Integrand size = 21, antiderivative size = 250 \[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {\sqrt {3} \log \left (1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\sqrt {3} \log \left (1+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}} \]
-arctan(sin(b*x+a)^(1/3)/cos(b*x+a)^(1/3))/b-1/2*arctan(2*sin(b*x+a)^(1/3) /cos(b*x+a)^(1/3)-3^(1/2))/b-1/2*arctan(2*sin(b*x+a)^(1/3)/cos(b*x+a)^(1/3 )+3^(1/2))/b-3*cos(b*x+a)^(1/3)/b/sin(b*x+a)^(1/3)-1/4*ln(1+sin(b*x+a)^(2/ 3)/cos(b*x+a)^(2/3)-sin(b*x+a)^(1/3)*3^(1/2)/cos(b*x+a)^(1/3))*3^(1/2)/b+1 /4*ln(1+sin(b*x+a)^(2/3)/cos(b*x+a)^(2/3)+sin(b*x+a)^(1/3)*3^(1/2)/cos(b*x +a)^(1/3))*3^(1/2)/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.22 \[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=-\frac {3 \cos ^2(a+b x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},-\frac {1}{6},\frac {5}{6},\sin ^2(a+b x)\right )}{b \cos ^{\frac {5}{3}}(a+b x) \sqrt [3]{\sin (a+b x)}} \]
(-3*(Cos[a + b*x]^2)^(5/6)*Hypergeometric2F1[-1/6, -1/6, 5/6, Sin[a + b*x] ^2])/(b*Cos[a + b*x]^(5/3)*Sin[a + b*x]^(1/3))
Time = 0.43 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3047, 3042, 3054, 824, 27, 216, 1142, 25, 1083, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (a+b x)^{4/3}}{\sin (a+b x)^{4/3}}dx\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle -\int \frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}dx-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {\sin (a+b x)^{2/3}}{\cos (a+b x)^{2/3}}dx-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle -\frac {3 \int \frac {\sin ^{\frac {4}{3}}(a+b x)}{\cos ^{\frac {4}{3}}(a+b x) \left (\tan ^2(a+b x)+1\right )}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle -\frac {3 \left (\frac {1}{3} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{2 \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}{2 \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {1}{3} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {3 \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {1}{3} \arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {3 \left (\frac {1}{6} \left (-\int \frac {1}{-\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1}d\left (\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{6} \left (-\int \frac {1}{-\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1}d\left (\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )\right )\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\) |
(-3*(ArcTan[Sin[a + b*x]^(1/3)/Cos[a + b*x]^(1/3)]/3 + (-ArcTan[Sqrt[3] - (2*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3)] + (Sqrt[3]*Log[1 - (Sqrt[3]*Sin [a + b*x]^(1/3))/Cos[a + b*x]^(1/3) + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3 )])/2)/6 + (ArcTan[Sqrt[3] + (2*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3)] - (Sqrt[3]*Log[1 + (Sqrt[3]*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3) + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)])/2)/6))/b - (3*Cos[a + b*x]^(1/3))/(b*Sin[ a + b*x]^(1/3))
3.4.31.3.1 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[(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] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ (b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Cos[e + f*x] )^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ [m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f) Subst[Int[x^(k *(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
\[\int \frac {\cos ^{\frac {4}{3}}\left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {4}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (200) = 400\).
Time = 0.35 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.78 \[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=\frac {\sqrt {\frac {1}{2}} b \sqrt {\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} + 1}{b^{2}}} \log \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} b \sqrt {\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} + 1}{b^{2}}} \sin \left (b x + a\right ) + \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}\right )}}{\sin \left (b x + a\right )}\right ) \sin \left (b x + a\right ) - \sqrt {\frac {1}{2}} b \sqrt {\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} + 1}{b^{2}}} \log \left (-\frac {2 \, {\left (\sqrt {\frac {1}{2}} b \sqrt {\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} + 1}{b^{2}}} \sin \left (b x + a\right ) - \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}\right )}}{\sin \left (b x + a\right )}\right ) \sin \left (b x + a\right ) + \sqrt {\frac {1}{2}} b \sqrt {-\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} - 1}{b^{2}}} \log \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} b \sqrt {-\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} - 1}{b^{2}}} \sin \left (b x + a\right ) + \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}\right )}}{\sin \left (b x + a\right )}\right ) \sin \left (b x + a\right ) - \sqrt {\frac {1}{2}} b \sqrt {-\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} - 1}{b^{2}}} \log \left (-\frac {2 \, {\left (\sqrt {\frac {1}{2}} b \sqrt {-\frac {\sqrt {3} b^{2} \sqrt {-\frac {1}{b^{4}}} - 1}{b^{2}}} \sin \left (b x + a\right ) - \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}\right )}}{\sin \left (b x + a\right )}\right ) \sin \left (b x + a\right ) + 2 \, \arctan \left (\frac {\cos \left (b x + a\right )^{\frac {1}{3}}}{\sin \left (b x + a\right )^{\frac {1}{3}}}\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}}{2 \, b \sin \left (b x + a\right )} \]
1/2*(sqrt(1/2)*b*sqrt((sqrt(3)*b^2*sqrt(-1/b^4) + 1)/b^2)*log(2*(sqrt(1/2) *b*sqrt((sqrt(3)*b^2*sqrt(-1/b^4) + 1)/b^2)*sin(b*x + a) + cos(b*x + a)^(1 /3)*sin(b*x + a)^(2/3))/sin(b*x + a))*sin(b*x + a) - sqrt(1/2)*b*sqrt((sqr t(3)*b^2*sqrt(-1/b^4) + 1)/b^2)*log(-2*(sqrt(1/2)*b*sqrt((sqrt(3)*b^2*sqrt (-1/b^4) + 1)/b^2)*sin(b*x + a) - cos(b*x + a)^(1/3)*sin(b*x + a)^(2/3))/s in(b*x + a))*sin(b*x + a) + sqrt(1/2)*b*sqrt(-(sqrt(3)*b^2*sqrt(-1/b^4) - 1)/b^2)*log(2*(sqrt(1/2)*b*sqrt(-(sqrt(3)*b^2*sqrt(-1/b^4) - 1)/b^2)*sin(b *x + a) + cos(b*x + a)^(1/3)*sin(b*x + a)^(2/3))/sin(b*x + a))*sin(b*x + a ) - sqrt(1/2)*b*sqrt(-(sqrt(3)*b^2*sqrt(-1/b^4) - 1)/b^2)*log(-2*(sqrt(1/2 )*b*sqrt(-(sqrt(3)*b^2*sqrt(-1/b^4) - 1)/b^2)*sin(b*x + a) - cos(b*x + a)^ (1/3)*sin(b*x + a)^(2/3))/sin(b*x + a))*sin(b*x + a) + 2*arctan(cos(b*x + a)^(1/3)/sin(b*x + a)^(1/3))*sin(b*x + a) - 6*cos(b*x + a)^(1/3)*sin(b*x + a)^(2/3))/(b*sin(b*x + a))
\[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=\int \frac {\cos ^{\frac {4}{3}}{\left (a + b x \right )}}{\sin ^{\frac {4}{3}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {4}{3}}}{\sin \left (b x + a\right )^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {4}{3}}}{\sin \left (b x + a\right )^{\frac {4}{3}}} \,d x } \]
Time = 1.60 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.18 \[ \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx=-\frac {3\,{\cos \left (a+b\,x\right )}^{7/3}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{1/6}\,{{}}_2{\mathrm {F}}_1\left (\frac {7}{6},\frac {7}{6};\ \frac {13}{6};\ {\cos \left (a+b\,x\right )}^2\right )}{7\,b\,{\sin \left (a+b\,x\right )}^{1/3}} \]