\(\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [79]

Optimal result

Integrand size = 36, antiderivative size = 125 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {a^{3/2} (3 i A+2 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {2} a^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a A \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3674, 3681, 3561, 212, 3680, 65, 214} \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {a^{3/2} (2 B+3 i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {2} a^{3/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a A \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d} \]

[In]

Int[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

-((a^(3/2)*((3*I)*A + 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d) + (2*Sqrt[2]*a^(3/2)*(I*A + B)*ArcT
anh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (a*A*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3674

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^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x]
)^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c
+ d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m
- 1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && E
qQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3680

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

Rule 3681

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

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

Mathematica [A] (verified)

Time = 3.44 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {-i a^{3/2} (3 A-2 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+2 \sqrt {2} a^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-a A \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d} \]

[In]

Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

((-I)*a^(3/2)*(3*A - (2*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 2*Sqrt[2]*a^(3/2)*(I*A + B)*ArcTan
h[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] - a*A*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {2 i a^{2} \left (-\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 \sqrt {a}}+\frac {i A \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}-\frac {\left (-2 i B +3 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{d}\)	\(111\)
default	\(\frac {2 i a^{2} \left (-\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 \sqrt {a}}+\frac {i A \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}-\frac {\left (-2 i B +3 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{d}\)	\(111\)

[In]

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

[Out]

2*I/d*a^2*(-1/2*(-2*A+2*I*B)*2^(1/2)/a^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+1/2*I*A*(a+
I*a*tan(d*x+c))^(1/2)/a/tan(d*x+c)-1/2*(3*A-2*I*B)/a^(1/2)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (98) = 196\).

Time = 0.28 (sec) , antiderivative size = 685, normalized size of antiderivative = 5.48 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {4 \, \sqrt {2} \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - 4 \, \sqrt {2} \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - \sqrt {-\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {16 \, {\left (3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-3 i \, A - 2 \, B\right )} a^{2} + 2 \, \sqrt {2} \sqrt {-\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{3 i \, A + 2 \, B}\right ) + \sqrt {-\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {16 \, {\left (3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-3 i \, A - 2 \, B\right )} a^{2} - 2 \, \sqrt {2} \sqrt {-\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{3 i \, A + 2 \, B}\right ) + 4 \, \sqrt {2} {\left (i \, A a e^{\left (3 i \, d x + 3 i \, c\right )} + i \, A a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{4 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

[In]

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

[Out]

-1/4*(4*sqrt(2)*sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*log(4*((-I*A - B)*a^2*e^(I*d*
x + I*c) + sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))
*e^(-I*d*x - I*c)/((-I*A - B)*a)) - 4*sqrt(2)*sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)
*log(4*((-I*A - B)*a^2*e^(I*d*x + I*c) - sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt
(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a)) - sqrt(-(9*A^2 - 12*I*A*B - 4*B^2)*a^3/d^2)*(d
*e^(2*I*d*x + 2*I*c) - d)*log(-16*(3*(-3*I*A - 2*B)*a^2*e^(2*I*d*x + 2*I*c) + (-3*I*A - 2*B)*a^2 + 2*sqrt(2)*s
qrt(-(9*A^2 - 12*I*A*B - 4*B^2)*a^3/d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*
c) + 1)))*e^(-2*I*d*x - 2*I*c)/(3*I*A + 2*B)) + sqrt(-(9*A^2 - 12*I*A*B - 4*B^2)*a^3/d^2)*(d*e^(2*I*d*x + 2*I*
c) - d)*log(-16*(3*(-3*I*A - 2*B)*a^2*e^(2*I*d*x + 2*I*c) + (-3*I*A - 2*B)*a^2 - 2*sqrt(2)*sqrt(-(9*A^2 - 12*I
*A*B - 4*B^2)*a^3/d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*
d*x - 2*I*c)/(3*I*A + 2*B)) + 4*sqrt(2)*(I*A*a*e^(3*I*d*x + 3*I*c) + I*A*a*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x
+ 2*I*c) + 1)))/(d*e^(2*I*d*x + 2*I*c) - d)

Sympy [F]

\[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

Integral((I*a*(tan(c + d*x) - I))**(3/2)*(A + B*tan(c + d*x))*cot(c + d*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.16 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {i \, {\left (2 \, \sqrt {2} {\left (A - i \, B\right )} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) - {\left (3 \, A - 2 i \, B\right )} \sqrt {a} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right ) - \frac {2 i \, \sqrt {i \, a \tan \left (d x + c\right ) + a} A}{\tan \left (d x + c\right )}\right )} a}{2 \, d} \]

[In]

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

[Out]

-1/2*I*(2*sqrt(2)*(A - I*B)*sqrt(a)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqr
t(I*a*tan(d*x + c) + a))) - (3*A - 2*I*B)*sqrt(a)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x
+ c) + a) + sqrt(a))) - 2*I*sqrt(I*a*tan(d*x + c) + a)*A/tan(d*x + c))*a/d

Giac [F]

\[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 2338, normalized size of antiderivative = 18.70 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

- 2*atanh((6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((3*B^2*a^3)/(2*d^2) - (17*A^2*a^3)/(8*d^2) - ((A^4*a^18)/d^4 +
(16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(8*a^6) + (A*B*a
^3*7i)/(2*d^2))^(1/2)*((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3
*B*a^18*8i)/d^4)^(1/2))/(A^3*a^11*d*10i + 32*B^3*a^11*d + A*B^2*a^11*d*72i - 32*A^2*B*a^11*d + A*a^2*d^3*((A^4
*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)
+ (2*A^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((3*B^2*a^3)/(2*d^2) - (17*A^2*a^3)/(8*d^2) - ((A^4*a^18)/d^4 +
(16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(8*a^6) + (A*B*a
^3*7i)/(2*d^2))^(1/2))/(A^3*a^8*d*10i + 32*B^3*a^8*d + A*B^2*a^8*d*72i - 32*A^2*B*a^8*d + (A*d^3*((A^4*a^18)/d
^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)/a) + (8*
B^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((3*B^2*a^3)/(2*d^2) - (17*A^2*a^3)/(8*d^2) - ((A^4*a^18)/d^4 + (16*
B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(8*a^6) + (A*B*a^3*7i
)/(2*d^2))^(1/2))/(A^3*a^8*d*10i + 32*B^3*a^8*d + A*B^2*a^8*d*72i - 32*A^2*B*a^8*d + (A*d^3*((A^4*a^18)/d^4 +
(16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)/a) + (A*B*a^6
*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((3*B^2*a^3)/(2*d^2) - (17*A^2*a^3)/(8*d^2) - ((A^4*a^18)/d^4 + (16*B^4*a^1
8)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(8*a^6) + (A*B*a^3*7i)/(2*d^
2))^(1/2)*8i)/(A^3*a^8*d*10i + 32*B^3*a^8*d + A*B^2*a^8*d*72i - 32*A^2*B*a^8*d + (A*d^3*((A^4*a^18)/d^4 + (16*
B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)/a))*((3*B^2*a^3)/
(2*d^2) - (17*A^2*a^3)/(8*d^2) - ((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)
/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(8*a^6) + (A*B*a^3*7i)/(2*d^2))^(1/2) - 2*atanh((2*A^2*a^6*d^2*(a + a*tan(c
+ d*x)*1i)^(1/2)*(((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a
^18*8i)/d^4)^(1/2)/(8*a^6) - (17*A^2*a^3)/(8*d^2) + (3*B^2*a^3)/(2*d^2) + (A*B*a^3*7i)/(2*d^2))^(1/2))/(A^3*a^
8*d*10i + 32*B^3*a^8*d + A*B^2*a^8*d*72i - 32*A^2*B*a^8*d - (A*d^3*((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^
2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)/a) - (6*d^4*(a + a*tan(c + d*x)*1i)^(1
/2)*(((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^
(1/2)/(8*a^6) - (17*A^2*a^3)/(8*d^2) + (3*B^2*a^3)/(2*d^2) + (A*B*a^3*7i)/(2*d^2))^(1/2)*((A^4*a^18)/d^4 + (16
*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2))/(A^3*a^11*d*10i + 3
2*B^3*a^11*d + A*B^2*a^11*d*72i - 32*A^2*B*a^11*d - A*a^2*d^3*((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2
*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i) + (8*B^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1
/2)*(((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^
(1/2)/(8*a^6) - (17*A^2*a^3)/(8*d^2) + (3*B^2*a^3)/(2*d^2) + (A*B*a^3*7i)/(2*d^2))^(1/2))/(A^3*a^8*d*10i + 32*
B^3*a^8*d + A*B^2*a^8*d*72i - 32*A^2*B*a^8*d - (A*d^3*((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d
^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)/a) + (A*B*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((
A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(
8*a^6) - (17*A^2*a^3)/(8*d^2) + (3*B^2*a^3)/(2*d^2) + (A*B*a^3*7i)/(2*d^2))^(1/2)*8i)/(A^3*a^8*d*10i + 32*B^3*
a^8*d + A*B^2*a^8*d*72i - 32*A^2*B*a^8*d - (A*d^3*((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a^18)/d^4 +
(A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)*2i)/a))*(((A^4*a^18)/d^4 + (16*B^4*a^18)/d^4 - (8*A^2*B^2*a
^18)/d^4 + (A*B^3*a^18*32i)/d^4 + (A^3*B*a^18*8i)/d^4)^(1/2)/(8*a^6) - (17*A^2*a^3)/(8*d^2) + (3*B^2*a^3)/(2*d
^2) + (A*B*a^3*7i)/(2*d^2))^(1/2) - (A*a*cot(c + d*x)*(a + a*tan(c + d*x)*1i)^(1/2))/d