\(\int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\) [369]

Optimal result

Integrand size = 34, antiderivative size = 154 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {21, 3650, 3734, 3620, 3618, 65, 214, 3715} \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^2 B}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]

[In]

Int[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

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

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 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(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 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}& = B \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx \\ & = \frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(2 B) \int \frac {\cot (c+d x) \left (\frac {1}{2} \left (a^2+b^2\right )-\frac {1}{2} a b \tan (c+d x)+\frac {1}{2} b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )} \\ & = \frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{a}+\frac {(2 B) \int \frac {-\frac {a b}{2}-\frac {1}{2} a^2 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )} \\ & = \frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a-b)}-\frac {B \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a+b)}+\frac {B \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {B \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b) 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 (a+i b) 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 )}{a b d} \\ & = -\frac {2 B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {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 )}{(i a-b) b d}+\frac {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 (i a+b) d} \\ & = -\frac {2 B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.08 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {B \left (-\frac {2 \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {a (a+i b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {a (a-i b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}+\frac {2 b^2}{\sqrt {a+b \tan (c+d x)}}\right )}{a \left (a^2+b^2\right ) d} \]

[In]

Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1778\) vs. \(2(130)=260\).

Time = 0.33 (sec) , antiderivative size = 1779, normalized size of antiderivative = 11.55

method	result	size
default	\(\text {Expression too large to display}\)	\(1779\)

[In]

int(cot(d*x+c)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2281 vs. \(2 (126) = 252\).

Time = 0.39 (sec) , antiderivative size = 4578, normalized size of antiderivative = 29.73 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*(4*sqrt(b*tan(d*x + c) + a)*B*a*b^2 + ((a^4*b + a^2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^
3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^
12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)
*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^
8 + 6*a^2*b^10 + b^12)*d^4)) - (3*B^2*a^4 - 4*B^2*a^2*b^2 + B^2*b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b
^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - ((a^4*b
+ a^2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4
*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*
tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^
6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*B^2*a^4 - 4*B^2*
a^2*b^2 + B^2*b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b
^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*
d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - ((a^4*b + a^2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)*sq
rt((B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4
*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^
4 + a*b^6)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +
15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^2*a^4 - 4*B^2*a^2*b^2 + B^2*b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2
- (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2)))
+ ((a^4*b + a^2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^
6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^
2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^
4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^2*a^
4 - 4*B^2*a^2*b^2 + B^2*b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9
*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^1
0 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + 2*(B*a^3 + B*a*b^2 + (B*a^2*b + B*b^3)*tan(d*x
+ c))*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^4*b + a^2*b^3
)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d), 1/2*(4*sqrt(b*tan(d*x + c) + a)*B*a*b^2 + 4*(B*a^3 + B*a*b^2 + (B*a^2*b
+ B*b^3)*tan(d*x + c))*sqrt(-a)*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a) + ((a^4*b + a^2*b^3)*d*tan(d*x +
c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4
*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12
)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*tan(d*x + c) + a) + (2*(
a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 +
15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*B^2*a^4 - 4*B^2*a^2*b^2 + B^2*b^4)*d)*sq
rt((B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4
*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6)*d^2))) - ((a^4*b + a^2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^
2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^
2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2)
)*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*
B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4)) - (3*B^2*a^4 - 4*B^2*a^2*b^2 + B^2*b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^
6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - ((a^4*b + a^2*b^3)*d*
tan(d*x + c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-
(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b
^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*tan(d*x + c) +
a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*
a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^2*a^4 - 4*B^2*a^2*b^2 + B^2*
b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2
*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + ((a^4*b + a^2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)*sqrt((B^2*a^3 -
3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 +
6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^2))*log(-(3*B^3*a^2 - B^3*b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3
*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 +
6*a^2*b^10 + b^12)*d^4)) + (3*B^2*a^4 - 4*B^2*a^2*b^2 + B^2*b^4)*d)*sqrt((B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))))/((a^4*b + a^
2*b^3)*d*tan(d*x + c) + (a^5 + a^3*b^2)*d)]

Sympy [F]

\[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=B \int \frac {\cot {\left (c + d x \right )}}{a \sqrt {a + b \tan {\left (c + d x \right )}} + b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}}\, dx \]

[In]

integrate(cot(d*x+c)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c))**(5/2),x)

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 14.09 (sec) , antiderivative size = 7172, normalized size of antiderivative = 46.57 \[ \int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

int((cot(c + d*x)*(B*a + B*b*tan(c + d*x)))/(a + b*tan(c + d*x))^(5/2),x)

[Out]

(log(((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/
(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(((((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^
4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^
(1/2)*(512*B*a^8*b^28*d^8 - ((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d
^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(
512*a^9*b^28*d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^24*d^9 + 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161
280*a^19*b^18*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23*b^14*d^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 76
8*a^29*b^8*d^9))/4 + 5248*B*a^10*b^26*d^8 + 23936*B*a^12*b^24*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16*b^20
*d^8 + 130816*B*a^18*b^18*d^8 + 105728*B*a^20*b^16*d^8 + 57856*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 + 4224*
B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^8))/4 + (a + b*tan(c + d*x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*b^
24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272*B^2*a^14*b^20*d^7 + 9856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7 +
15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^22*b^12*d^7 + 3712*B^2*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(((96*B^4*
a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d
^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - 128*B^3*a^7*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*a^11*b^
22*d^6 + 10976*B^3*a^13*b^20*d^6 + 20384*B^3*a^15*b^18*d^6 + 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b^14*d^6
+ 3232*B^3*a^21*b^12*d^6 + 96*B^3*a^23*b^10*d^6 - 96*B^3*a^25*b^8*d^6))/4 - (a + b*tan(c + d*x))^(1/2)*(1120*
B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^22*d^5 - 672*B^4*a^11*b^20*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24*d^5
+ 2016*B^4*a^17*b^14*d^5 + 1568*B^4*a^19*b^12*d^5 + 608*B^4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(((96*B^4*a
^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^
4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 + (log(((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2
*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(((
((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6
*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(512*B*a^8*b^28*d^8 - ((-((96*B^4*a^2*b^4*d^4 - 16*B^4*
b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 +
3*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^9*b^28*d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^24*d^9
+ 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161280*a^19*b^18*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23*b^14*d
^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 768*a^29*b^8*d^9))/4 + 5248*B*a^10*b^26*d^8 + 23936*B*a^12*b^2
4*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16*b^20*d^8 + 130816*B*a^18*b^18*d^8 + 105728*B*a^20*b^16*d^8 + 578
56*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 + 4224*B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^8))/4 + (a + b*tan(c + d*
x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*b^24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272*B^2*a^14*b^20*d^7 + 9
856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7 + 15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^22*b^12*d^7 + 3712*B^2
*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) -
4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - 128*B^3*a^7
*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*a^11*b^22*d^6 + 10976*B^3*a^13*b^20*d^6 + 20384*B^3*a^15*b^18*d^6
+ 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b^14*d^6 + 3232*B^3*a^21*b^12*d^6 + 96*B^3*a^23*b^10*d^6 - 96*B^3*a
^25*b^8*d^6))/4 - (a + b*tan(c + d*x))^(1/2)*(1120*B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^22*d^5 - 672*B^4*a^11*b^2
0*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24*d^5 + 2016*B^4*a^17*b^14*d^5 + 1568*B^4*a^19*b^12*d^5 + 608*B^
4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) -
4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - log((((96*B
^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 +
16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b
^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))
^(1/2)*((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2
)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^9*b^28*
d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^24*d^9 + 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161280*a^19*b^18
*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23*b^14*d^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 768*a^29*b^8*d^
9) + 512*B*a^8*b^28*d^8 + 5248*B*a^10*b^26*d^8 + 23936*B*a^12*b^24*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16
*b^20*d^8 + 130816*B*a^18*b^18*d^8 + 105728*B*a^20*b^16*d^8 + 57856*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 +
4224*B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^8) - (a + b*tan(c + d*x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*
b^24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272*B^2*a^14*b^20*d^7 + 9856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7
+ 15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^22*b^12*d^7 + 3712*B^2*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(((96*B^
4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 +
16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 128*B^3*a^7*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*
a^11*b^22*d^6 + 10976*B^3*a^13*b^20*d^6 + 20384*B^3*a^15*b^18*d^6 + 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b
^14*d^6 + 3232*B^3*a^21*b^12*d^6 + 96*B^3*a^23*b^10*d^6 - 96*B^3*a^25*b^8*d^6) + (a + b*tan(c + d*x))^(1/2)*(1
120*B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^22*d^5 - 672*B^4*a^11*b^20*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24
*d^5 + 2016*B^4*a^17*b^14*d^5 + 1568*B^4*a^19*b^12*d^5 + 608*B^4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(((96*B
^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 +
16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - log((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*
a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*
d^4))^(1/2)*(((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*
b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b
^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*
d^4 + 48*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^9*b^28*d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^
24*d^9 + 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161280*a^19*b^18*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23
*b^14*d^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 768*a^29*b^8*d^9) + 512*B*a^8*b^28*d^8 + 5248*B*a^10*b^
26*d^8 + 23936*B*a^12*b^24*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16*b^20*d^8 + 130816*B*a^18*b^18*d^8 + 105
728*B*a^20*b^16*d^8 + 57856*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 + 4224*B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^
8) - (a + b*tan(c + d*x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*b^24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272
*B^2*a^14*b^20*d^7 + 9856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7 + 15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^
22*b^12*d^7 + 3712*B^2*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^
4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^
2*d^4))^(1/2) - 128*B^3*a^7*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*a^11*b^22*d^6 + 10976*B^3*a^13*b^20*d^6
+ 20384*B^3*a^15*b^18*d^6 + 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b^14*d^6 + 3232*B^3*a^21*b^12*d^6 + 96*B
^3*a^23*b^10*d^6 - 96*B^3*a^25*b^8*d^6) + (a + b*tan(c + d*x))^(1/2)*(1120*B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^2
2*d^5 - 672*B^4*a^11*b^20*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24*d^5 + 2016*B^4*a^17*b^14*d^5 + 1568*B^
4*a^19*b^12*d^5 + 608*B^4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*
B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*
b^2*d^4))^(1/2) + (B*atan((B^5*a^3*b^28*d^4*(a + b*tan(c + d*x))^(1/2)*1024i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*
d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 3
02400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640
*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^5*b^26*d^4*(a + b*tan(c + d*x))^(1/2)*10240i)/((a^3)^(1/2
)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*
B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5
*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^7*b^24*d^4*(a + b*tan(c + d*x))^(1/2
)*47616i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a
^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16
*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^9*b^22*d^4*(a +
b*tan(c + d*x))^(1/2)*133184i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b
^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16
*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) +
(B^5*a^11*b^20*d^4*(a + b*tan(c + d*x))^(1/2)*244160i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^2
6*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4
+ 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 57
6*B^5*a^22*b^8*d^4)) + (B^5*a^13*b^18*d^4*(a + b*tan(c + d*x))^(1/2)*302400i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*
d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 3
02400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640
*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^15*b^16*d^4*(a + b*tan(c + d*x))^(1/2)*253120i)/((a^3)^(1
/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 24416
0*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B
^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^17*b^14*d^4*(a + b*tan(c + d*x))^(
1/2)*139456i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B
^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*
a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^19*b^12*d^4
*(a + b*tan(c + d*x))^(1/2)*47424i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a
^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*
b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)
) + (B^5*a^21*b^10*d^4*(a + b*tan(c + d*x))^(1/2)*8640i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b
^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d
^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 +
576*B^5*a^22*b^8*d^4)) + (B^5*a^23*b^8*d^4*(a + b*tan(c + d*x))^(1/2)*576i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^
4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302
400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B
^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)))*2i)/(d*(a^3)^(1/2)) + (2*B*b^2)/(d*(a*b^2 + a^3)*(a + b*tan(c + d*x
))^(1/2))