Integrand size = 12, antiderivative size = 242 \[ \int (c \cot (a+b x))^{4/3} \, dx=\frac {c^{4/3} \arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {c^{4/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}+\frac {c^{4/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac {\sqrt {3} c^{4/3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{4/3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b} \]
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
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.85 \[ \int (c \cot (a+b x))^{4/3} \, dx=-\frac {c \sqrt [3]{c \cot (a+b x)} \left (6 \sqrt [6]{\cot ^2(a+b x)}-i \log \left (1-i \sqrt [6]{\cot ^2(a+b x)}\right )+i \log \left (1+i \sqrt [6]{\cot ^2(a+b x)}\right )-(-1)^{5/6} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+(-1)^{5/6} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )-\sqrt [6]{-1} \log \left (1-(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )+\sqrt [6]{-1} \log \left (1+(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )\right )}{2 b \sqrt [6]{\cot ^2(a+b x)}} \]
-1/2*(c*(c*Cot[a + b*x])^(1/3)*(6*(Cot[a + b*x]^2)^(1/6) - I*Log[1 - I*(Co t[a + b*x]^2)^(1/6)] + I*Log[1 + I*(Cot[a + b*x]^2)^(1/6)] - (-1)^(5/6)*Lo g[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))
Time = 0.48 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.90, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3954, 3042, 3957, 266, 753, 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.
| \(\displaystyle \int (c \cot (a+b x))^{4/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{4/3}dx\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle c^2 \left (-\int \frac {1}{(c \cot (a+b x))^{2/3}}dx\right )-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c^2 \left (-\int \frac {1}{\left (-c \tan \left (a+b x+\frac {\pi }{2}\right )\right )^{2/3}}dx\right )-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {c^3 \int \frac {1}{(c \cot (a+b x))^{2/3} \left (\cot ^2(a+b x) c^2+c^2\right )}d(c \cot (a+b x))}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 c^3 \int \frac {1}{c^6 \cot ^6(a+b x)+c^2}d\sqrt [3]{c \cot (a+b x)}}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\int \frac {1}{c^2 \cot ^2(a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{3 c^{4/3}}+\frac {\int \frac {2 \sqrt [3]{c}-\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{2 \left (c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}\right )}d\sqrt [3]{c \cot (a+b x)}}{3 c^{5/3}}+\frac {\int \frac {2 \sqrt [3]{c}+\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{2 \left (c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}\right )}d\sqrt [3]{c \cot (a+b x)}}{3 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\int \frac {1}{c^2 \cot ^2(a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{3 c^{4/3}}+\frac {\int \frac {2 \sqrt [3]{c}-\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\int \frac {2 \sqrt [3]{c}+\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\int \frac {2 \sqrt [3]{c}+\sqrt {3} \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\frac {1}{2} \sqrt [3]{c} \int \frac {1}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-\frac {1}{3}}d\left (1-\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}}{6 c^{5/3}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\frac {\int \frac {1}{-c^2 \cot ^2(a+b x)-\frac {1}{3}}d\left (\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}+1\right )}{\sqrt {3}}}{6 c^{5/3}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}-2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)-\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}-\arctan \left (\sqrt {3} \left (1-\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}\right )\right )}{6 c^{5/3}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{c}+2 \sqrt [3]{c \cot (a+b x)}}{c^2 \cot ^2(a+b x)+\sqrt {3} c^{4/3} \cot (a+b x)+c^{2/3}}d\sqrt [3]{c \cot (a+b x)}+\arctan \left (\sqrt {3} \left (\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}+1\right )\right )}{6 c^{5/3}}+\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 c^3 \left (\frac {\arctan \left (c^{2/3} \cot (a+b x)\right )}{3 c^{5/3}}+\frac {-\arctan \left (\sqrt {3} \left (1-\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}\right )\right )-\frac {1}{2} \sqrt {3} \log \left (-\sqrt {3} c^{4/3} \cot (a+b x)+c^2 \cot ^2(a+b x)+c^{2/3}\right )}{6 c^{5/3}}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 c^{2/3} \cot (a+b x)}{\sqrt {3}}+1\right )\right )+\frac {1}{2} \sqrt {3} \log \left (\sqrt {3} c^{4/3} \cot (a+b x)+c^2 \cot ^2(a+b x)+c^{2/3}\right )}{6 c^{5/3}}\right )}{b}-\frac {3 c \sqrt [3]{c \cot (a+b x)}}{b}\) |
(-3*c*(c*Cot[a + b*x])^(1/3))/b + (3*c^3*(ArcTan[c^(2/3)*Cot[a + b*x]]/(3* c^(5/3)) + (-ArcTan[Sqrt[3]*(1 - (2*c^(2/3)*Cot[a + b*x])/Sqrt[3])] - (Sqr t[3]*Log[c^(2/3) - Sqrt[3]*c^(4/3)*Cot[a + b*x] + c^2*Cot[a + b*x]^2])/2)/ (6*c^(5/3)) + (ArcTan[Sqrt[3]*(1 + (2*c^(2/3)*Cot[a + b*x])/Sqrt[3])] + (S qrt[3]*Log[c^(2/3) + Sqrt[3]*c^(4/3)*Cot[a + b*x] + c^2*Cot[a + b*x]^2])/2 )/(6*c^(5/3))))/b
3.1.17.3.1 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[((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] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && 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*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
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 = 0.10 (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 | \(-\frac {3 c \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{b}-\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}+\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 b}\) | \(214\) |
-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))
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} \]
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)*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)) + 2*(-c^8/b^6)^(1/6)*b*log(c*((c*cos(2*b*x + 2*a) + c)/si n(2*b*x + 2*a))^(1/3) + (-c^8/b^6)^(1/6)*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) - 1 2*c*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3))/b
\[ \int (c \cot (a+b x))^{4/3} \, dx=\int \left (c \cot {\left (a + b x \right )}\right )^{\frac {4}{3}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.81 \[ \int (c \cot (a+b x))^{4/3} \, dx=\frac {{\left (\sqrt {3} c^{\frac {1}{3}} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right ) - \sqrt {3} c^{\frac {1}{3}} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right ) + 2 \, c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) + 4 \, c^{\frac {1}{3}} \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) - 12 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}\right )} c}{4 \, b} \]
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((sqrt (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
\[ \int (c \cot (a+b x))^{4/3} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {4}{3}} \,d x } \]
Time = 12.89 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int (c \cot (a+b x))^{4/3} \, dx=-\frac {3\,c\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{5/6}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left ({\left (-1\right )}^{1/6}\,c^{1/3}-2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}-\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}-{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}-{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]
((-1)^(1/6)*c^(4/3)*atan(((-1)^(5/6)*(c*cot(a + b*x))^(1/3)*1i)/c^(1/3))*1 i)/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