Integrand size = 21, antiderivative size = 155 \[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}}{\sqrt {3}}\right )}{2 b}+\frac {\log \left (1+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{2 b}-\frac {\log \left (1-\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sin ^{\frac {4}{3}}(a+b x)}{\cos ^{\frac {4}{3}}(a+b x)}\right )}{4 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)} \]
1/2*ln(1+sin(b*x+a)^(2/3)/cos(b*x+a)^(2/3))/b-1/4*ln(1-sin(b*x+a)^(2/3)/co s(b*x+a)^(2/3)+sin(b*x+a)^(4/3)/cos(b*x+a)^(4/3))/b-3/2*cos(b*x+a)^(2/3)/b /sin(b*x+a)^(2/3)+1/2*arctan(1/3*(1-2*sin(b*x+a)^(2/3)/cos(b*x+a)^(2/3))*3 ^(1/2))*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 = 57, normalized size of antiderivative = 0.37 \[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=-\frac {3 \cos ^2(a+b x)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{3},\frac {2}{3},\sin ^2(a+b x)\right )}{2 b \cos ^{\frac {4}{3}}(a+b x) \sin ^{\frac {2}{3}}(a+b x)} \]
(-3*(Cos[a + b*x]^2)^(2/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, Sin[a + b*x] ^2])/(2*b*Cos[a + b*x]^(4/3)*Sin[a + b*x]^(2/3))
Time = 0.38 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.65, 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, 807, 821, 16, 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.
\(\displaystyle \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (a+b x)^{5/3}}{\sin (a+b x)^{5/3}}dx\) |
\(\Big \downarrow \) 3047 |
\(\displaystyle -\int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}dx-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}dx-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 3054 |
\(\displaystyle -\frac {3 \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {3 \int \frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x) (\tan (a+b x)+1)}d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle -\frac {3 \left (\frac {1}{3} \int \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {1}{3} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1}d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {3 \left (\frac {1}{3} \int \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle -\frac {3 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {1}{2} \int \left (\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {3 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {1}{2} \int \left (1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {3 \left (\frac {1}{3} \left (-3 \int \frac {1}{-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-2}d\left (\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1\right )-\frac {1}{2} \int \left (1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {3 \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \int \left (1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {3 \left (\frac {\arctan \left (\frac {\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}-\frac {3 \cos ^{\frac {2}{3}}(a+b x)}{2 b \sin ^{\frac {2}{3}}(a+b x)}\) |
(-3*(ArcTan[(-1 + (2*Sin[a + b*x]^(2/3))/Cos[a + b*x]^(2/3))/Sqrt[3]]/Sqrt [3] - Log[1 + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)]/3))/(2*b) - (3*Cos[a + b*x]^(2/3))/(2*b*Sin[a + b*x]^(2/3))
3.4.32.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
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]
\[\int \frac {\cos ^{\frac {5}{3}}\left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {5}{3}}}d x\]
Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} \cos \left (b x + a\right ) - 2 \, \sqrt {3} \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}}{3 \, \cos \left (b x + a\right )}\right ) \sin \left (b x + a\right ) - 2 \, \log \left (\frac {\cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}} + \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) \sin \left (b x + a\right ) + \log \left (\frac {\cos \left (b x + a\right )^{2} - \cos \left (b x + a\right )^{\frac {4}{3}} \sin \left (b x + a\right )^{\frac {2}{3}} + \cos \left (b x + a\right )^{\frac {2}{3}} \sin \left (b x + a\right )^{\frac {4}{3}}}{\cos \left (b x + a\right )^{2}}\right ) \sin \left (b x + a\right ) + 6 \, \cos \left (b x + a\right )^{\frac {2}{3}} \sin \left (b x + a\right )^{\frac {1}{3}}}{4 \, b \sin \left (b x + a\right )} \]
-1/4*(2*sqrt(3)*arctan(-1/3*(sqrt(3)*cos(b*x + a) - 2*sqrt(3)*cos(b*x + a) ^(1/3)*sin(b*x + a)^(2/3))/cos(b*x + a))*sin(b*x + a) - 2*log((cos(b*x + a )^(1/3)*sin(b*x + a)^(2/3) + cos(b*x + a))/cos(b*x + a))*sin(b*x + a) + lo g((cos(b*x + a)^2 - cos(b*x + a)^(4/3)*sin(b*x + a)^(2/3) + cos(b*x + a)^( 2/3)*sin(b*x + a)^(4/3))/cos(b*x + a)^2)*sin(b*x + a) + 6*cos(b*x + a)^(2/ 3)*sin(b*x + a)^(1/3))/(b*sin(b*x + a))
Timed out. \[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {5}{3}}}{\sin \left (b x + a\right )^{\frac {5}{3}}} \,d x } \]
\[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {5}{3}}}{\sin \left (b x + a\right )^{\frac {5}{3}}} \,d x } \]
Time = 1.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.28 \[ \int \frac {\cos ^{\frac {5}{3}}(a+b x)}{\sin ^{\frac {5}{3}}(a+b x)} \, dx=-\frac {3\,{\cos \left (a+b\,x\right )}^{8/3}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {4}{3},\frac {4}{3};\ \frac {7}{3};\ {\cos \left (a+b\,x\right )}^2\right )}{8\,b\,{\sin \left (a+b\,x\right )}^{2/3}} \]