A*b^2*c^18*d^3*f^4 + 192*A*a*b*c*d^20*f^4 + 1472*A*a*b*c^3*d^18*f^4 + 4864*A*a*b*c

^5*d^16*f^4 + 8960*A*a*b*c^7*d^14*f^4 + 9856*A*a*b*c^9*d^12*f^4 + 6272*A*a*b*c^11*d^10*f^4 + 1792*A*a*b*c^13*d

^8*f^4 - 256*A*a*b*c^15*d^6*f^4 - 320*A*a*b*c^17*d^4*f^4 - 64*A*a*b*c^19*d^2*f^4))*((((8*A^2*a^4*c^5*f^2 + 8*A

^2*b^4*c^5*f^2 - 48*A^2*a^2*b^2*c^5*f^2 - 80*A^2*a^4*c^3*d^2*f^2 - 80*A^2*b^4*c^3*d^2*f^2 - 32*A^2*a*b^3*d^5*f

^2 + 32*A^2*a^3*b*d^5*f^2 + 40*A^2*a^4*c*d^4*f^2 + 40*A^2*b^4*c*d^4*f^2 - 160*A^2*a*b^3*c^4*d*f^2 + 160*A^2*a^

3*b*c^4*d*f^2 + 320*A^2*a*b^3*c^2*d^3*f^2 - 240*A^2*a^2*b^2*c*d^4*f^2 - 320*A^2*a^3*b*c^2*d^3*f^2 + 480*A^2*a^

2*b^2*c^3*d^2*f^2)^2/4 - (A^4*a^8 + A^4*b^8 + 4*A^4*a^2*b^6 + 6*A^4*a^4*b^4 + 4*A^4*a^6*b^2)*(16*c^10*f^4 + 16

*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - 4*A^2*a^4*c^5*f^2 -

4*A^2*b^4*c^5*f^2 + 24*A^2*a^2*b^2*c^5*f^2 + 40*A^2*a^4*c^3*d^2*f^2 + 40*A^2*b^4*c^3*d^2*f^2 + 16*A^2*a*b^3*d^

5*f^2 - 16*A^2*a^3*b*d^5*f^2 - 20*A^2*a^4*c*d^4*f^2 - 20*A^2*b^4*c*d^4*f^2 + 80*A^2*a*b^3*c^4*d*f^2 - 80*A^2*a

^3*b*c^4*d*f^2 - 160*A^2*a*b^3*c^2*d^3*f^2 + 120*A^2*a^2*b^2*c*d^4*f^2 + 160*A^2*a^3*b*c^2*d^3*f^2 - 240*A^2*a

^2*b^2*c^3*d^2*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4

)))^(1/2) - ((c + d*tan(e + f*x))^(1/2)*(96*A^2*a^2*b^2*d^18*f^3 - 16*A^2*b^4*d^18*f^3 - 16*A^2*a^4*d^18*f^3 +