\(\int \cot (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) [321]

Optimal result

Integrand size = 31, antiderivative size = 131 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {2 \sqrt {a} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {\sqrt {a-i b} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3693, 3620, 3618, 65, 214, 3715} \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {a-i b} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 \sqrt {a} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d} \]

[In]

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

[Out]

(-2*Sqrt[a]*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + (Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c
+ d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/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 3693

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

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.67 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {2 \sqrt {a} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )-\frac {\left (A \left (b^2+a \sqrt {-b^2}\right )+b \left (a-\sqrt {-b^2}\right ) B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}+\frac {\left (A \left (b^2-a \sqrt {-b^2}\right )+b \left (a+\sqrt {-b^2}\right ) B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}}{d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(975\) vs. \(2(107)=214\).

Time = 0.24 (sec) , antiderivative size = 976, normalized size of antiderivative = 7.45

method	result	size
derivativedivides	\(\frac {\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 ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\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 ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}+\frac {\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 ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\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 ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\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 ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \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 ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\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 ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\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 ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}-\frac {\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 ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\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 ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\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 ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \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 ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {a}}\right ) \sqrt {a}}{d}\)	\(976\)
default	\(\frac {\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 ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\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 ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}+\frac {\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 ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\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 ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\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 ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \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 ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\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 ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\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 ) B \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d b}-\frac {\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 ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\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 ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\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 ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \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 ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {a}}\right ) \sqrt {a}}{d}\)	\(976\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1218 vs. \(2 (101) = 202\).

Time = 0.53 (sec) , antiderivative size = 2452, normalized size of antiderivative = 18.72 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(cot(d*x+c)*(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), x)

Maxima [F]

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

[In]

integrate(cot(d*x+c)*(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), x)

Giac [F(-1)]

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

[In]

integrate(cot(d*x+c)*(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 = 10.09 (sec) , antiderivative size = 9785, normalized size of antiderivative = 74.69 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(A*a^(1/2)*atan(((A*a^(1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*
b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*A^3*B*a^3*b^9))/d^4 + (A*a^(1/2)*((32*(3*A^3*a^2*b
^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 -
9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*d^2))/d^5 + (A*a^(1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*
d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*A*B*a^2*b^9*d^2))/d^4 -
(A*a^(1/2)*((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 - (32*A*a^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a +
b*tan(c + d*x))^(1/2))/d^5))/d))/d))/d)*1i)/d + (A*a^(1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^1
2 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*A^3*B*a^3*b^9))/d^4
- (A*a^(1/2)*((32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^1
1*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*d^2))/d^5 - (A*a^(1/2)*((32*(a + b*tan(c
+ d*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^
2 + 24*A*B*a^2*b^9*d^2))/d^4 + (A*a^(1/2)*((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 + (32*A*a^(1/2)*(16*b
^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d))/d))/d)*1i)/d)/((64*(A^5*a*b^12 + A^5*a^3*b^10
+ A^2*B^3*a^2*b^11 + A^2*B^3*a^4*b^9 + 3*A^3*B^2*a^3*b^10 + A^3*B^2*a^5*b^8 + A*B^4*a*b^12 + 2*A*B^4*a^3*b^10
+ A*B^4*a^5*b^8 + 2*A^3*B^2*a*b^12 + A^4*B*a^2*b^11 + A^4*B*a^4*b^9))/d^5 - (A*a^(1/2)*((32*(a + b*tan(c + d*x
))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*
b^10 - 8*A^3*B*a^3*b^9))/d^4 + (A*a^(1/2)*((32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3
*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*d^2))/d^5 +
(A*a^(1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*
a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*A*B*a^2*b^9*d^2))/d^4 - (A*a^(1/2)*((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4
))/d^5 - (32*A*a^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d))/d))/d))/d + (A*a^(
1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 +
B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*A^3*B*a^3*b^9))/d^4 - (A*a^(1/2)*((32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8
*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 1
5*A^2*B*a^3*b^9*d^2))/d^5 - (A*a^(1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^
2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*A*B*a^2*b^9*d^2))/d^4 + (A*a^(1/2)*((32*(12*A*a
*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 + (32*A*a^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/
d^5))/d))/d))/d))/d))*2i)/d - atan(((((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 - (32*(16*b^10*d^4 + 24*
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) - (32*(a + b*tan(c + d*x))^(1/2)*(
10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*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) - (32*(3*A^3*a^2*b^10*d^2
+ 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*
a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*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
) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^
4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*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 - (((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 + (32*(16*b^10*d^4 + 24*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) + (32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*
d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*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) - (32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8
*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 1
5*A^2*B*a^3*b^9*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) + (32*(a + b*
tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A
^2*B^2*a^2*b^10 - 8*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)/((((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 - (32*(16*b^10*d^4 + 24*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) - (32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^
3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*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) - (32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*
b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9
*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) - (32*(a + b*tan(c + d*x))^(
1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10
- 8*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) + (((((32*(12*
A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 + (32*(16*b^10*d^4 + 24*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) + (32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B
*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*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) - (32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^
11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*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) + (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B
^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*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) + (64*(A^5*a*b^12 + A^5*a^3*b^10
+ A^2*B^3*a^2*b^11 + A^2*B^3*a^4*b^9 + 3*A^3*B^2*a^3*b^10 + A^3*B^2*a^5*b^8 + A*B^4*a*b^12 + 2*A*B^4*a^3*b^10
+ A*B^4*a^5*b^8 + 2*A^3*B^2*a*b^12 + A^4*B*a^2*b^11 + A^4*B*a^4*b^9))/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)*2i - atan(((((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 - (32*(16*b^
10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*
d^2))^(1/2))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(a + b*tan(c + d
*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 +
24*A*B*a^2*b^9*d^2))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(3*A^3*a
^2*b^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^
2 - 9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*d^2))/d^5)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(
2*d^2))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a
^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*A^3*B*a^3*b^9))/d^4)*((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) - (B^2*a)/(4*d^
2) - (A*B*b)/(2*d^2))^(1/2)*1i - (((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a^2
*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/d^
4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(10*
B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*A*B*a^2*b^9*
d^2))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(3*A^3*a^2*b^10*d^2 + 3
*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4
*b^8*d^2 - 15*A^2*B*a^3*b^9*d^2))/d^5)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) +
(32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a
^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*A^3*B*a^3*b^9))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2
*d^2))^(1/2)*1i)/((((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a +
b*tan(c + d*x))^(1/2)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*d^2
- 18*A^2*a^3*b^8*d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*A*B*a^2*b^9*d^2))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8*d^
2 + B^3*a^3*b^9*d^2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 15*A
^2*B*a^3*b^9*d^2))/d^5)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(a + b*tan
(c + d*x))^(1/2)*(A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*
B^2*a^2*b^10 - 8*A^3*B*a^3*b^9))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) +
(((((32*(12*A*a*b^10*d^4 + 12*A*a^3*b^8*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1
/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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2))/d^4)*((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) - (
B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(10*B^2*a^3*b^8*d^2 - 18*A^2*a^3*b^8*
d^2 + 16*A*B*b^11*d^2 - 6*A^2*a*b^10*d^2 + 6*B^2*a*b^10*d^2 + 24*A*B*a^2*b^9*d^2))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) - (32*(3*A^3*a^2*b^10*d^2 + 3*A^3*a^4*b^8*d^2 + B^3*a^3*b^9*d^
2 + B^3*a*b^11*d^2 - 15*A^2*B*a*b^11*d^2 - 9*A*B^2*a^2*b^10*d^2 - 9*A*B^2*a^4*b^8*d^2 - 15*A^2*B*a^3*b^9*d^2))
/d^5)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(
A^4*b^12 + B^4*b^12 + 2*A^2*B^2*b^12 + 3*A^4*a^4*b^8 + 2*B^4*a^2*b^10 + B^4*a^4*b^8 + 6*A^2*B^2*a^2*b^10 - 8*A
^3*B*a^3*b^9))/d^4)*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2) + (64*(A^5*a*b^12 +
A^5*a^3*b^10 + A^2*B^3*a^2*b^11 + A^2*B^3*a^4*b^9 + 3*A^3*B^2*a^3*b^10 + A^3*B^2*a^5*b^8 + A*B^4*a*b^12 + 2*A*
B^4*a^3*b^10 + A*B^4*a^5*b^8 + 2*A^3*B^2*a*b^12 + A^4*B*a^2*b^11 + A^4*B*a^4*b^9))/d^5))*((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) - (B^2*a)/(4*d^2) - (A*B*b)/(2*d^2))^(1/2)*2i