Integrand size = 20, antiderivative size = 112 \[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {a+2 \sqrt [3]{a^3-b^3 \cos ^n(x)}}{\sqrt {3} a}\right )}{a^4 n}-\frac {3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}+\frac {\log (\cos (x))}{2 a^4}-\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n} \]
-3/a^3/n/(a^3-b^3*cos(x)^n)^(1/3)+1/2*ln(cos(x))/a^4-3/2*ln(a-(a^3-b^3*cos (x)^n)^(1/3))/a^4/n-arctan(1/3*(a+2*(a^3-b^3*cos(x)^n)^(1/3))/a*3^(1/2))*3 ^(1/2)/a^4/n
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.42 \[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},1-\frac {b^3 \cos ^n(x)}{a^3}\right )}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}} \]
(-3*Hypergeometric2F1[-1/3, 1, 2/3, 1 - (b^3*Cos[x]^n)/a^3])/(a^3*n*(a^3 - b^3*Cos[x]^n)^(1/3))
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 25, 3709, 798, 61, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \left (a^3-b^3 \sin \left (x+\frac {\pi }{2}\right )^n\right )^{4/3}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (a^3-b^3 \sin \left (x+\frac {\pi }{2}\right )^n\right )^{4/3} \tan \left (x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3709 |
| \(\displaystyle -\int \frac {\sec (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}}d\cos (x)\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\frac {\int \frac {\sec (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}}d\cos ^n(x)}{n}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {\frac {\int \frac {\sec (x)}{\sqrt [3]{a^3-b^3 \cos ^n(x)}}d\cos ^n(x)}{a^3}+\frac {3}{a^3 \sqrt [3]{a^3-b^3 \cos ^n(x)}}}{n}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {\frac {-\frac {3 \int \frac {1}{a-\sqrt [3]{a^3-b^3 \cos ^n(x)}}d\sqrt [3]{a^3-b^3 \cos ^n(x)}}{2 a}+\frac {3}{2} \int \frac {1}{\cos ^{2 n}(x)+a^2+a \sqrt [3]{a^3-b^3 \cos ^n(x)}}d\sqrt [3]{a^3-b^3 \cos ^n(x)}-\frac {\log \left (\cos ^n(x)\right )}{2 a}}{a^3}+\frac {3}{a^3 \sqrt [3]{a^3-b^3 \cos ^n(x)}}}{n}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {\frac {\frac {3}{2} \int \frac {1}{\cos ^{2 n}(x)+a^2+a \sqrt [3]{a^3-b^3 \cos ^n(x)}}d\sqrt [3]{a^3-b^3 \cos ^n(x)}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a}-\frac {\log \left (\cos ^n(x)\right )}{2 a}}{a^3}+\frac {3}{a^3 \sqrt [3]{a^3-b^3 \cos ^n(x)}}}{n}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {-\frac {3 \int \frac {1}{-\cos ^{2 n}(x)-3}d\left (\frac {2 \sqrt [3]{a^3-b^3 \cos ^n(x)}}{a}+1\right )}{a}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a}-\frac {\log \left (\cos ^n(x)\right )}{2 a}}{a^3}+\frac {3}{a^3 \sqrt [3]{a^3-b^3 \cos ^n(x)}}}{n}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a^3-b^3 \cos ^n(x)}}{a}+1}{\sqrt {3}}\right )}{a}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a}-\frac {\log \left (\cos ^n(x)\right )}{2 a}}{a^3}+\frac {3}{a^3 \sqrt [3]{a^3-b^3 \cos ^n(x)}}}{n}\) |
-((3/(a^3*(a^3 - b^3*Cos[x]^n)^(1/3)) + ((Sqrt[3]*ArcTan[(1 + (2*(a^3 - b^ 3*Cos[x]^n)^(1/3))/a)/Sqrt[3]])/a - Log[Cos[x]^n]/(2*a) + (3*Log[a - (a^3 - b^3*Cos[x]^n)^(1/3)])/(2*a))/a^3)/n)
3.5.48.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
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] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff^(m + 1)/f Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
Time = 0.69 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(-\frac {\frac {\ln \left (a -\left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}\right )}{a^{4}}+\frac {-\frac {\ln \left (a^{2}+a \left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}+\left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{4}}+\frac {3}{a^{3} \left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}}}{n}\) | \(127\) |
| default | \(-\frac {\frac {\ln \left (a -\left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}\right )}{a^{4}}+\frac {-\frac {\ln \left (a^{2}+a \left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}+\left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{4}}+\frac {3}{a^{3} \left (a^{3}-b^{3} \left (\cos ^{n}\left (x \right )\right )\right )^{\frac {1}{3}}}}{n}\) | \(127\) |
-1/n*(1/a^4*ln(a-(a^3-b^3*cos(x)^n)^(1/3))+1/a^4*(-1/2*ln(a^2+a*(a^3-b^3*c os(x)^n)^(1/3)+(a^3-b^3*cos(x)^n)^(2/3))+3^(1/2)*arctan(1/3*(a+2*(a^3-b^3* cos(x)^n)^(1/3))/a*3^(1/2)))+3/a^3/(a^3-b^3*cos(x)^n)^(1/3))
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.65 \[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=-\frac {2 \, {\left (\sqrt {3} b^{3} \cos \left (x\right )^{n} - \sqrt {3} a^{3}\right )} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}}}{3 \, a}\right ) - {\left (b^{3} \cos \left (x\right )^{n} - a^{3}\right )} \log \left (a^{2} + {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} \cos \left (x\right )^{n} - a^{3}\right )} \log \left (-a + {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}}\right ) - 6 \, {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {2}{3}} a}{2 \, {\left (a^{4} b^{3} n \cos \left (x\right )^{n} - a^{7} n\right )}} \]
-1/2*(2*(sqrt(3)*b^3*cos(x)^n - sqrt(3)*a^3)*arctan(1/3*(sqrt(3)*a + 2*sqr t(3)*(-b^3*cos(x)^n + a^3)^(1/3))/a) - (b^3*cos(x)^n - a^3)*log(a^2 + (-b^ 3*cos(x)^n + a^3)^(1/3)*a + (-b^3*cos(x)^n + a^3)^(2/3)) + 2*(b^3*cos(x)^n - a^3)*log(-a + (-b^3*cos(x)^n + a^3)^(1/3)) - 6*(-b^3*cos(x)^n + a^3)^(2 /3)*a)/(a^4*b^3*n*cos(x)^n - a^7*n)
\[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=\int \frac {\tan {\left (x \right )}}{\left (a^{3} - b^{3} \cos ^{n}{\left (x \right )}\right )^{\frac {4}{3}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21 \[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{4} n} + \frac {\log \left (a^{2} + {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a^{4} n} - \frac {\log \left (-a + {\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}}\right )}{a^{4} n} - \frac {3}{{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {1}{3}} a^{3} n} \]
-sqrt(3)*arctan(1/3*sqrt(3)*(a + 2*(-b^3*cos(x)^n + a^3)^(1/3))/a)/(a^4*n) + 1/2*log(a^2 + (-b^3*cos(x)^n + a^3)^(1/3)*a + (-b^3*cos(x)^n + a^3)^(2/ 3))/(a^4*n) - log(-a + (-b^3*cos(x)^n + a^3)^(1/3))/(a^4*n) - 3/((-b^3*cos (x)^n + a^3)^(1/3)*a^3*n)
\[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx=\int \frac {\mathrm {tan}\left (x\right )}{{\left (a^3-b^3\,{\cos \left (x\right )}^n\right )}^{4/3}} \,d x \]