Integrand size = 21, antiderivative size = 128 \[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}}{\sqrt {3}}\right )}{2 b}-\frac {\log \left (1+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (1-\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sin ^{\frac {4}{3}}(a+b x)}{\cos ^{\frac {4}{3}}(a+b x)}\right )}{4 b} \]
-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)/c os(b*x+a)^(2/3)+sin(b*x+a)^(4/3)/cos(b*x+a)^(4/3))/b-1/2*arctan(1/3*(1-2*s in(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 = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=\frac {3 \cos ^2(a+b x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {2}{3},\frac {5}{3},\sin ^2(a+b x)\right ) \sin ^{\frac {4}{3}}(a+b x)}{4 b \cos ^{\frac {4}{3}}(a+b x)} \]
(3*(Cos[a + b*x]^2)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, Sin[a + b*x]^2] *Sin[a + b*x]^(4/3))/(4*b*Cos[a + b*x]^(4/3))
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {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.
| \(\displaystyle \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}dx\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {3 \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}}{b}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3 \int \frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x) (\tan (a+b x)+1)}d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}}{2 b}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \int \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {1}{3} \int \frac {1}{\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1}d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{2 b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \int \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {1}{2} \int \left (\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {1}{2} \int \left (1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (-3 \int \frac {1}{-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-2}d\left (\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1\right )-\frac {1}{2} \int \left (1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \int \left (1-\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )d\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {\arctan \left (\frac {\frac {2 \sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+1\right )\right )}{2 b}\) |
(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)
3.4.24.3.1 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_)]*(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]
\[\int \frac {\sin ^{\frac {1}{3}}\left (b x +a \right )}{\cos \left (b x +a \right )^{\frac {1}{3}}}d x\]
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} \cos \left (b x + a\right ) - 2 \, \sqrt {3} \cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}}}{3 \, \cos \left (b x + a\right )}\right ) - 2 \, \log \left (\frac {\cos \left (b x + a\right )^{\frac {1}{3}} \sin \left (b x + a\right )^{\frac {2}{3}} + \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) + \log \left (\frac {\cos \left (b x + a\right )^{2} - \cos \left (b x + a\right )^{\frac {4}{3}} \sin \left (b x + a\right )^{\frac {2}{3}} + \cos \left (b x + a\right )^{\frac {2}{3}} \sin \left (b x + a\right )^{\frac {4}{3}}}{\cos \left (b x + a\right )^{2}}\right )}{4 \, b} \]
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)) - 2*log((cos(b*x + a)^(1/3)*sin(b* x + a)^(2/3) + cos(b*x + a))/cos(b*x + a)) + log((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))/co s(b*x + a)^2))/b
\[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=\int \frac {\sqrt [3]{\sin {\left (a + b x \right )}}}{\sqrt [3]{\cos {\left (a + b x \right )}}}\, dx \]
\[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )^{\frac {1}{3}}}{\cos \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )^{\frac {1}{3}}}{\cos \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
Time = 1.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx=-\frac {3\,{\cos \left (a+b\,x\right )}^{2/3}\,{\sin \left (a+b\,x\right )}^{4/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ {\cos \left (a+b\,x\right )}^2\right )}{2\,b\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{2/3}} \]