Integrand size = 36, antiderivative size = 62 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=\frac {B \sec ^3(e+f x)}{3 a^2 c^2 f}+\frac {A \tan (e+f x)}{a^2 c^2 f}+\frac {A \tan ^3(e+f x)}{3 a^2 c^2 f} \]
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=\frac {B \sec ^3(e+f x)}{3 a^2 c^2 f}+\frac {A \left (\tan (e+f x)+\frac {1}{3} \tan ^3(e+f x)\right )}{a^2 c^2 f} \]
Time = 0.43 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3042, 3446, 3042, 3148, 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.
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^2}dx\) |
\(\Big \downarrow \) 3446 |
\(\displaystyle \frac {\int \sec ^4(e+f x) (A+B \sin (e+f x))dx}{a^2 c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {A+B \sin (e+f x)}{\cos (e+f x)^4}dx}{a^2 c^2}\) |
\(\Big \downarrow \) 3148 |
\(\displaystyle \frac {A \int \sec ^4(e+f x)dx+\frac {B \sec ^3(e+f x)}{3 f}}{a^2 c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \int \csc \left (e+f x+\frac {\pi }{2}\right )^4dx+\frac {B \sec ^3(e+f x)}{3 f}}{a^2 c^2}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {\frac {B \sec ^3(e+f x)}{3 f}-\frac {A \int \left (\tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{f}}{a^2 c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {B \sec ^3(e+f x)}{3 f}-\frac {A \left (-\frac {1}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{f}}{a^2 c^2}\) |
3.1.66.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
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]
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {4 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+\frac {8 B \,{\mathrm e}^{3 i \left (f x +e \right )}}{3}+\frac {4 i A}{3}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} a^{2} c^{2} f}\) | \(70\) |
parallelrisch | \(\frac {6 A \sin \left (f x +e \right )+2 A \sin \left (3 f x +3 e \right )+3 \cos \left (f x +e \right ) B +\cos \left (3 f x +3 e \right ) B +4 B}{3 a^{2} c^{2} f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(77\) |
derivativedivides | \(\frac {-\frac {2 \left (\frac {A}{2}+\frac {B}{2}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {\frac {A}{2}+\frac {B}{2}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {A}{2}+\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {-\frac {A}{2}+\frac {B}{2}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{2}-\frac {B}{2}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {A}{2}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} c^{2} f}\) | \(145\) |
default | \(\frac {-\frac {2 \left (\frac {A}{2}+\frac {B}{2}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {\frac {A}{2}+\frac {B}{2}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {A}{2}+\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {-\frac {A}{2}+\frac {B}{2}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{2}-\frac {B}{2}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {A}{2}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} c^{2} f}\) | \(145\) |
norman | \(\frac {-\frac {2 B}{3 a c f}-\frac {2 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a c f}-\frac {2 A \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 A \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 A \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f c}-\frac {2 B \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 B \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 B \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\) | \(221\) |
4/3*(3*I*A*exp(2*I*(f*x+e))+2*B*exp(3*I*(f*x+e))+I*A)/(exp(I*(f*x+e))-I)^3 /(exp(I*(f*x+e))+I)^3/a^2/c^2/f
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=\frac {{\left (2 \, A \cos \left (f x + e\right )^{2} + A\right )} \sin \left (f x + e\right ) + B}{3 \, a^{2} c^{2} f \cos \left (f x + e\right )^{3}} \]
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (56) = 112\).
Time = 2.04 (sec) , antiderivative size = 469, normalized size of antiderivative = 7.56 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 A \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c^{2} f} + \frac {4 A \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c^{2} f} - \frac {6 A \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c^{2} f} - \frac {6 B \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c^{2} f} - \frac {2 B}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\left (e \right )}\right )}{\left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right )^{2}} & \text {otherwise} \end {cases} \]
Piecewise((-6*A*tan(e/2 + f*x/2)**5/(3*a**2*c**2*f*tan(e/2 + f*x/2)**6 - 9 *a**2*c**2*f*tan(e/2 + f*x/2)**4 + 9*a**2*c**2*f*tan(e/2 + f*x/2)**2 - 3*a **2*c**2*f) + 4*A*tan(e/2 + f*x/2)**3/(3*a**2*c**2*f*tan(e/2 + f*x/2)**6 - 9*a**2*c**2*f*tan(e/2 + f*x/2)**4 + 9*a**2*c**2*f*tan(e/2 + f*x/2)**2 - 3 *a**2*c**2*f) - 6*A*tan(e/2 + f*x/2)/(3*a**2*c**2*f*tan(e/2 + f*x/2)**6 - 9*a**2*c**2*f*tan(e/2 + f*x/2)**4 + 9*a**2*c**2*f*tan(e/2 + f*x/2)**2 - 3* a**2*c**2*f) - 6*B*tan(e/2 + f*x/2)**4/(3*a**2*c**2*f*tan(e/2 + f*x/2)**6 - 9*a**2*c**2*f*tan(e/2 + f*x/2)**4 + 9*a**2*c**2*f*tan(e/2 + f*x/2)**2 - 3*a**2*c**2*f) - 2*B/(3*a**2*c**2*f*tan(e/2 + f*x/2)**6 - 9*a**2*c**2*f*ta n(e/2 + f*x/2)**4 + 9*a**2*c**2*f*tan(e/2 + f*x/2)**2 - 3*a**2*c**2*f), Ne (f, 0)), (x*(A + B*sin(e))/((a*sin(e) + a)**2*(-c*sin(e) + c)**2), True))
Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=\frac {\frac {{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} A}{a^{2} c^{2}} + \frac {B}{a^{2} c^{2} \cos \left (f x + e\right )^{3}}}{3 \, f} \]
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (3 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{2} c^{2} f} \]
-2/3*(3*A*tan(1/2*f*x + 1/2*e)^5 + 3*B*tan(1/2*f*x + 1/2*e)^4 - 2*A*tan(1/ 2*f*x + 1/2*e)^3 + 3*A*tan(1/2*f*x + 1/2*e) + B)/((tan(1/2*f*x + 1/2*e)^2 - 1)^3*a^2*c^2*f)
Time = 12.46 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx=-\frac {2\,\left (3\,A\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+3\,B\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,A\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+3\,A\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+B\right )}{3\,a^2\,c^2\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^3} \]