\(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx\) [349]

Optimal result

Integrand size = 33, antiderivative size = 224 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\left (8 a^2 A-3 A b^2+4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {(A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}+\frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d} \]

[Out]

1/4*(8*A*a^2-3*A*b^2+4*B*a*b)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-(A-I*B)*arctanh((a+b*tan(d*x+c
))^(1/2)/(a-I*b)^(1/2))/d/(a-I*b)^(1/2)-(A+I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d/(a+I*b)^(1/2)+
1/4*(3*A*b-4*B*a)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a^2/d-1/2*A*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/a/d

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3690, 3730, 3734, 3620, 3618, 65, 214, 3715} \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}+\frac {\left (8 a^2 A+4 a b B-3 A b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {(A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d} \]

[In]

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

((8*a^2*A - 3*A*b^2 + 4*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(5/2)*d) - ((A - I*B)*ArcTanh[S
qrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a
+ I*b]])/(Sqrt[a + I*b]*d) + ((3*A*b - 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*a^2*d) - (A*Cot[c + d
*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*a*d)

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 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 3690

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

Rule 3715

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

Rule 3730

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*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

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

Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}-\frac {\int \frac {\cot ^2(c+d x) \left (\frac {1}{2} (3 A b-4 a B)+2 a A \tan (c+d x)+\frac {3}{2} A b \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a} \\ & = \frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {\int \frac {\cot (c+d x) \left (\frac {1}{4} \left (-8 a^2 A+3 A b^2-4 a b B\right )-2 a^2 B \tan (c+d x)+\frac {1}{4} b (3 A b-4 a B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a^2} \\ & = \frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {\int \frac {-2 a^2 B+2 a^2 A \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a^2}-\frac {\left (8 a^2 A-3 A b^2+4 a b B\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{8 a^2} \\ & = \frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {1}{2} (-i A-B) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (i A-B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (8 a^2 A-3 A b^2+4 a b B\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 a^2 d} \\ & = \frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}+\frac {(A-i B) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac {(A+i B) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {\left (8 a^2 A-3 A b^2+4 a b B\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{4 a^2 b d} \\ & = \frac {\left (8 a^2 A-3 A b^2+4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}+\frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d}-\frac {(i A-B) \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}+\frac {(i A+B) \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d} \\ & = \frac {\left (8 a^2 A-3 A b^2+4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {(A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}+\frac {(3 A b-4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.32 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {2 b^3 \left (\frac {A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} b^3}-\frac {3 A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} b}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{2 a^{3/2} b^2}-\frac {\left (A \sqrt {-b^2}+b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{2 b^3 \sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}-\frac {b \left (A \sqrt {-b^2}-b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{2 \left (-b^2\right )^{5/2} \sqrt {a+\sqrt {-b^2}}}+\frac {3 A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 b^2}-\frac {B \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{2 a b^3}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 a b^3}\right )}{d} \]

[In]

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

(2*b^3*((A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*b^3) - (3*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sq
rt[a]])/(8*a^(5/2)*b) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(2*a^(3/2)*b^2) - ((A*Sqrt[-b^2] + b*B)*
ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*b^3*Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - (b*(A*Sqrt[-
b^2] - b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*(-b^2)^(5/2)*Sqrt[a + Sqrt[-b^2]]) + (3
*A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*a^2*b^2) - (B*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(2*a*b^3) -
(A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(4*a*b^3)))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4174\) vs. \(2(190)=380\).

Time = 0.24 (sec) , antiderivative size = 4175, normalized size of antiderivative = 18.64

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4175\)
default	\(\text {Expression too large to display}\)	\(4175\)

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/d/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1
/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a^3-1/4/d/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a
^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/4/d/(a^2+b^2)*ln((a+b*tan(d*x+
c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-2/d/
(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^
2+b^2)^(1/2)-2*a)^(1/2))*A*a^2+1/d/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1
/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a+1/d/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2
*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a^3
+1/4/d/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))
*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/d/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1
/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a-1/4/d/(a^2+b^2)*ln(b*tan(d*x+c)+a+
(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+2/d/(a
^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+
b^2)^(1/2)-2*a)^(1/2))*A*a^2-2/d*b^3/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^
(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B-1/4/d*b^2/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c
)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/4/
d/b^2*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(
1/2)+2*a)^(1/2)*a-1/d/b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*
a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a^2-1/d*b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*
(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B+2/d*b^3/(a^2+b^2)^(3/2)/
(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/
2)-2*a)^(1/2))*B+1/4/d*b^2/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c
)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B+1/d/b^2/(2*(a^2+b^
2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1
/2))*A*a^2-1/4/d/b^2*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A
*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d*b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)
+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b^2/(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*a
rctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A+1/4/d*b/(a^2+b
^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/
2)+2*a)^(1/2)-1/d*b^2/(a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan
(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A+2*A*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)+1/d/b^2
*(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))
/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a-1/d/b^2/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)
^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a^3-1/d/b/(a^2+b^2)^(1/2)/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)
^(1/2))*B*a^2+1/4/d/b^2/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+
b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/4/d/b/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)
+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/d/b^2/(a^2+b^2)/(2*(a^2+b^2)
^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2
))*A*a^4-1/4/d/b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1
/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/4/d/b^2/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-3/d*b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^
(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)
)*B*a^2+1/d/b/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2
)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2+1/d*b^2/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a+3/d*b/(a^2+b^2)^
(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^
2)^(1/2)-2*a)^(1/2))*B*a^2-1/4/d/b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*t
an(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/4/d*b/(a^2+b^2)^(3/2)*ln((a+b*tan(d*x+c))^(
1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/d/b^2/(
a^2+b^2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2))*A*a^4+1/4/d/b/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/4/d*b/(a^2+b^2)^(3/2)*ln(b*tan(d*x+c)+a+
(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/d/b^
2*(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)
)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a+1/d/b^2/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(
d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a^3-1/d/b/(a^2+b^2)^(3/2)/(2*(a^
2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a
)^(1/2))*B*a^4-1/d*b^2/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+
b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a+1/d/b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*
arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^4+3/4/d/tan
(d*x+c)^2/a^2*(a+b*tan(d*x+c))^(3/2)*A-5/4/d/tan(d*x+c)^2/a*(a+b*tan(d*x+c))^(1/2)*A+1/d/b/tan(d*x+c)^2*(a+b*t
an(d*x+c))^(1/2)*B-3/4/d*b^2/a^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*A+1/d*b/a^(3/2)*arctanh((a+b*tan(
d*x+c))^(1/2)/a^(1/2))*B-1/d/b/tan(d*x+c)^2/a*(a+b*tan(d*x+c))^(3/2)*B

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1836 vs. \(2 (184) = 368\).

Time = 7.51 (sec) , antiderivative size = 3688, normalized size of antiderivative = 16.46 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[1/8*(4*a^3*d*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2
)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4
- B^4)*b)*sqrt(b*tan(d*x + c) + a) + ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B
- A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a
*b - (A^2*B - B^3)*b^2)*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B
^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 -
4*a^3*d*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^
4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4
)*b)*sqrt(b*tan(d*x + c) + a) - ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B
^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b - (
A^2*B - B^3)*b^2)*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B
^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 - 4*a^3*
d*sqrt(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2
*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*
sqrt(b*tan(d*x + c) + a) + ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a
*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b - (A^2*B
- B^3)*b^2)*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*
b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 + 4*a^3*d*sq
rt(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2
*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*sqrt
(b*tan(d*x + c) + a) - ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b +
(A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b - (A^2*B - B
^3)*b^2)*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)
/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 - (8*A*a^2 + 4*B
*a*b - 3*A*b^2)*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))*tan(d*x
+ c)^2 - 2*(2*A*a^2 + (4*B*a^2 - 3*A*a*b)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(a^3*d*tan(d*x + c)^2), 1/4*
(2*a^3*d*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a
^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^
4)*b)*sqrt(b*tan(d*x + c) + a) + ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*
B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b -
(A^2*B - B^3)*b^2)*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 +
B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 - 2*a^3
*d*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2
*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*
sqrt(b*tan(d*x + c) + a) - ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a
*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b - (A^2*B
- B^3)*b^2)*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b
^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + 2*A*B*b + (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 - 2*a^3*d*sqr
t(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*
b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*sqrt(
b*tan(d*x + c) + a) + ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b +
(A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b - (A^2*B - B^
3)*b^2)*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/
((a^4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 + 2*a^3*d*sqrt(-(
(a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2
+ b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2))*log((2*(A^3*B + A*B^3)*a - (A^4 - B^4)*b)*sqrt(b*ta
n(d*x + c) + a) - ((B*a^3 - A*a^2*b + B*a*b^2 - A*b^3)*d^3*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (2*A^2*B*a^2 - (A^3 - 3*A*B^2)*a*b - (A^2*B - B^3)*b
^2)*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-(4*A^2*B^2*a^2 - 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/((a^
4 + 2*a^2*b^2 + b^4)*d^4)) - 2*A*B*b - (A^2 - B^2)*a)/((a^2 + b^2)*d^2)))*tan(d*x + c)^2 - (8*A*a^2 + 4*B*a*b
- 3*A*b^2)*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a)*tan(d*x + c)^2 - (2*A*a^2 + (4*B*a^2 - 3*A*a*b
)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/(a^3*d*tan(d*x + c)^2)]

Sympy [F]

\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]

[In]

integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**(1/2),x)

[Out]

Integral((A + B*tan(c + d*x))*cot(c + d*x)**3/sqrt(a + b*tan(c + d*x)), x)

Maxima [F]

\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{3}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^3/sqrt(b*tan(d*x + c) + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 10.91 (sec) , antiderivative size = 13182, normalized size of antiderivative = 58.85 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^(1/2),x)

[Out]

- (((5*A*b^2 - 4*B*a*b)*(a + b*tan(c + d*x))^(1/2))/(4*a) - ((3*A*b^2 - 4*B*a*b)*(a + b*tan(c + d*x))^(3/2))/(
4*a^2))/(d*(a + b*tan(c + d*x))^2 + a^2*d - 2*a*d*(a + b*tan(c + d*x))) - atan(((((((640*A*a^4*b^10*d^4 - 384*
A*a^2*b^12*d^4 + 768*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) + ((512*a^4*b^10*d^4
+ 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4
+ 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^
4)))^(1/2))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2
*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*ta
n(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 36
*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A
*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b
*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*a^4*b^9*d^2 + 36*
A^2*B*b^13*d^2 + 36*A^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2*a^5*b^8*d^2 + 96
*A^2*B*a^2*b^11*d^2 - 768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 -
(16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*
d^4 + b^2*d^4)))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a
^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3
*a^3*b^9 + 64*A^3*B*a^3*b^9))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16
*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^
(1/2)*1i - (((((640*A*a^4*b^10*d^4 - 384*A*a^2*b^12*d^4 + 768*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b
^9*d^4)/(2*a^4*d^5) - ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^
2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a
*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)
^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16
*(a^2*d^4 + b^2*d^4)))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^
2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 36*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^
4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1
/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^
3*a^5*b^8*d^2 + 256*B^3*a^4*b^9*d^2 + 36*A^2*B*b^13*d^2 + 36*A^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*
a^3*b^10*d^2 + 576*A*B^2*a^5*b^8*d^2 + 96*A^2*B*a^2*b^11*d^2 - 768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))*(-(((8*A^2*
a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d
^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*
A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*
A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d
^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2
- 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*1i)/((((((640*A*a^4*b^10*d^4 - 384*A*a^2*b^12*d^4 + 768*A*a^6*b
^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*ta
n(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*
B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^4*d^4))*(-(((
8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A
^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(576*A^2*a
^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 36*A^2*a*b^12*d^2 - 96*A*B*a^2
*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4
+ 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4
)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*a^4*b^9*d^2 + 36*A^2*B*b^13*d^2 + 36*A^3*a*b^
12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2*a^5*b^8*d^2 + 96*A^2*B*a^2*b^11*d^2 - 768*A^
2*B*a^4*b^9*d^2)/(2*a^4*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(
A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + ((a
+ b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 3
2*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9)
)/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 +
B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (((((640*A*a^4*b^10*
d^4 - 384*A*a^2*b^12*d^4 + 768*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) - ((512*a^4
*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (
16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^
4 + b^2*d^4)))^(1/2))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4
)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) +
((a + b*tan(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^
8*d^2 + 36*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*
d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2
- 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*a^4*b^9
*d^2 + 36*A^2*B*b^13*d^2 + 36*A^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2*a^5*b^
8*d^2 + 96*A^2*B*a^2*b^11*d^2 - 768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*
d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)
/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10
+ 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11
- 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(a^4*d^4))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^
2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b
^2*d^4)))^(1/2) + (24*A^5*a^2*b^10 - 9*A^3*B^2*b^12 - 9*A^5*b^12 + 32*B^5*a^3*b^9 - 16*A^3*B^2*a^2*b^10 + 64*A
^3*B^2*a^4*b^8 + 24*A^4*B*a*b^11 - 40*A*B^4*a^2*b^10 + 64*A*B^4*a^4*b^8 + 24*A^2*B^3*a*b^11 - 32*A^4*B*a^3*b^9
)/(a^4*d^5)))*(-(((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2
+ B^4))^(1/2) - 4*A^2*a*d^2 + 4*B^2*a*d^2 - 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*2i - atan(((((((640*A
*a^4*b^10*d^4 - 384*A*a^2*b^12*d^4 + 768*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) +
((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2
)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(1
6*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16
*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^
(1/2) - ((a + b*tan(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^
2*a^5*b^8*d^2 + 36*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8
*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^
2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*
a^4*b^9*d^2 + 36*A^2*B*b^13*d^2 + 36*A^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2
*a^5*b^8*d^2 + 96*A^2*B*a^2*b^11*d^2 - 768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*
A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*
b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2
*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a
*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (
16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^
4 + b^2*d^4)))^(1/2)*1i - (((((640*A*a^4*b^10*d^4 - 384*A*a^2*b^12*d^4 + 768*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^
4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) - ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^
2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a
*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 +
16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A
*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^1
0*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 36*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*
d^2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^
2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*
d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*a^4*b^9*d^2 + 36*A^2*B*b^13*d^2 + 36*A^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2
- 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2*a^5*b^8*d^2 + 96*A^2*B*a^2*b^11*d^2 - 768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))
*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2)
+ 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(9*A^
4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2
*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(a^4*d^4))*((((8*A^2*a*d^2 -
8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*
B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*1i)/((((((640*A*a^4*b^10*d^4 - 384*A*a^2*b^12*d^4 + 7
68*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)
*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4
+ 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2))/(a^4*d^
4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/
2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(5
76*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 36*A^2*a*b^12*d^2 - 96
*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a
^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 +
b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*a^4*b^9*d^2 + 36*A^2*B*b^13*d^2 + 36*A
^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2*a^5*b^8*d^2 + 96*A^2*B*a^2*b^11*d^2 -
768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*
d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)
+ ((a + b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^
10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^
3*b^9))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*
B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (((((640*A*a^4*
b^10*d^4 - 384*A*a^2*b^12*d^4 + 768*A*a^6*b^8*d^4 + 512*B*a^3*b^11*d^4 + 256*B*a^5*b^9*d^4)/(2*a^4*d^5) - ((51
2*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4
- (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^
2*d^4 + b^2*d^4)))^(1/2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*
d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)
+ ((a + b*tan(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5
*b^8*d^2 + 36*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*
a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d
^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) + (64*B^3*a^2*b^11*d^2 - 192*A^3*a^5*b^8*d^2 + 256*B^3*a^4*b
^9*d^2 + 36*A^2*B*b^13*d^2 + 36*A^3*a*b^12*d^2 - 96*A*B^2*a*b^12*d^2 - 384*A*B^2*a^3*b^10*d^2 + 576*A*B^2*a^5*
b^8*d^2 + 96*A^2*B*a^2*b^11*d^2 - 768*A^2*B*a^4*b^9*d^2)/(2*a^4*d^5))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b
*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2
)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10
+ 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11
- 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(a^4*d^4))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^
2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 + B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b
^2*d^4)))^(1/2) + (24*A^5*a^2*b^10 - 9*A^3*B^2*b^12 - 9*A^5*b^12 + 32*B^5*a^3*b^9 - 16*A^3*B^2*a^2*b^10 + 64*A
^3*B^2*a^4*b^8 + 24*A^4*B*a*b^11 - 40*A*B^4*a^2*b^10 + 64*A*B^4*a^4*b^8 + 24*A^2*B^3*a*b^11 - 32*A^4*B*a^3*b^9
)/(a^4*d^5)))*((((8*A^2*a*d^2 - 8*B^2*a*d^2 + 16*A*B*b*d^2)^2/4 - (16*a^2*d^4 + 16*b^2*d^4)*(A^4 + 2*A^2*B^2 +
B^4))^(1/2) + 4*A^2*a*d^2 - 4*B^2*a*d^2 + 8*A*B*b*d^2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*2i - (atan(-((((((32*B
^3*a^2*b^11*d^2 - 96*A^3*a^5*b^8*d^2 + 128*B^3*a^4*b^9*d^2 + 18*A^2*B*b^13*d^2 + 18*A^3*a*b^12*d^2 - 48*A*B^2*
a*b^12*d^2 - 192*A*B^2*a^3*b^10*d^2 + 288*A*B^2*a^5*b^8*d^2 + 48*A^2*B*a^2*b^11*d^2 - 384*A^2*B*a^4*b^9*d^2)/(
8*a^4*d^5) + (((((320*A*a^4*b^10*d^4 - 192*A*a^2*b^12*d^4 + 384*A*a^6*b^8*d^4 + 256*B*a^3*b^11*d^4 + 128*B*a^5
*b^9*d^4)/(8*a^4*d^5) - ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^9 + 9*A^2*a
^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(64*a^9*d^5))*(64*A^2*a^9 + 9
*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) + ((a + b*tan
(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 36*
A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(8*a^4*d^4))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 48*A^2
*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 48*
A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) - ((a + b*tan(c + d*x))^(1/2)*(
9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B^2
*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(8*a^4*d^4))*(64*A^2*a^9
+ 9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2)*1i)/(a^5*d) - (((((3
2*B^3*a^2*b^11*d^2 - 96*A^3*a^5*b^8*d^2 + 128*B^3*a^4*b^9*d^2 + 18*A^2*B*b^13*d^2 + 18*A^3*a*b^12*d^2 - 48*A*B
^2*a*b^12*d^2 - 192*A*B^2*a^3*b^10*d^2 + 288*A*B^2*a^5*b^8*d^2 + 48*A^2*B*a^2*b^11*d^2 - 384*A^2*B*a^4*b^9*d^2
)/(8*a^4*d^5) + (((((320*A*a^4*b^10*d^4 - 192*A*a^2*b^12*d^4 + 384*A*a^6*b^8*d^4 + 256*B*a^3*b^11*d^4 + 128*B*
a^5*b^9*d^4)/(8*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^9 + 9*A^
2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(64*a^9*d^5))*(64*A^2*a^9
+ 9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) - ((a + b*
tan(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 +
36*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(8*a^4*d^4))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 48*
A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d))*(64*A^2*a^9 + 9*A^2*a^5*b^4 -
48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) + ((a + b*tan(c + d*x))^(1/2
)*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*
B^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(8*a^4*d^4))*(64*A^2*
a^9 + 9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2)*1i)/(a^5*d))/((24
*A^5*a^2*b^10 - 9*A^3*B^2*b^12 - 9*A^5*b^12 + 32*B^5*a^3*b^9 - 16*A^3*B^2*a^2*b^10 + 64*A^3*B^2*a^4*b^8 + 24*A
^4*B*a*b^11 - 40*A*B^4*a^2*b^10 + 64*A*B^4*a^4*b^8 + 24*A^2*B^3*a*b^11 - 32*A^4*B*a^3*b^9)/(a^4*d^5) + (((((32
*B^3*a^2*b^11*d^2 - 96*A^3*a^5*b^8*d^2 + 128*B^3*a^4*b^9*d^2 + 18*A^2*B*b^13*d^2 + 18*A^3*a*b^12*d^2 - 48*A*B^
2*a*b^12*d^2 - 192*A*B^2*a^3*b^10*d^2 + 288*A*B^2*a^5*b^8*d^2 + 48*A^2*B*a^2*b^11*d^2 - 384*A^2*B*a^4*b^9*d^2)
/(8*a^4*d^5) + (((((320*A*a^4*b^10*d^4 - 192*A*a^2*b^12*d^4 + 384*A*a^6*b^8*d^4 + 256*B*a^3*b^11*d^4 + 128*B*a
^5*b^9*d^4)/(8*a^4*d^5) - ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^9 + 9*A^2
*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(64*a^9*d^5))*(64*A^2*a^9 +
9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) + ((a + b*t
an(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 3
6*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(8*a^4*d^4))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 48*A
^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 4
8*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) - ((a + b*tan(c + d*x))^(1/2)
*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B
^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(8*a^4*d^4))*(64*A^2*a
^9 + 9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(a^5*d) + (((((32
*B^3*a^2*b^11*d^2 - 96*A^3*a^5*b^8*d^2 + 128*B^3*a^4*b^9*d^2 + 18*A^2*B*b^13*d^2 + 18*A^3*a*b^12*d^2 - 48*A*B^
2*a*b^12*d^2 - 192*A*B^2*a^3*b^10*d^2 + 288*A*B^2*a^5*b^8*d^2 + 48*A^2*B*a^2*b^11*d^2 - 384*A^2*B*a^4*b^9*d^2)
/(8*a^4*d^5) + (((((320*A*a^4*b^10*d^4 - 192*A*a^2*b^12*d^4 + 384*A*a^6*b^8*d^4 + 256*B*a^3*b^11*d^4 + 128*B*a
^5*b^9*d^4)/(8*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^9 + 9*A^2
*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(64*a^9*d^5))*(64*A^2*a^9 +
9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) - ((a + b*t
an(c + d*x))^(1/2)*(576*A^2*a^5*b^8*d^2 - 192*A^2*a^3*b^10*d^2 + 64*B^2*a^3*b^10*d^2 - 320*B^2*a^5*b^8*d^2 + 3
6*A^2*a*b^12*d^2 - 96*A*B*a^2*b^11*d^2 + 768*A*B*a^4*b^9*d^2))/(8*a^4*d^4))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 48*A
^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d))*(64*A^2*a^9 + 9*A^2*a^5*b^4 - 4
8*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(8*a^5*d) + ((a + b*tan(c + d*x))^(1/2)
*(9*A^4*b^12 - 9*A^2*B^2*b^12 - 48*A^4*a^2*b^10 + 96*A^4*a^4*b^8 - 16*B^4*a^2*b^10 + 32*B^4*a^4*b^8 + 64*A^2*B
^2*a^2*b^10 + 24*A*B^3*a*b^11 - 24*A^3*B*a*b^11 - 64*A*B^3*a^3*b^9 + 64*A^3*B*a^3*b^9))/(8*a^4*d^4))*(64*A^2*a
^9 + 9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2))/(a^5*d)))*(64*A^2
*a^9 + 9*A^2*a^5*b^4 - 48*A^2*a^7*b^2 + 16*B^2*a^7*b^2 + 64*A*B*a^8*b - 24*A*B*a^6*b^3)^(1/2)*1i)/(4*a^5*d)