\(\int (c \cot (a+b x))^{4/3} \, dx\) [17]

Optimal result

Integrand size = 12, antiderivative size = 242 \[ \int (c \cot (a+b x))^{4/3} \, dx=\frac {c^{4/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {c^{4/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}+\frac {c^{4/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac {\sqrt {3} c^{4/3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{4/3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b} \]

[Out]

c^(4/3)*arctan((c*cot(b*x+a))^(1/3)/c^(1/3))/b+1/2*c^(4/3)*arctan(2*(c*cot(b*x+a))^(1/3)/c^(1/3)-3^(1/2))/b+1/
2*c^(4/3)*arctan(2*(c*cot(b*x+a))^(1/3)/c^(1/3)+3^(1/2))/b-3*c*(c*cot(b*x+a))^(1/3)/b-1/4*c^(4/3)*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^(4/3)*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 = 0.44 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3554, 3557, 335, 215, 648, 632, 210, 642, 209} \[ \int (c \cot (a+b x))^{4/3} \, dx=\frac {c^{4/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {c^{4/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}+\frac {c^{4/3} \arctan \left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b}-\frac {\sqrt {3} c^{4/3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{4/3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b} \]

[In]

Int[(c*Cot[a + b*x])^(4/3),x]

[Out]

(c^(4/3)*ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)])/b - (c^(4/3)*ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1
/3)])/(2*b) + (c^(4/3)*ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)])/(2*b) - (3*c*(c*Cot[a + b*x])^(1/
3))/b - (Sqrt[3]*c^(4/3)*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^(4/3)*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 215

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] +
Int[(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/
(r^2 + s^2*x^2), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && 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 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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 {3 c \sqrt [3]{c \cot (a+b x)}}{b}-c^2 \int \frac {1}{(c \cot (a+b x))^{2/3}} \, dx \\ & = -\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}+\frac {c^3 \text {Subst}\left (\int \frac {1}{x^{2/3} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b} \\ & = -\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b} \\ & = -\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}+\frac {c^{4/3} \text {Subst}\left (\int \frac {\sqrt [3]{c}-\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^{4/3} \text {Subst}\left (\int \frac {\sqrt [3]{c}+\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^{5/3} \text {Subst}\left (\int \frac {1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b} \\ & = \frac {c^{4/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac {\left (\sqrt {3} c^{4/3}\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^{4/3}\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^{5/3} \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^{5/3} \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^{4/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac {\sqrt {3} c^{4/3} \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^{4/3} \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^{4/3} \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^{4/3} \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^{4/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {c^{4/3} \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^{4/3} \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 {3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac {\sqrt {3} c^{4/3} \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^{4/3} \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 = 0.39 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.85 \[ \int (c \cot (a+b x))^{4/3} \, dx=-\frac {c \sqrt [3]{c \cot (a+b x)} \left (6 \sqrt [6]{\cot ^2(a+b x)}-i \log \left (1-i \sqrt [6]{\cot ^2(a+b x)}\right )+i \log \left (1+i \sqrt [6]{\cot ^2(a+b x)}\right )-(-1)^{5/6} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+(-1)^{5/6} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )-\sqrt [6]{-1} \log \left (1-(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )+\sqrt [6]{-1} \log \left (1+(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )\right )}{2 b \sqrt [6]{\cot ^2(a+b x)}} \]

[In]

Integrate[(c*Cot[a + b*x])^(4/3),x]

[Out]

-1/2*(c*(c*Cot[a + b*x])^(1/3)*(6*(Cot[a + b*x]^2)^(1/6) - I*Log[1 - I*(Cot[a + b*x]^2)^(1/6)] + I*Log[1 + I*(
Cot[a + b*x]^2)^(1/6)] - (-1)^(5/6)*Log[1 - (-1)^(1/6)*(Cot[a + b*x]^2)^(1/6)] + (-1)^(5/6)*Log[1 + (-1)^(1/6)
*(Cot[a + b*x]^2)^(1/6)] - (-1)^(1/6)*Log[1 - (-1)^(5/6)*(Cot[a + b*x]^2)^(1/6)] + (-1)^(1/6)*Log[1 + (-1)^(5/
6)*(Cot[a + b*x]^2)^(1/6)]))/(b*(Cot[a + b*x]^2)^(1/6))

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-\frac {3 c \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{b}-\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}+\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 b}\)	\(214\)
default	\(-\frac {3 c \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{b}-\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}+\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 b}\)	\(214\)

[In]

int((c*cot(b*x+a))^(4/3),x,method=_RETURNVERBOSE)

[Out]

-3*c*(c*cot(b*x+a))^(1/3)/b-1/4/b*c*3^(1/2)*(c^2)^(1/6)*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))+1/2/b*c*(c^2)^(1/6)*arctan(2*(c*cot(b*x+a))^(1/3)/(c^2)^(1/6)-3^(1/2))+1/b*c*(c^2)^(1/
6)*arctan((c*cot(b*x+a))^(1/3)/(c^2)^(1/6))+1/4/b*c*3^(1/2)*(c^2)^(1/6)*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))+1/2/b*c*(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. 431 vs. \(2 (184) = 368\).

Time = 0.30 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.78 \[ \int (c \cot (a+b x))^{4/3} \, dx=\frac {\left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b + b\right )} \log \left (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}} + \frac {1}{2} \, \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b + b\right )}\right ) - \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b + b\right )} \log \left (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}} - \frac {1}{2} \, \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b + b\right )}\right ) + \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b - b\right )} \log \left (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}} + \frac {1}{2} \, \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b - b\right )}\right ) - \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b - b\right )} \log \left (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}} - \frac {1}{2} \, \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} b - b\right )}\right ) + 2 \, \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} b \log \left (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}} + \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} b\right ) - 2 \, \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} b \log \left (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}} - \left (-\frac {c^{8}}{b^{6}}\right )^{\frac {1}{6}} b\right ) - 12 \, 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}}}{4 \, b} \]

[In]

integrate((c*cot(b*x+a))^(4/3),x, algorithm="fricas")

[Out]

1/4*((-c^8/b^6)^(1/6)*(sqrt(-3)*b + b)*log(c*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3) + 1/2*(-c^8/b^6
)^(1/6)*(sqrt(-3)*b + b)) - (-c^8/b^6)^(1/6)*(sqrt(-3)*b + b)*log(c*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a)
)^(1/3) - 1/2*(-c^8/b^6)^(1/6)*(sqrt(-3)*b + b)) + (-c^8/b^6)^(1/6)*(sqrt(-3)*b - b)*log(c*((c*cos(2*b*x + 2*a
) + c)/sin(2*b*x + 2*a))^(1/3) + 1/2*(-c^8/b^6)^(1/6)*(sqrt(-3)*b - b)) - (-c^8/b^6)^(1/6)*(sqrt(-3)*b - b)*lo
g(c*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3) - 1/2*(-c^8/b^6)^(1/6)*(sqrt(-3)*b - b)) + 2*(-c^8/b^6)^
(1/6)*b*log(c*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3) + (-c^8/b^6)^(1/6)*b) - 2*(-c^8/b^6)^(1/6)*b*l
og(c*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3) - (-c^8/b^6)^(1/6)*b) - 12*c*((c*cos(2*b*x + 2*a) + c)/
sin(2*b*x + 2*a))^(1/3))/b

Sympy [F]

\[ \int (c \cot (a+b x))^{4/3} \, dx=\int \left (c \cot {\left (a + b x \right )}\right )^{\frac {4}{3}}\, dx \]

[In]

integrate((c*cot(b*x+a))**(4/3),x)

[Out]

Integral((c*cot(a + b*x))**(4/3), x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.81 \[ \int (c \cot (a+b x))^{4/3} \, dx=\frac {{\left (\sqrt {3} c^{\frac {1}{3}} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right ) - \sqrt {3} c^{\frac {1}{3}} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right ) + 2 \, c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) + 4 \, c^{\frac {1}{3}} \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) - 12 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}\right )} c}{4 \, b} \]

[In]

integrate((c*cot(b*x+a))^(4/3),x, algorithm="maxima")

[Out]

1/4*(sqrt(3)*c^(1/3)*log(sqrt(3)*c^(1/3)*(c/tan(b*x + a))^(1/3) + c^(2/3) + (c/tan(b*x + a))^(2/3)) - sqrt(3)*
c^(1/3)*log(-sqrt(3)*c^(1/3)*(c/tan(b*x + a))^(1/3) + c^(2/3) + (c/tan(b*x + a))^(2/3)) + 2*c^(1/3)*arctan((sq
rt(3)*c^(1/3) + 2*(c/tan(b*x + a))^(1/3))/c^(1/3)) + 2*c^(1/3)*arctan(-(sqrt(3)*c^(1/3) - 2*(c/tan(b*x + a))^(
1/3))/c^(1/3)) + 4*c^(1/3)*arctan((c/tan(b*x + a))^(1/3)/c^(1/3)) - 12*(c/tan(b*x + a))^(1/3))*c/b

Giac [F]

\[ \int (c \cot (a+b x))^{4/3} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((c*cot(b*x+a))^(4/3),x, algorithm="giac")

[Out]

integrate((c*cot(b*x + a))^(4/3), x)

Mupad [B] (verification not implemented)

Time = 12.89 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int (c \cot (a+b x))^{4/3} \, dx=-\frac {3\,c\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{5/6}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left ({\left (-1\right )}^{1/6}\,c^{1/3}-2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}-\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}-{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}-{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]

[In]

int((c*cot(a + b*x))^(4/3),x)

[Out]

((-1)^(1/6)*c^(4/3)*atan(((-1)^(5/6)*(c*cot(a + b*x))^(1/3)*1i)/c^(1/3))*1i)/b - (3*c*(c*cot(a + b*x))^(1/3))/
b - ((-1)^(1/6)*c^(4/3)*log((-1)^(1/6)*c^(1/3) - 2*(c*cot(a + b*x))^(1/3) + (-1)^(2/3)*3^(1/2)*c^(1/3))*((3^(1
/2)*1i)/2 + 1/2))/(2*b) - ((-1)^(1/6)*c^(4/3)*log(2*(c*cot(a + b*x))^(1/3) + (-1)^(1/6)*c^(1/3) - (-1)^(2/3)*3
^(1/2)*c^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(2*b) + ((-1)^(1/6)*c^(4/3)*log(2*(c*cot(a + b*x))^(1/3) + (-1)^(1/6)*
c^(1/3) + (-1)^(2/3)*3^(1/2)*c^(1/3))*((3^(1/2)*1i)/4 + 1/4))/b + ((-1)^(1/6)*c^(4/3)*log(2*(c*cot(a + b*x))^(
1/3) - (-1)^(1/6)*c^(1/3) + (-1)^(2/3)*3^(1/2)*c^(1/3))*((3^(1/2)*1i)/4 - 1/4))/b