\(\int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) [322]

Optimal result

Integrand size = 33, antiderivative size = 167 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {(A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\sqrt {a-i b} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3689, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {(2 a B+A b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\sqrt {a-i b} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

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

[Out]

-(((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) + (Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sq
rt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a +
I*b]])/d - (A*Cot[c + d*x]*Sqrt[a + b*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 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 3689

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

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

Mathematica [A] (verified)

Time = 2.65 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.41 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {-\frac {(A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\frac {\left (A \left (b^2+a \sqrt {-b^2}\right )+b \left (a-\sqrt {-b^2}\right ) B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}+\frac {\left (A \left (b^2-a \sqrt {-b^2}\right )+b \left (a+\sqrt {-b^2}\right ) B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}-A b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{b}}{d} \]

[In]

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

[Out]

(-(((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a]) + (((A*(b^2 + a*Sqrt[-b^2]) + b*(a - Sqr
t[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] + ((A*(b^2 - a*Sqrt[-
b^2]) + b*(a + Sqrt[-b^2])*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] - A
*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/b)/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1028\) vs. \(2(141)=282\).

Time = 0.22 (sec) , antiderivative size = 1029, normalized size of antiderivative = 6.16

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1029\)
default	\(\text {Expression too large to display}\)	\(1029\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (135) = 270\).

Time = 0.88 (sec) , antiderivative size = 2579, normalized size of antiderivative = 15.44 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((A + B*tan(c + d*x))*sqrt(a + b*tan(c + d*x))*cot(c + d*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

integrate((B*tan(d*x + c) + A)*sqrt(b*tan(d*x + c) + a)*cot(d*x + c)^2, x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 8.83 (sec) , antiderivative size = 10987, normalized size of antiderivative = 65.79 \[ \int \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((((16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4
*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*
b^9))/d^4 + ((A*b + 2*B*a)*((8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*b^12*
d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^4*b^8
*d^2))/d^5 - (((16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^2 - 8*A
^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4 + ((A*b + 2*B*a)*((8*(32*A*b^11*d^4 + 48*B*a*b^10
*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 - (8*(A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(
c + d*x))^(1/2))/(a^(1/2)*d^5)))/(2*a^(1/2)*d))*(A*b + 2*B*a))/(2*a^(1/2)*d)))/(2*a^(1/2)*d))*(A*b + 2*B*a)*1i
)/(2*a^(1/2)*d) + (((16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10
+ 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4
*A^3*B*a^3*b^9))/d^4 - ((A*b + 2*B*a)*((8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*
A^2*B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^
2*B*a^4*b^8*d^2))/d^5 + (((16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^1
1*d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4 - ((A*b + 2*B*a)*((8*(32*A*b^11*d^4 +
48*B*a*b^10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 + (8*(A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*
(a + b*tan(c + d*x))^(1/2))/(a^(1/2)*d^5)))/(2*a^(1/2)*d))*(A*b + 2*B*a))/(2*a^(1/2)*d)))/(2*a^(1/2)*d))*(A*b
+ 2*B*a)*1i)/(2*a^(1/2)*d))/((16*(A^5*b^13 + 2*A*B^4*b^13 + 4*B^5*a*b^12 + 3*A^3*B^2*b^13 + 3*A^5*a^2*b^11 + 2
*A^5*a^4*b^9 + 4*B^5*a^3*b^10 + 7*A^2*B^3*a^3*b^10 + 4*A^2*B^3*a^5*b^8 - A^3*B^2*a^2*b^11 - 4*A^3*B^2*a^4*b^9
- A^4*B*a*b^12 - 4*A*B^4*a^2*b^11 - 6*A*B^4*a^4*b^9 + 3*A^2*B^3*a*b^12 + 3*A^4*B*a^3*b^10 + 4*A^4*B*a^5*b^8))/
d^5 - (((16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*
b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b
^9))/d^4 + ((A*b + 2*B*a)*((8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*b^12*d
^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^4*b^8*
d^2))/d^5 - (((16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^2 - 8*A^
2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4 + ((A*b + 2*B*a)*((8*(32*A*b^11*d^4 + 48*B*a*b^10*
d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 - (8*(A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c
+ d*x))^(1/2))/(a^(1/2)*d^5)))/(2*a^(1/2)*d))*(A*b + 2*B*a))/(2*a^(1/2)*d)))/(2*a^(1/2)*d))*(A*b + 2*B*a))/(2
*a^(1/2)*d) + (((16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*
A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3
*B*a^3*b^9))/d^4 - ((A*b + 2*B*a)*((8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*
B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*
a^4*b^8*d^2))/d^5 + (((16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^
2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4 - ((A*b + 2*B*a)*((8*(32*A*b^11*d^4 + 48*B
*a*b^10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 + (8*(A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a +
b*tan(c + d*x))^(1/2))/(a^(1/2)*d^5)))/(2*a^(1/2)*d))*(A*b + 2*B*a))/(2*a^(1/2)*d)))/(2*a^(1/2)*d))*(A*b + 2*
B*a))/(2*a^(1/2)*d)))*(A*b + 2*B*a)*1i)/(a^(1/2)*d) - atan(((((((8*(32*A*b^11*d^4 + 48*B*a*b^10*d^4 + 32*A*a^2
*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*
b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (
A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*
a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) +
(A*B*b)/(2*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^1
1*d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4
*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/
(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*
B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*
a^4*b^8*d^2))/d^5)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A
^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*(a + b*tan(c +
d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^
2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b^9))/d^4)*((2*A^2*B^2*b^2*d
^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a
)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*1i - (((((8*(32*A*b^11*d^4 + 48*B*a*b^10*d^4 + 32*A*a^2*b
^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^
2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^
2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^
2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (
A*B*b)/(2*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*
d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A
^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4
*d^2) + (A*B*b)/(2*d^2))^(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*
b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^
4*b^8*d^2))/d^5)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3
*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + (16*(a + b*tan(c + d*
x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*
B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b^9))/d^4)*((2*A^2*B^2*b^2*d^4
- B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/
(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*1i)/((16*(A^5*b^13 + 2*A*B^4*b^13 + 4*B^5*a*b^12 + 3*A^3*B^
2*b^13 + 3*A^5*a^2*b^11 + 2*A^5*a^4*b^9 + 4*B^5*a^3*b^10 + 7*A^2*B^3*a^3*b^10 + 4*A^2*B^3*a^5*b^8 - A^3*B^2*a^
2*b^11 - 4*A^3*B^2*a^4*b^9 - A^4*B*a*b^12 - 4*A*B^4*a^2*b^11 - 6*A*B^4*a^4*b^9 + 3*A^2*B^3*a*b^12 + 3*A^4*B*a^
3*b^10 + 4*A^4*B*a^5*b^8))/d^5 + (((((8*(32*A*b^11*d^4 + 48*B*a*b^10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4
))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A
^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4
*d^2) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A
*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + (
16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^2 - 8*A^2*a*b^10*d^2 +
12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*
d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^
(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*b^12*d^2 - 20*A^3*a*b^11*
d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^4*b^8*d^2))/d^5)*((2*A^2*
B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4)
- (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2
*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*
b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b^9))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^
2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2)
+ (A*B*b)/(2*d^2))^(1/2) + (((((8*(32*A*b^11*d^4 + 48*B*a*b^10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^
5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^
2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2)
+ (A*B*b)/(2*d^2))^(1/2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*
a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*(a
+ b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^2 - 8*A^2*a*b^10*d^2 + 12*B^
2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 +
4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)
- (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*b^12*d^2 - 20*A^3*a*b^11*d^2 +
60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^4*b^8*d^2))/d^5)*((2*A^2*B^2*b
^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A
^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*
b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11
+ 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b^9))/d^4)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2
*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A
*B*b)/(2*d^2))^(1/2)))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 -
4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*2i - atan(((((((8
*(32*A*b^11*d^4 + 48*B*a*b^10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*
d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*
B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((
B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^
3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a
^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2)
)/d^4)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4
+ 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^
3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^
3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^4*b^8*d^2))/d^5)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2
*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) +
(A*B*b)/(2*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*
b^10 + 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^
9 - 4*A^3*B*a^3*b^9))/d^4)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a
^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*1i - (((((8*(
32*A*b^11*d^4 + 48*B*a*b^10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^
4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^
2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((B^
2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*
a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3
*b^8*d^2 - 20*A^2*a^3*b^8*d^2 + 32*A*B*b^11*d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/
d^4)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 +
4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*
a^3*b^9*d^2 + 12*B^3*a^4*b^8*d^2 + 28*A^2*B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*
b^9*d^2 - 8*A^2*B*a^2*b^10*d^2 - 36*A^2*B*a^4*b^8*d^2))/d^5)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b
^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A
*B*b)/(2*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^
10 + 2*A^4*a^4*b^8 + 6*B^4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9
- 4*A^3*B*a^3*b^9))/d^4)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2
*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*1i)/((16*(A^5*b
^13 + 2*A*B^4*b^13 + 4*B^5*a*b^12 + 3*A^3*B^2*b^13 + 3*A^5*a^2*b^11 + 2*A^5*a^4*b^9 + 4*B^5*a^3*b^10 + 7*A^2*B
^3*a^3*b^10 + 4*A^2*B^3*a^5*b^8 - A^3*B^2*a^2*b^11 - 4*A^3*B^2*a^4*b^9 - A^4*B*a*b^12 - 4*A*B^4*a^2*b^11 - 6*A
*B^4*a^4*b^9 + 3*A^2*B^3*a*b^12 + 3*A^4*B*a^3*b^10 + 4*A^4*B*a^5*b^8))/d^5 + (((((8*(32*A*b^11*d^4 + 48*B*a*b^
10*d^4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(
1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 +
4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((B^2*a)/(4*d^2) - (A^2*a)/(4
*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4
)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8
*d^2 + 32*A*B*b^11*d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4)*((B^2*a)/(4*d^2) - (
A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*
B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*
b^8*d^2 + 28*A^2*B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^1
0*d^2 - 36*A^2*B*a^4*b^8*d^2))/d^5)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*
A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (1
6*(a + b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*b^8 + 6*B^
4*a^4*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b^9))/d^4)*
((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*
B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + (((((8*(32*A*b^11*d^4 + 48*B*a*b^10*d^
4 + 32*A*a^2*b^9*d^4 + 48*B*a^3*b^8*d^4))/d^5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*
((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*
B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/d^4)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2)
- (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/
2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(36*B^2*a^3*b^8*d^2 - 20*A^2*a^3*b^8*d^2
+ 32*A*B*b^11*d^2 - 8*A^2*a*b^10*d^2 + 12*B^2*a*b^10*d^2 + 64*A*B*a^2*b^9*d^2))/d^4)*((B^2*a)/(4*d^2) - (A^2*a
)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b
*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (8*(12*B^3*a^2*b^10*d^2 - 20*A^3*a^3*b^9*d^2 + 12*B^3*a^4*b^8*d
^2 + 28*A^2*B*b^12*d^2 - 20*A^3*a*b^11*d^2 + 60*A*B^2*a*b^11*d^2 + 60*A*B^2*a^3*b^9*d^2 - 8*A^2*B*a^2*b^10*d^2
- 36*A^2*B*a^4*b^8*d^2))/d^5)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B
^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + (16*(a
+ b*tan(c + d*x))^(1/2)*(3*A^4*b^12 + 2*B^4*b^12 + 3*A^2*B^2*b^12 + 3*A^4*a^2*b^10 + 2*A^4*a^4*b^8 + 6*B^4*a^4
*b^8 + 29*A^2*B^2*a^2*b^10 - 4*A*B^3*a*b^11 + 8*A^3*B*a*b^11 + 20*A*B^3*a^3*b^9 - 4*A^3*B*a^3*b^9))/d^4)*((B^2
*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a
*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2
*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4
) + (A*B*b)/(2*d^2))^(1/2)*2i + (A*b*(a + b*tan(c + d*x))^(1/2))/(a*d - d*(a + b*tan(c + d*x)))