Integrand size = 16, antiderivative size = 33 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {3 x}{8 a}-\frac {3 \cos (x) \sin (x)}{8 a}-\frac {\cos (x) \sin ^3(x)}{4 a} \] Output:
3/8*x/a-3/8*cos(x)*sin(x)/a-1/4*cos(x)*sin(x)^3/a
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {\frac {3 x}{8}-\frac {1}{4} \sin (2 x)+\frac {1}{32} \sin (4 x)}{a} \] Input:
Integrate[Sin[x]^6/(a - a*Cos[x]^2),x]
Output:
((3*x)/8 - Sin[2*x]/4 + Sin[4*x]/32)/a
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 3654, 3042, 3115, 3042, 3115, 24}
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.
\(\displaystyle \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (x+\frac {\pi }{2}\right )^6}{a-a \sin \left (x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle \frac {\int \sin ^4(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sin (x)^4dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {3}{4} \int \sin ^2(x)dx-\frac {1}{4} \sin ^3(x) \cos (x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3}{4} \int \sin (x)^2dx-\frac {1}{4} \sin ^3(x) \cos (x)}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)}{a}\) |
Input:
Int[Sin[x]^6/(a - a*Cos[x]^2),x]
Output:
(-1/4*(Cos[x]*Sin[x]^3) + (3*(x/2 - (Cos[x]*Sin[x])/2))/4)/a
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
Time = 0.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {12 x +\sin \left (4 x \right )-8 \sin \left (2 x \right )}{32 a}\) | \(20\) |
risch | \(\frac {3 x}{8 a}+\frac {\sin \left (4 x \right )}{32 a}-\frac {\sin \left (2 x \right )}{4 a}\) | \(26\) |
default | \(\frac {\frac {-\frac {5 \tan \left (x \right )^{3}}{8}-\frac {3 \tan \left (x \right )}{8}}{\left (1+\tan \left (x \right )^{2}\right )^{2}}+\frac {3 \arctan \left (\tan \left (x \right )\right )}{8}}{a}\) | \(31\) |
norman | \(\frac {-\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{4 a}-\frac {17 \tan \left (\frac {x}{2}\right )^{4}}{4 a}-\frac {7 \tan \left (\frac {x}{2}\right )^{6}}{2 a}+\frac {7 \tan \left (\frac {x}{2}\right )^{8}}{2 a}+\frac {17 \tan \left (\frac {x}{2}\right )^{10}}{4 a}+\frac {3 \tan \left (\frac {x}{2}\right )^{12}}{4 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{8 a}+\frac {9 x \tan \left (\frac {x}{2}\right )^{3}}{4 a}+\frac {45 x \tan \left (\frac {x}{2}\right )^{5}}{8 a}+\frac {15 x \tan \left (\frac {x}{2}\right )^{7}}{2 a}+\frac {45 x \tan \left (\frac {x}{2}\right )^{9}}{8 a}+\frac {9 x \tan \left (\frac {x}{2}\right )^{11}}{4 a}+\frac {3 x \tan \left (\frac {x}{2}\right )^{13}}{8 a}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{6} \tan \left (\frac {x}{2}\right )}\) | \(167\) |
Input:
int(sin(x)^6/(a-a*cos(x)^2),x,method=_RETURNVERBOSE)
Output:
1/32*(12*x+sin(4*x)-8*sin(2*x))/a
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {{\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 3 \, x}{8 \, a} \] Input:
integrate(sin(x)^6/(a-a*cos(x)^2),x, algorithm="fricas")
Output:
1/8*((2*cos(x)^3 - 5*cos(x))*sin(x) + 3*x)/a
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (29) = 58\).
Time = 1.98 (sec) , antiderivative size = 473, normalized size of antiderivative = 14.33 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {3 x \tan ^{8}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {12 x \tan ^{6}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {18 x \tan ^{4}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {12 x \tan ^{2}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {3 x}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {6 \tan ^{7}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} + \frac {22 \tan ^{5}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} - \frac {22 \tan ^{3}{\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} - \frac {6 \tan {\left (\frac {x}{2} \right )}}{8 a \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a} \] Input:
integrate(sin(x)**6/(a-a*cos(x)**2),x)
Output:
3*x*tan(x/2)**8/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 3 2*a*tan(x/2)**2 + 8*a) + 12*x*tan(x/2)**6/(8*a*tan(x/2)**8 + 32*a*tan(x/2) **6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 18*x*tan(x/2)**4/(8*a*t an(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 12*x*tan(x/2)**2/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 3*x/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48* a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 6*tan(x/2)**7/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) + 22*tan(x/ 2)**5/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/ 2)**2 + 8*a) - 22*tan(x/2)**3/(8*a*tan(x/2)**8 + 32*a*tan(x/2)**6 + 48*a*t an(x/2)**4 + 32*a*tan(x/2)**2 + 8*a) - 6*tan(x/2)/(8*a*tan(x/2)**8 + 32*a* tan(x/2)**6 + 48*a*tan(x/2)**4 + 32*a*tan(x/2)**2 + 8*a)
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=-\frac {5 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{8 \, {\left (a \tan \left (x\right )^{4} + 2 \, a \tan \left (x\right )^{2} + a\right )}} + \frac {3 \, x}{8 \, a} \] Input:
integrate(sin(x)^6/(a-a*cos(x)^2),x, algorithm="maxima")
Output:
-1/8*(5*tan(x)^3 + 3*tan(x))/(a*tan(x)^4 + 2*a*tan(x)^2 + a) + 3/8*x/a
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {3 \, x}{8 \, a} - \frac {5 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{8 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{2} a} \] Input:
integrate(sin(x)^6/(a-a*cos(x)^2),x, algorithm="giac")
Output:
3/8*x/a - 1/8*(5*tan(x)^3 + 3*tan(x))/((tan(x)^2 + 1)^2*a)
Time = 0.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {\sin \left (4\,x\right )}{32\,a}-\frac {\sin \left (2\,x\right )}{4\,a}+\frac {3\,x}{8\,a} \] Input:
int(sin(x)^6/(a - a*cos(x)^2),x)
Output:
sin(4*x)/(32*a) - sin(2*x)/(4*a) + (3*x)/(8*a)
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^6(x)}{a-a \cos ^2(x)} \, dx=\frac {-2 \cos \left (x \right ) \sin \left (x \right )^{3}-3 \cos \left (x \right ) \sin \left (x \right )+3 x}{8 a} \] Input:
int(sin(x)^6/(a-a*cos(x)^2),x)
Output:
( - 2*cos(x)*sin(x)**3 - 3*cos(x)*sin(x) + 3*x)/(8*a)