\(\int (c \cot (a+b x))^{2/3} \, dx\) [18]

Optimal result

Integrand size = 12, antiderivative size = 225 \[ \int (c \cot (a+b x))^{2/3} \, dx=-\frac {c^{2/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}+\frac {c^{2/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {c^{2/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {\sqrt {3} c^{2/3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{2/3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3557, 335, 301, 648, 632, 210, 642, 209} \[ \int (c \cot (a+b x))^{2/3} \, dx=-\frac {c^{2/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}+\frac {c^{2/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {c^{2/3} \arctan \left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b}-\frac {\sqrt {3} c^{2/3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{2/3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b} \]

[In]

Int[(c*Cot[a + b*x])^(2/3),x]

[Out]

-((c^(2/3)*ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)])/b) + (c^(2/3)*ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c
^(1/3)])/(2*b) - (c^(2/3)*ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)])/(2*b) - (Sqrt[3]*c^(2/3)*Log[c
^(2/3) - Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b) + (Sqrt[3]*c^(2/3)*Log[c^(2/3
) + Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b)

Rule 209

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

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 301

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] +
Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 {c \text {Subst}\left (\int \frac {x^{2/3}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b} \\ & = -\frac {(3 c) \text {Subst}\left (\int \frac {x^4}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b} \\ & = -\frac {c^{2/3} \text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c}}{2}+\frac {\sqrt {3} x}{2}}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}-\frac {c^{2/3} \text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c}}{2}-\frac {\sqrt {3} x}{2}}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}-\frac {c \text {Subst}\left (\int \frac {1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b} \\ & = -\frac {c^{2/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {\left (\sqrt {3} c^{2/3}\right ) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [3]{c}+2 x}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b}+\frac {\left (\sqrt {3} c^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [3]{c}+2 x}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b}-\frac {c \text {Subst}\left (\int \frac {1}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b}-\frac {c \text {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b} \\ & = -\frac {c^{2/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{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}-\frac {c^{2/3} \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{2 \sqrt {3} b}+\frac {c^{2/3} \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{2 \sqrt {3} b} \\ & = -\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 (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b}-\frac {c^{2/3} \arctan \left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\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} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.82 \[ \int (c \cot (a+b x))^{2/3} \, dx=\frac {(c \cot (a+b x))^{5/3} \left (-i \log \left (1-i \sqrt [6]{\cot ^2(a+b x)}\right )+i \log \left (1+i \sqrt [6]{\cot ^2(a+b x)}\right )+\sqrt [6]{-1} \left (-\log \left (1-\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+\log \left (1+\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+(-1)^{2/3} \left (-\log \left (1-(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )+\log \left (1+(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )\right )\right )\right )}{2 b c \cot ^2(a+b x)^{5/6}} \]

[In]

Integrate[(c*Cot[a + b*x])^(2/3),x]

[Out]

((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)*(-
Log[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))

Maple [A] (verified)

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	\(-\frac {3 c \left (\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}\)	\(191\)

[In]

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

[Out]

-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)))

Fricas [B] (verification not implemented)

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 ) \]

[In]

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

[Out]

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)) + 1/4*(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
))

Sympy [F]

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

[In]

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

[Out]

Integral((c*cot(a + b*x))**(2/3), x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.81 \[ \int (c \cot (a+b x))^{2/3} \, dx=\frac {{\left (\frac {\sqrt {3} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c}{4 \, b} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((c*cot(b*x + a))^(2/3), x)

Mupad [B] (verification not implemented)

Time = 12.60 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.16 \[ \int (c \cot (a+b x))^{2/3} \, dx=-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}-\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}-\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}+\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}+\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]

[In]

int((c*cot(a + b*x))^(2/3),x)

[Out]

((-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