Integrand size = 31, antiderivative size = 27 \[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=-\log (\tan (x))+\frac {3}{2} \log \left (1-\sqrt [3]{1-8 \tan ^2(x)}\right ) \] Output:
-ln(tan(x))+3/2*ln(1-(1-8*tan(x)^2)^(1/3))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=\frac {3 \cos ^2(x) \sqrt [3]{\sec ^2(x)-9 \tan ^2(x)} \left (\operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},\frac {2 \cos ^2(x)}{-7+9 \cos (2 x)}\right )+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {2 \cos ^2(x)}{-7+9 \cos (2 x)}\right ) \sqrt [3]{\sec ^2(x)-9 \tan ^2(x)}\right )}{-4+36 \sin ^2(x)} \] Input:
Integrate[(Csc[x]*Sec[x]*(1 + (1 - 8*Tan[x]^2)^(1/3)))/(1 - 8*Tan[x]^2)^(2 /3),x]
Output:
(3*Cos[x]^2*(Sec[x]^2 - 9*Tan[x]^2)^(1/3)*(Hypergeometric2F1[2/3, 1, 5/3, (2*Cos[x]^2)/(-7 + 9*Cos[2*x])] + 2*Hypergeometric2F1[1/3, 1, 4/3, (2*Cos[ x]^2)/(-7 + 9*Cos[2*x])]*(Sec[x]^2 - 9*Tan[x]^2)^(1/3)))/(-4 + 36*Sin[x]^2 )
Time = 1.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 4866, 7276, 2009}
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.
\(\displaystyle \int \frac {\left (\sqrt [3]{1-8 \tan ^2(x)}+1\right ) \csc (x) \sec (x)}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (\sqrt [3]{1-8 \tan (x)^2}+1\right ) \sec (x)}{\sin (x) \left (1-8 \tan (x)^2\right )^{2/3}}dx\) |
\(\Big \downarrow \) 4866 |
\(\displaystyle -\int \frac {\sec (x) \left (\sqrt [3]{9-8 \sec ^2(x)}+1\right )}{\left (1-\cos ^2(x)\right ) \left (9-8 \sec ^2(x)\right )^{2/3}}d\cos (x)\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\int \left (-\frac {\sec (x)}{\left (\cos ^2(x)-1\right ) \sqrt [3]{9-8 \sec ^2(x)}}-\frac {\sec (x)}{\left (\cos ^2(x)-1\right ) \left (9-8 \sec ^2(x)\right )^{2/3}}\right )d\cos (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} \log \left (1-\sqrt [3]{9-8 \sec ^2(x)}\right )-\frac {1}{2} \log \left (1-\sec ^2(x)\right )\) |
Input:
Int[(Csc[x]*Sec[x]*(1 + (1 - 8*Tan[x]^2)^(1/3)))/(1 - 8*Tan[x]^2)^(2/3),x]
Output:
-1/2*Log[1 - Sec[x]^2] + (3*Log[1 - (9 - 8*Sec[x]^2)^(1/3)])/2
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = Free Factors[Cos[c*(a + b*x)], x]}, Simp[-d/(b*c) Subst[Int[SubstFor[(1 - d^2* x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && Intege rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \frac {\cot \left (x \right ) \left (1+\left (1-8 \tan \left (x \right )^{2}\right )^{\frac {1}{3}}\right )}{\cos \left (x \right )^{2} \left (1-8 \tan \left (x \right )^{2}\right )^{\frac {2}{3}}}d x\]
Input:
int(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x)
Output:
int(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (23) = 46\).
Time = 0.81 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.44 \[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=-\frac {1}{2} \, \log \left (\frac {16 \, {\left (145 \, \cos \left (x\right )^{4} - 200 \, \cos \left (x\right )^{2} + 3 \, {\left (11 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2}\right )} \left (\frac {9 \, \cos \left (x\right )^{2} - 8}{\cos \left (x\right )^{2}}\right )^{\frac {2}{3}} + 3 \, {\left (19 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2}\right )} \left (\frac {9 \, \cos \left (x\right )^{2} - 8}{\cos \left (x\right )^{2}}\right )^{\frac {1}{3}} + 64\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}\right ) \] Input:
integrate(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x, algorithm="fricas")
Output:
-1/2*log(16*(145*cos(x)^4 - 200*cos(x)^2 + 3*(11*cos(x)^4 - 8*cos(x)^2)*(( 9*cos(x)^2 - 8)/cos(x)^2)^(2/3) + 3*(19*cos(x)^4 - 16*cos(x)^2)*((9*cos(x) ^2 - 8)/cos(x)^2)^(1/3) + 64)/(cos(x)^4 - 2*cos(x)^2 + 1))
\[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=\int \frac {\left (\sqrt [3]{1 - 8 \tan ^{2}{\left (x \right )}} + 1\right ) \cot {\left (x \right )}}{\left (1 - 8 \tan ^{2}{\left (x \right )}\right )^{\frac {2}{3}} \cos ^{2}{\left (x \right )}}\, dx \] Input:
integrate(cot(x)*(1+(1-8*tan(x)**2)**(1/3))/cos(x)**2/(1-8*tan(x)**2)**(2/ 3),x)
Output:
Integral(((1 - 8*tan(x)**2)**(1/3) + 1)*cot(x)/((1 - 8*tan(x)**2)**(2/3)*c os(x)**2), x)
\[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=\int { \frac {{\left ({\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} + 1\right )} \cot \left (x\right )}{{\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} \cos \left (x\right )^{2}} \,d x } \] Input:
integrate(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x, algorithm="maxima")
Output:
integrate(((-8*tan(x)^2 + 1)^(1/3) + 1)*cot(x)/((-8*tan(x)^2 + 1)^(2/3)*co s(x)^2), x)
Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=-\frac {1}{2} \, \log \left ({\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} + {\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \log \left ({\left | {\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:
integrate(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x, algorithm="giac")
Output:
-1/2*log((-8*tan(x)^2 + 1)^(2/3) + (-8*tan(x)^2 + 1)^(1/3) + 1) + log(abs( (-8*tan(x)^2 + 1)^(1/3) - 1))
Timed out. \[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=\int \frac {\mathrm {cot}\left (x\right )\,\left ({\left (1-8\,{\mathrm {tan}\left (x\right )}^2\right )}^{1/3}+1\right )}{{\cos \left (x\right )}^2\,{\left (1-8\,{\mathrm {tan}\left (x\right )}^2\right )}^{2/3}} \,d x \] Input:
int((cot(x)*((1 - 8*tan(x)^2)^(1/3) + 1))/(cos(x)^2*(1 - 8*tan(x)^2)^(2/3) ),x)
Output:
int((cot(x)*((1 - 8*tan(x)^2)^(1/3) + 1))/(cos(x)^2*(1 - 8*tan(x)^2)^(2/3) ), x)
\[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx=\int \frac {\cot \left (x \right )}{\left (-8 \tan \left (x \right )^{2}+1\right )^{\frac {2}{3}} \cos \left (x \right )^{2}}d x +\int \frac {\cot \left (x \right )}{\left (-8 \tan \left (x \right )^{2}+1\right )^{\frac {1}{3}} \cos \left (x \right )^{2}}d x \] Input:
int(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x)
Output:
int(cot(x)/(( - 8*tan(x)**2 + 1)**(2/3)*cos(x)**2),x) + int(cot(x)/(( - 8* tan(x)**2 + 1)**(1/3)*cos(x)**2),x)