\(\int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx\) [14]

Optimal result

Integrand size = 12, antiderivative size = 212 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}} \]

[Out]

-1/2*arctan(1-2^(1/2)*(c*cot(b*x+a))^(1/2)/c^(1/2))/b/c^(3/2)*2^(1/2)+1/2*arctan(1+2^(1/2)*(c*cot(b*x+a))^(1/2
)/c^(1/2))/b/c^(3/2)*2^(1/2)+1/4*ln(c^(1/2)+cot(b*x+a)*c^(1/2)-2^(1/2)*(c*cot(b*x+a))^(1/2))/b/c^(3/2)*2^(1/2)
-1/4*ln(c^(1/2)+cot(b*x+a)*c^(1/2)+2^(1/2)*(c*cot(b*x+a))^(1/2))/b/c^(3/2)*2^(1/2)+2/b/c/(c*cot(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}+1\right )}{\sqrt {2} b c^{3/2}}+\frac {\log \left (\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}} \]

[In]

Int[(c*Cot[a + b*x])^(-3/2),x]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(3/2))) + ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a +
b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(3/2)) + 2/(b*c*Sqrt[c*Cot[a + b*x]]) + Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqr
t[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(3/2)) - Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a +
b*x]]]/(2*Sqrt[2]*b*c^(3/2))

Rule 210

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3555

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\int \sqrt {c \cot (a+b x)} \, dx}{c^2} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\text {Subst}\left (\int \frac {c-x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {\text {Subst}\left (\int \frac {c+x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c}+2 x}{-c-\sqrt {2} \sqrt {c} x-x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c}-2 x}{-c+\sqrt {2} \sqrt {c} x-x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{c-\sqrt {2} \sqrt {c} x+x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 b c}+\frac {\text {Subst}\left (\int \frac {1}{c+\sqrt {2} \sqrt {c} x+x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}}{b c \sqrt {c \cot (a+b x)}} \]

[In]

Integrate[(c*Cot[a + b*x])^(-3/2),x]

[Out]

(2 + ArcTan[(-Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x]^2)^(1/4) - ArcTanh[(-Cot[a + b*x]^2)^(1/4)]*(-Cot[a + b*x]
^2)^(1/4))/(b*c*Sqrt[c*Cot[a + b*x]])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(-\frac {2 c \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{2} \left (c^{2}\right )^{\frac {1}{4}}}-\frac {1}{c^{2} \sqrt {c \cot \left (b x +a \right )}}\right )}{b}\)	\(157\)
default	\(-\frac {2 c \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{2} \left (c^{2}\right )^{\frac {1}{4}}}-\frac {1}{c^{2} \sqrt {c \cot \left (b x +a \right )}}\right )}{b}\)	\(157\)

[In]

int(1/(c*cot(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/b*c*(-1/8/c^2/(c^2)^(1/4)*2^(1/2)*(ln((c*cot(b*x+a)-(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)*2^(1/2)+(c^2)^(1/2))/(
c*cot(b*x+a)+(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)*2^(1/2)+(c^2)^(1/2)))+2*arctan(2^(1/2)/(c^2)^(1/4)*(c*cot(b*x+a)
)^(1/2)+1)-2*arctan(-2^(1/2)/(c^2)^(1/4)*(c*cot(b*x+a))^(1/2)+1))-1/c^2/(c*cot(b*x+a))^(1/2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {{\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + {\left (-i \, b c^{2} \cos \left (2 \, b x + 2 \, a\right ) - i \, b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (i \, b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + {\left (i \, b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + i \, b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) - {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (-b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + 4 \, \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right )}{2 \, {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )}} \]

[In]

integrate(1/(c*cot(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

1/2*((b*c^2*cos(2*b*x + 2*a) + b*c^2)*(-1/(b^4*c^6))^(1/4)*log(b^3*c^5*(-1/(b^4*c^6))^(3/4) + sqrt((c*cos(2*b*
x + 2*a) + c)/sin(2*b*x + 2*a))) + (-I*b*c^2*cos(2*b*x + 2*a) - I*b*c^2)*(-1/(b^4*c^6))^(1/4)*log(I*b^3*c^5*(-
1/(b^4*c^6))^(3/4) + sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))) + (I*b*c^2*cos(2*b*x + 2*a) + I*b*c^2)*(
-1/(b^4*c^6))^(1/4)*log(-I*b^3*c^5*(-1/(b^4*c^6))^(3/4) + sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))) - (
b*c^2*cos(2*b*x + 2*a) + b*c^2)*(-1/(b^4*c^6))^(1/4)*log(-b^3*c^5*(-1/(b^4*c^6))^(3/4) + sqrt((c*cos(2*b*x + 2
*a) + c)/sin(2*b*x + 2*a))) + 4*sqrt((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))*sin(2*b*x + 2*a))/(b*c^2*cos(2
*b*x + 2*a) + b*c^2)

Sympy [F]

\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(c*cot(b*x+a))**(3/2),x)

[Out]

Integral((c*cot(a + b*x))**(-3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {c {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}}}{c^{2}} + \frac {8}{c^{2} \sqrt {\frac {c}{\tan \left (b x + a\right )}}}\right )}}{4 \, b} \]

[In]

integrate(1/(c*cot(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

1/4*c*((2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(c) + 2*sqrt(c/tan(b*x + a)))/sqrt(c))/sqrt(c) + 2*sqrt(2)*a
rctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(c) - 2*sqrt(c/tan(b*x + a)))/sqrt(c))/sqrt(c) - sqrt(2)*log(sqrt(2)*sqrt(c)*s
qrt(c/tan(b*x + a)) + c + c/tan(b*x + a))/sqrt(c) + sqrt(2)*log(-sqrt(2)*sqrt(c)*sqrt(c/tan(b*x + a)) + c + c/
tan(b*x + a))/sqrt(c))/c^2 + 8/(c^2*sqrt(c/tan(b*x + a))))/b

Giac [F]

\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(c*cot(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((c*cot(b*x + a))^(-3/2), x)

Mupad [B] (verification not implemented)

Time = 12.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {2}{b\,c\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{3/2}} \]

[In]

int(1/(c*cot(a + b*x))^(3/2),x)

[Out]

2/(b*c*(c*cot(a + b*x))^(1/2)) + ((-1)^(1/4)*atan(((-1)^(1/4)*(c*cot(a + b*x))^(1/2))/c^(1/2)))/(b*c^(3/2)) -
((-1)^(1/4)*atanh(((-1)^(1/4)*(c*cot(a + b*x))^(1/2))/c^(1/2)))/(b*c^(3/2))