Integrand size = 11, antiderivative size = 37 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\tan (x)}{a^4}+\frac {\tan ^3(x)}{a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^7(x)}{7 a^4} \]
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\tan (x)+\tan ^3(x)+\frac {3 \tan ^5(x)}{5}+\frac {\tan ^7(x)}{7}}{a^4} \]
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3654, 3042, 4254, 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.
\(\displaystyle \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a-a \sin (x)^2\right )^4}dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle \frac {\int \sec ^8(x)dx}{a^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \csc \left (x+\frac {\pi }{2}\right )^8dx}{a^4}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\int \left (\tan ^6(x)+3 \tan ^4(x)+3 \tan ^2(x)+1\right )d(-\tan (x))}{a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{7} \tan ^7(x)-\frac {3 \tan ^5(x)}{5}-\tan ^3(x)-\tan (x)}{a^4}\) |
3.1.62.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 0.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {\frac {\left (\tan ^{7}\left (x \right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (x \right )\right )}{5}+\tan ^{3}\left (x \right )+\tan \left (x \right )}{a^{4}}\) | \(24\) |
parallelrisch | \(\frac {\tan \left (x \right ) \left (\sec ^{6}\left (x \right )\right ) \left (32+\cos \left (6 x \right )+8 \cos \left (4 x \right )+29 \cos \left (2 x \right )\right )}{70 a^{4}}\) | \(30\) |
risch | \(\frac {32 i \left (35 \,{\mathrm e}^{6 i x}+21 \,{\mathrm e}^{4 i x}+7 \,{\mathrm e}^{2 i x}+1\right )}{35 \left ({\mathrm e}^{2 i x}+1\right )^{7} a^{4}}\) | \(39\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {86 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {424 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{35 a}-\frac {86 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {4 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{7}}\) | \(91\) |
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {{\left (16 \, \cos \left (x\right )^{6} + 8 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} + 5\right )} \sin \left (x\right )}{35 \, a^{4} \cos \left (x\right )^{7}} \]
Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (36) = 72\).
Time = 5.40 (sec) , antiderivative size = 675, normalized size of antiderivative = 18.24 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=- \frac {70 \tan ^{13}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {140 \tan ^{11}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {602 \tan ^{9}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {424 \tan ^{7}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {602 \tan ^{5}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {140 \tan ^{3}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {70 \tan {\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} \]
-70*tan(x/2)**13/(35*a**4*tan(x/2)**14 - 245*a**4*tan(x/2)**12 + 735*a**4* tan(x/2)**10 - 1225*a**4*tan(x/2)**8 + 1225*a**4*tan(x/2)**6 - 735*a**4*ta n(x/2)**4 + 245*a**4*tan(x/2)**2 - 35*a**4) + 140*tan(x/2)**11/(35*a**4*ta n(x/2)**14 - 245*a**4*tan(x/2)**12 + 735*a**4*tan(x/2)**10 - 1225*a**4*tan (x/2)**8 + 1225*a**4*tan(x/2)**6 - 735*a**4*tan(x/2)**4 + 245*a**4*tan(x/2 )**2 - 35*a**4) - 602*tan(x/2)**9/(35*a**4*tan(x/2)**14 - 245*a**4*tan(x/2 )**12 + 735*a**4*tan(x/2)**10 - 1225*a**4*tan(x/2)**8 + 1225*a**4*tan(x/2) **6 - 735*a**4*tan(x/2)**4 + 245*a**4*tan(x/2)**2 - 35*a**4) + 424*tan(x/2 )**7/(35*a**4*tan(x/2)**14 - 245*a**4*tan(x/2)**12 + 735*a**4*tan(x/2)**10 - 1225*a**4*tan(x/2)**8 + 1225*a**4*tan(x/2)**6 - 735*a**4*tan(x/2)**4 + 245*a**4*tan(x/2)**2 - 35*a**4) - 602*tan(x/2)**5/(35*a**4*tan(x/2)**14 - 245*a**4*tan(x/2)**12 + 735*a**4*tan(x/2)**10 - 1225*a**4*tan(x/2)**8 + 12 25*a**4*tan(x/2)**6 - 735*a**4*tan(x/2)**4 + 245*a**4*tan(x/2)**2 - 35*a** 4) + 140*tan(x/2)**3/(35*a**4*tan(x/2)**14 - 245*a**4*tan(x/2)**12 + 735*a **4*tan(x/2)**10 - 1225*a**4*tan(x/2)**8 + 1225*a**4*tan(x/2)**6 - 735*a** 4*tan(x/2)**4 + 245*a**4*tan(x/2)**2 - 35*a**4) - 70*tan(x/2)/(35*a**4*tan (x/2)**14 - 245*a**4*tan(x/2)**12 + 735*a**4*tan(x/2)**10 - 1225*a**4*tan( x/2)**8 + 1225*a**4*tan(x/2)**6 - 735*a**4*tan(x/2)**4 + 245*a**4*tan(x/2) **2 - 35*a**4)
Time = 0.50 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {5 \, \tan \left (x\right )^{7} + 21 \, \tan \left (x\right )^{5} + 35 \, \tan \left (x\right )^{3} + 35 \, \tan \left (x\right )}{35 \, a^{4}} \]
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {5 \, \tan \left (x\right )^{7} + 21 \, \tan \left (x\right )^{5} + 35 \, \tan \left (x\right )^{3} + 35 \, \tan \left (x\right )}{35 \, a^{4}} \]
Time = 12.85 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\mathrm {tan}\left (x\right )}{a^4}+\frac {{\mathrm {tan}\left (x\right )}^3}{a^4}+\frac {3\,{\mathrm {tan}\left (x\right )}^5}{5\,a^4}+\frac {{\mathrm {tan}\left (x\right )}^7}{7\,a^4} \]