Integrand size = 12, antiderivative size = 131 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b \sqrt [3]{c}}-\frac {\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac {\log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b \sqrt [3]{c}} \]
-1/2*ln(c^(2/3)+(c*cot(b*x+a))^(2/3))/b/c^(1/3)+1/4*ln(c^(4/3)-c^(2/3)*(c* cot(b*x+a))^(2/3)+(c*cot(b*x+a))^(4/3))/b/c^(1/3)+1/2*arctan(1/3*(c^(2/3)- 2*(c*cot(b*x+a))^(2/3))/c^(2/3)*3^(1/2))*3^(1/2)/b/c^(1/3)
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\frac {\sqrt [3]{\cot (a+b x)} \left (-2 \sqrt {3} \arctan \left (\frac {-1+2 \cot ^{\frac {2}{3}}(a+b x)}{\sqrt {3}}\right )-2 \log \left (1+\cot ^{\frac {2}{3}}(a+b x)\right )+\log \left (1-\cot ^{\frac {2}{3}}(a+b x)+\cot ^{\frac {4}{3}}(a+b x)\right )\right )}{4 b \sqrt [3]{c \cot (a+b x)}} \]
(Cot[a + b*x]^(1/3)*(-2*Sqrt[3]*ArcTan[(-1 + 2*Cot[a + b*x]^(2/3))/Sqrt[3] ] - 2*Log[1 + Cot[a + b*x]^(2/3)] + Log[1 - Cot[a + b*x]^(2/3) + Cot[a + b *x]^(4/3)]))/(4*b*(c*Cot[a + b*x])^(1/3))
Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3957, 266, 807, 750, 16, 1142, 25, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{-c \tan \left (a+b x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {c \int \frac {1}{\sqrt [3]{c \cot (a+b x)} \left (\cot ^2(a+b x) c^2+c^2\right )}d(c \cot (a+b x))}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {3 c \int \frac {\sqrt [3]{c \cot (a+b x)}}{c^6 \cot ^6(a+b x)+c^2}d\sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle -\frac {3 c \int \frac {1}{c^3 \cot ^3(a+b x)+c^2}d\left (c^2 \cot ^2(a+b x)\right )}{2 b}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {3 c \left (\frac {\int \frac {1}{c^2 \cot ^2(a+b x)+c^{2/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{4/3}}+\frac {\int \frac {2 c^{2/3}-c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{4/3}}\right )}{2 b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3 c \left (\frac {\int \frac {2 c^{2/3}-c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{4/3}}+\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{4/3}}\right )}{2 b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 c \left (\frac {\frac {3}{2} c^{2/3} \int \frac {1}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )-\frac {1}{2} \int -\frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{4/3}}+\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{4/3}}\right )}{2 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 c \left (\frac {\frac {3}{2} c^{2/3} \int \frac {1}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )+\frac {1}{2} \int \frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{4/3}}+\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{4/3}}\right )}{2 b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 c \left (\frac {\frac {1}{2} \int \frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )+3 \int \frac {1}{2 \sqrt [3]{c} \cot (a+b x)-4}d\left (1-2 \sqrt [3]{c} \cot (a+b x)\right )}{3 c^{4/3}}+\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{4/3}}\right )}{2 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 c \left (\frac {\frac {1}{2} \int \frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )-\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \cot (a+b x)}{\sqrt {3}}\right )}{3 c^{4/3}}+\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{4/3}}\right )}{2 b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 c \left (\frac {-\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \cot (a+b x)}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-c^{5/3} \cot (a+b x)+c^2 \cot ^2(a+b x)+c^{4/3}\right )}{3 c^{4/3}}+\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{4/3}}\right )}{2 b}\) |
(-3*c*(Log[c^(2/3) + c^2*Cot[a + b*x]^2]/(3*c^(4/3)) + (-(Sqrt[3]*ArcTan[( 1 - 2*c^(1/3)*Cot[a + b*x])/Sqrt[3]]) - Log[c^(4/3) - c^(5/3)*Cot[a + b*x] + c^2*Cot[a + b*x]^2]/2)/(3*c^(4/3))))/(2*b)
3.1.20.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(-\frac {3 c \left (\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}} \left (c^{2}\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{12 \left (c^{2}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}\right )}{b}\) | \(108\) |
| default | \(-\frac {3 c \left (\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}} \left (c^{2}\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{12 \left (c^{2}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}\right )}{b}\) | \(108\) |
-3/b*c*(1/6/(c^2)^(2/3)*ln((c*cot(b*x+a))^(2/3)+(c^2)^(1/3))-1/12/(c^2)^(2 /3)*ln((c*cot(b*x+a))^(4/3)-(c*cot(b*x+a))^(2/3)*(c^2)^(1/3)+(c^2)^(2/3))+ 1/6/(c^2)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2*(c*cot(b*x+a))^(2/3)/(c^2)^( 1/3)-1)))
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (100) = 200\).
Time = 0.27 (sec) , antiderivative size = 639, normalized size of antiderivative = 4.88 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\left [\frac {\sqrt {3} c \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \log \left (\frac {1}{2} \, \sqrt {3} {\left (\left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} {\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} - 2 \, c \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}} \sin \left (2 \, b x + 2 \, a\right ) + {\left (c \cos \left (2 \, b x + 2 \, a\right ) - c\right )} \left (-c\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - \frac {3}{2} \, \left (-c\right )^{\frac {1}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} {\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} + \frac {3}{2} \, c \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, c\right ) - 2 \, \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) + \left (-c\right )^{\frac {2}{3}} \log \left (-\frac {\left (-c\right )^{\frac {1}{3}} c \sin \left (2 \, b x + 2 \, a\right ) + \left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) - {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b c}, -\frac {2 \, \sqrt {3} c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \arctan \left (\frac {\sqrt {3} \left (-c\right )^{\frac {1}{3}} c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} + 2 \, \sqrt {3} \left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}}}{3 \, c}\right ) + 2 \, \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) - \left (-c\right )^{\frac {2}{3}} \log \left (-\frac {\left (-c\right )^{\frac {1}{3}} c \sin \left (2 \, b x + 2 \, a\right ) + \left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) - {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b c}\right ] \]
[1/4*(sqrt(3)*c*sqrt((-c)^(1/3)/c)*log(1/2*sqrt(3)*((-c)^(2/3)*((c*cos(2*b *x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)*(cos(2*b*x + 2*a) - 1) - 2*c*((c*co s(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3)*sin(2*b*x + 2*a) + (c*cos(2*b* x + 2*a) - c)*(-c)^(1/3))*sqrt((-c)^(1/3)/c) - 3/2*(-c)^(1/3)*((c*cos(2*b* x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)*(cos(2*b*x + 2*a) - 1) + 3/2*c*cos(2 *b*x + 2*a) + 1/2*c) - 2*(-c)^(2/3)*log((-c)^(2/3) + ((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)) + (-c)^(2/3)*log(-((-c)^(1/3)*c*sin(2*b*x + 2*a) + (-c)^(2/3)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)*sin(2* b*x + 2*a) - (c*cos(2*b*x + 2*a) + c)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3))/sin(2*b*x + 2*a)))/(b*c), -1/4*(2*sqrt(3)*c*sqrt(-(-c)^(1/3 )/c)*arctan(1/3*(sqrt(3)*(-c)^(1/3)*c*sqrt(-(-c)^(1/3)/c) + 2*sqrt(3)*(-c) ^(2/3)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)*sqrt(-(-c)^(1/3)/ c))/c) + 2*(-c)^(2/3)*log((-c)^(2/3) + ((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)) - (-c)^(2/3)*log(-((-c)^(1/3)*c*sin(2*b*x + 2*a) + (-c)^(2 /3)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)*sin(2*b*x + 2*a) - ( c*cos(2*b*x + 2*a) + c)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3)) /sin(2*b*x + 2*a)))/(b*c)]
\[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\int \frac {1}{\sqrt [3]{c \cot {\left (a + b x \right )}}}\, dx \]
Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=-\frac {c {\left (\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (c^{\frac {2}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} - \frac {\log \left (c^{\frac {4}{3}} - c^{\frac {2}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {4}{3}}\right )}{c^{\frac {4}{3}}} + \frac {2 \, \log \left (c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {4}{3}}}\right )}}{4 \, b} \]
-1/4*c*(2*sqrt(3)*arctan(-1/3*sqrt(3)*(c^(2/3) - 2*(c/tan(b*x + a))^(2/3)) /c^(2/3))/c^(4/3) - log(c^(4/3) - c^(2/3)*(c/tan(b*x + a))^(2/3) + (c/tan( b*x + a))^(4/3))/c^(4/3) + 2*log(c^(2/3) + (c/tan(b*x + a))^(2/3))/c^(4/3) )/b
\[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {1}{3}}} \,d x } \]
Time = 12.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=-\frac {\ln \left ({\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}+c^{2/3}\right )}{2\,b\,c^{1/3}}-\frac {\ln \left (\frac {81\,c^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^3}+\frac {162\,c^3\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,b\,c^{1/3}}+\frac {\ln \left (\frac {81\,c^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^3}-\frac {162\,c^3\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,b\,c^{1/3}} \]