\(\int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx\) [367]

Optimal result

Integrand size = 34, antiderivative size = 119 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {2 B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {21, 3655, 3620, 3618, 65, 214, 3715} \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {2 B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}} \]

[In]

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

[Out]

(-2*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*
b]])/(Sqrt[a - I*b]*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 3655

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.94 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {B \left (-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}\right )}{d} \]

[In]

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

[Out]

(B*((-2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/S
qrt[a - I*b] + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(97)=194\).

Time = 0.34 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.14

method	result	size
default	\(\frac {2 B \,b^{2} \left (\frac {\frac {\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \sqrt {a^{2}+b^{2}}}+\frac {-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \sqrt {a^{2}+b^{2}}}}{b^{2}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{b^{2} \sqrt {a}}\right )}{d}\)	\(374\)

[In]

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

[Out]

2*B/d*b^2*(1/b^2*(1/4/(a^2+b^2)^(1/2)*(1/2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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))+2*(a-(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)))+1/4/(a^2+b^2)^(1/2)*(
-1/2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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))+2*((a^2+b^2)^(1/2)-a)/(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))))-1/b^2/a^(1/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (93) = 186\).

Time = 0.29 (sec) , antiderivative size = 1674, normalized size of antiderivative = 14.07 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/2*(a*d*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))*log(
sqrt(b*tan(d*x + c) + a)*B^3 + ((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3 - B^2*a*d)*sqrt((
(a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))) - a*d*sqrt(((a^2 + b
^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*
B^3 - ((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3 - B^2*a*d)*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2
/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))) - a*d*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*B^3 + ((a^2 + b^2)*sqrt(
-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3 + B^2*a*d)*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^
4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))) + a*d*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)
)*d^2 - B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*B^3 - ((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*
b^2 + b^4)*d^4))*d^3 + B^2*a*d)*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*a)/(
(a^2 + b^2)*d^2))) - 2*B*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))
)/(a*d), -1/2*(a*d*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^
2))*log(sqrt(b*tan(d*x + c) + a)*B^3 + ((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3 - B^2*a*d
)*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))) - a*d*sqrt((
(a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x +
c) + a)*B^3 - ((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3 - B^2*a*d)*sqrt(((a^2 + b^2)*sqrt(
-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))) - a*d*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^
2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*B^3 + ((a^2 + b^
2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3 + B^2*a*d)*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*
b^2 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))) + a*d*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b
^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*B^3 - ((a^2 + b^2)*sqrt(-B^4*b^2/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*d^3 + B^2*a*d)*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 -
B^2*a)/((a^2 + b^2)*d^2))) - 4*B*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a))/(a*d)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 2142, normalized size of antiderivative = 18.00 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan(((((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*d^2))/d^3 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*
x))^(1/2)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2))/d^4)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) + (576*B^2*a*b^8*(a + b*
tan(c + d*x))^(1/2))/d^2)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) - (96*B^3*a*b^8)/d^3)*(B^2/(4*(a*d^2 - b*d^2*1i))
)^(1/2) - (96*B^4*b^8*(a + b*tan(c + d*x))^(1/2))/d^4)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2)*1i - (((((32*(16*B*b
^10*d^2 + 12*B*a^2*b^8*d^2))/d^3 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(B^2/(4*(a*d^
2 - b*d^2*1i)))^(1/2))/d^4)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) - (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^
2)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) - (96*B^3*a*b^8)/d^3)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) + (96*B^4*b^8*(
a + b*tan(c + d*x))^(1/2))/d^4)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2)*1i)/((((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*
d^2))/d^3 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2))/
d^4)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) + (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*(B^2/(4*(a*d^2 - b*d
^2*1i)))^(1/2) - (96*B^3*a*b^8)/d^3)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) - (96*B^4*b^8*(a + b*tan(c + d*x))^(1/
2))/d^4)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) + (((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*d^2))/d^3 + (32*(16*b^10*d
^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2))/d^4)*(B^2/(4*(a*d^2 - b*d^
2*1i)))^(1/2) - (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) - (96*B^3*a
*b^8)/d^3)*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2) + (96*B^4*b^8*(a + b*tan(c + d*x))^(1/2))/d^4)*(B^2/(4*(a*d^2 -
b*d^2*1i)))^(1/2)))*(B^2/(4*(a*d^2 - b*d^2*1i)))^(1/2)*2i - atan(((((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*d^2))/
d^3 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2))/d
^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) + (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*((B^2*1i)/(4*(a*
d^2*1i - b*d^2)))^(1/2) - (96*B^3*a*b^8)/d^3)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) - (96*B^4*b^8*(a + b*tan
(c + d*x))^(1/2))/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2)*1i - (((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*d^2)
)/d^3 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2))
/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) - (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*((B^2*1i)/(4*(
a*d^2*1i - b*d^2)))^(1/2) - (96*B^3*a*b^8)/d^3)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) + (96*B^4*b^8*(a + b*t
an(c + d*x))^(1/2))/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2)*1i)/((((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*d^
2))/d^3 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2
))/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) + (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*((B^2*1i)/(4
*(a*d^2*1i - b*d^2)))^(1/2) - (96*B^3*a*b^8)/d^3)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) - (96*B^4*b^8*(a + b
*tan(c + d*x))^(1/2))/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) + (((((32*(16*B*b^10*d^2 + 12*B*a^2*b^8*d^2
))/d^3 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2)
)/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) - (576*B^2*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*((B^2*1i)/(4*
(a*d^2*1i - b*d^2)))^(1/2) - (96*B^3*a*b^8)/d^3)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2) + (96*B^4*b^8*(a + b*
tan(c + d*x))^(1/2))/d^4)*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2)))*((B^2*1i)/(4*(a*d^2*1i - b*d^2)))^(1/2)*2i
- (2*B*atanh((576*B^5*b^8*(a + b*tan(c + d*x))^(1/2))/(a^(1/2)*(576*B^5*b^8 + (1024*B^5*b^10)/a^2)) + (1024*B
^5*b^10*(a + b*tan(c + d*x))^(1/2))/(a^(5/2)*(576*B^5*b^8 + (1024*B^5*b^10)/a^2))))/(a^(1/2)*d)