Integrand size = 12, antiderivative size = 225 \[ \int (c \cot (a+b x))^{2/3} \, dx=-\frac {c^{2/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}+\frac {c^{2/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {c^{2/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {\sqrt {3} c^{2/3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{2/3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b} \]
-c^(2/3)*arctan((c*cot(b*x+a))^(1/3)/c^(1/3))/b-1/2*c^(2/3)*arctan(2*(c*co t(b*x+a))^(1/3)/c^(1/3)-3^(1/2))/b-1/2*c^(2/3)*arctan(2*(c*cot(b*x+a))^(1/ 3)/c^(1/3)+3^(1/2))/b-1/4*c^(2/3)*ln(c^(2/3)+(c*cot(b*x+a))^(2/3)-c^(1/3)* (c*cot(b*x+a))^(1/3)*3^(1/2))*3^(1/2)/b+1/4*c^(2/3)*ln(c^(2/3)+(c*cot(b*x+ a))^(2/3)+c^(1/3)*(c*cot(b*x+a))^(1/3)*3^(1/2))*3^(1/2)/b
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.82 \[ \int (c \cot (a+b x))^{2/3} \, dx=\frac {(c \cot (a+b x))^{5/3} \left (-i \log \left (1-i \sqrt [6]{\cot ^2(a+b x)}\right )+i \log \left (1+i \sqrt [6]{\cot ^2(a+b x)}\right )+\sqrt [6]{-1} \left (-\log \left (1-\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+\log \left (1+\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+(-1)^{2/3} \left (-\log \left (1-(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )+\log \left (1+(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )\right )\right )\right )}{2 b c \cot ^2(a+b x)^{5/6}} \]
((c*Cot[a + b*x])^(5/3)*((-I)*Log[1 - I*(Cot[a + b*x]^2)^(1/6)] + I*Log[1 + I*(Cot[a + b*x]^2)^(1/6)] + (-1)^(1/6)*(-Log[1 - (-1)^(1/6)*(Cot[a + b*x ]^2)^(1/6)] + Log[1 + (-1)^(1/6)*(Cot[a + b*x]^2)^(1/6)] + (-1)^(2/3)*(-Lo g[1 - (-1)^(5/6)*(Cot[a + b*x]^2)^(1/6)] + Log[1 + (-1)^(5/6)*(Cot[a + b*x ]^2)^(1/6)]))))/(2*b*c*(Cot[a + b*x]^2)^(5/6))
Time = 0.39 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.87, 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, 824, 27, 216, 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.
| \(\displaystyle \int (c \cot (a+b x))^{2/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{2/3}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {c \int \frac {(c \cot (a+b x))^{2/3}}{\cot ^2(a+b x) c^2+c^2}d(c \cot (a+b x))}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {3 c \int \frac {c^4 \cot ^4(a+b x)}{c^6 \cot ^6(a+b x)+c^2}d\sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle -\frac {3 c \left (\frac {1}{3} \int \frac {1}{c^2 \cot ^2(a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}+\frac {\int -\frac {\sqrt [3]{c}-\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{2 \left (c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}\right )}d\sqrt [3]{c \cot (a+b x)}}{3 \sqrt [3]{c}}+\frac {\int -\frac {\sqrt [3]{c}+\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{2 \left (c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}\right )}d\sqrt [3]{c \cot (a+b x)}}{3 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 c \left (\frac {1}{3} \int \frac {1}{c^2 \cot ^2(a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {\int \frac {\sqrt [3]{c}-\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}-\frac {\int \frac {\sqrt [3]{c}+\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {3 c \left (-\frac {\int \frac {\sqrt [3]{c}-\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}-\frac {\int \frac {\sqrt [3]{c}+\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 c \left (-\frac {-\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-\frac {1}{3}}d\left (1-\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}\right )}{\sqrt {3}}}{6 \sqrt [3]{c}}-\frac {\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-\frac {1}{3}}d\left (\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}+1\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 \sqrt [3]{c}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}+\arctan \left (\sqrt {3} \left (1-\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}\right )\right )}{6 \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\arctan \left (\sqrt {3} \left (\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}+1\right )\right )}{6 \sqrt [3]{c}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 \sqrt [3]{c}}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 c \left (\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 \sqrt [3]{c}}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}\right )\right )-\frac {1}{2} \sqrt {3} \log \left (-\sqrt {3} c^{4/3} \cot (a+b x)+c^2 \cot ^2(a+b x)+c^{2/3}\right )}{6 \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \log \left (\sqrt {3} c^{4/3} \cot (a+b x)+c^2 \cot ^2(a+b x)+c^{2/3}\right )-\arctan \left (\sqrt {3} \left (\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}+1\right )\right )}{6 \sqrt [3]{c}}\right )}{b}\) |
(-3*c*(ArcTan[c^(2/3)*Cot[a + b*x]]/(3*c^(1/3)) - (ArcTan[Sqrt[3]*(1 - (2* c^(2/3)*Cot[a + b*x])/Sqrt[3])] - (Sqrt[3]*Log[c^(2/3) - Sqrt[3]*c^(4/3)*C ot[a + b*x] + c^2*Cot[a + b*x]^2])/2)/(6*c^(1/3)) - (-ArcTan[Sqrt[3]*(1 + (2*c^(2/3)*Cot[a + b*x])/Sqrt[3])] + (Sqrt[3]*Log[c^(2/3) + Sqrt[3]*c^(4/3 )*Cot[a + b*x] + c^2*Cot[a + b*x]^2])/2)/(6*c^(1/3))))/b
3.1.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
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 = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(-\frac {3 c \left (\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}\) | \(191\) |
| default | \(-\frac {3 c \left (\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}\) | \(191\) |
-3/b*c*(1/12/c^2*3^(1/2)*(c^2)^(5/6)*ln((c*cot(b*x+a))^(2/3)-3^(1/2)*(c^2) ^(1/6)*(c*cot(b*x+a))^(1/3)+(c^2)^(1/3))+1/6/(c^2)^(1/6)*arctan(2*(c*cot(b *x+a))^(1/3)/(c^2)^(1/6)-3^(1/2))+1/3/(c^2)^(1/6)*arctan((c*cot(b*x+a))^(1 /3)/(c^2)^(1/6))-1/12/c^2*3^(1/2)*(c^2)^(5/6)*ln((c*cot(b*x+a))^(2/3)+3^(1 /2)*(c^2)^(1/6)*(c*cot(b*x+a))^(1/3)+(c^2)^(1/3))+1/6/(c^2)^(1/6)*arctan(2 *(c*cot(b*x+a))^(1/3)/(c^2)^(1/6)+3^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (169) = 338\).
Time = 0.29 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.85 \[ \int (c \cot (a+b x))^{2/3} \, dx=\frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {1}{6}} \log \left (c^{3} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}} + \frac {1}{2} \, {\left (\sqrt {-3} b^{5} + b^{5}\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {5}{6}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {1}{6}} \log \left (c^{3} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}} - \frac {1}{2} \, {\left (\sqrt {-3} b^{5} + b^{5}\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {5}{6}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {1}{6}} \log \left (c^{3} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}} + \frac {1}{2} \, {\left (\sqrt {-3} b^{5} - b^{5}\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {5}{6}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {1}{6}} \log \left (c^{3} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}} - \frac {1}{2} \, {\left (\sqrt {-3} b^{5} - b^{5}\right )} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {5}{6}}\right ) - \frac {1}{2} \, \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {1}{6}} \log \left (b^{5} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {5}{6}} + c^{3} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {1}{6}} \log \left (-b^{5} \left (-\frac {c^{4}}{b^{6}}\right )^{\frac {5}{6}} + c^{3} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) \]
1/4*(sqrt(-3) - 1)*(-c^4/b^6)^(1/6)*log(c^3*((c*cos(2*b*x + 2*a) + c)/sin( 2*b*x + 2*a))^(1/3) + 1/2*(sqrt(-3)*b^5 + b^5)*(-c^4/b^6)^(5/6)) - 1/4*(sq rt(-3) - 1)*(-c^4/b^6)^(1/6)*log(c^3*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3) - 1/2*(sqrt(-3)*b^5 + b^5)*(-c^4/b^6)^(5/6)) + 1/4*(sqrt(-3) + 1)*(-c^4/b^6)^(1/6)*log(c^3*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^ (1/3) + 1/2*(sqrt(-3)*b^5 - b^5)*(-c^4/b^6)^(5/6)) - 1/4*(sqrt(-3) + 1)*(- c^4/b^6)^(1/6)*log(c^3*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3) - 1/2*(sqrt(-3)*b^5 - b^5)*(-c^4/b^6)^(5/6)) - 1/2*(-c^4/b^6)^(1/6)*log(b^5 *(-c^4/b^6)^(5/6) + c^3*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3)) + 1/2*(-c^4/b^6)^(1/6)*log(-b^5*(-c^4/b^6)^(5/6) + c^3*((c*cos(2*b*x + 2* a) + c)/sin(2*b*x + 2*a))^(1/3))
\[ \int (c \cot (a+b x))^{2/3} \, dx=\int \left (c \cot {\left (a + b x \right )}\right )^{\frac {2}{3}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.81 \[ \int (c \cot (a+b x))^{2/3} \, dx=\frac {{\left (\frac {\sqrt {3} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c}{4 \, b} \]
1/4*(sqrt(3)*log(sqrt(3)*c^(1/3)*(c/tan(b*x + a))^(1/3) + c^(2/3) + (c/tan (b*x + a))^(2/3))/c^(1/3) - sqrt(3)*log(-sqrt(3)*c^(1/3)*(c/tan(b*x + a))^ (1/3) + c^(2/3) + (c/tan(b*x + a))^(2/3))/c^(1/3) - 2*arctan((sqrt(3)*c^(1 /3) + 2*(c/tan(b*x + a))^(1/3))/c^(1/3))/c^(1/3) - 2*arctan(-(sqrt(3)*c^(1 /3) - 2*(c/tan(b*x + a))^(1/3))/c^(1/3))/c^(1/3) - 4*arctan((c/tan(b*x + a ))^(1/3)/c^(1/3))/c^(1/3))*c/b
\[ \int (c \cot (a+b x))^{2/3} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {2}{3}} \,d x } \]
Time = 12.60 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.16 \[ \int (c \cot (a+b x))^{2/3} \, dx=-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}-\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}-\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}+\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}+\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]
((-1)^(1/6)*c^(2/3)*log((972*c^9)/b^3 + (486*(-1)^(1/6)*c^(26/3)*(3^(1/2)* 1i - 1)*(c*cot(a + b*x))^(1/3))/b^3)*((3^(1/2)*1i)/4 - 1/4))/b - ((-1)^(1/ 6)*c^(2/3)*log((972*c^9)/b^3 - (486*(-1)^(1/6)*c^(26/3)*(3^(1/2)*1i - 1)*( c*cot(a + b*x))^(1/3))/b^3)*((3^(1/2)*1i)/2 - 1/2))/(2*b) - ((-1)^(1/6)*c^ (2/3)*log((972*c^9)/b^3 - (486*(-1)^(1/6)*c^(26/3)*(3^(1/2)*1i + 1)*(c*cot (a + b*x))^(1/3))/b^3)*((3^(1/2)*1i)/2 + 1/2))/(2*b) - ((-1)^(1/6)*c^(2/3) *atan(((-1)^(2/3)*(c*cot(a + b*x))^(1/3))/c^(1/3))*1i)/b + ((-1)^(1/6)*c^( 2/3)*log((972*c^9)/b^3 + (486*(-1)^(1/6)*c^(26/3)*(3^(1/2)*1i + 1)*(c*cot( a + b*x))^(1/3))/b^3)*((3^(1/2)*1i)/4 + 1/4))/b