Integrand size = 23, antiderivative size = 65 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {(A-B) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}-\frac {(A+2 B) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )} \]
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {\cos (e+f x) (2 A+B+(A+2 B) \sin (e+f x))}{3 a^2 f (1+\sin (e+f x))^2} \]
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3229, 3042, 3127}
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.
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^2}dx\) |
\(\Big \downarrow \) 3229 |
\(\displaystyle \frac {(A+2 B) \int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {(A-B) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(A+2 B) \int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {(A-B) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle -\frac {(A+2 B) \cos (e+f x)}{3 a f (a \sin (e+f x)+a)}-\frac {(A-B) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\) |
-1/3*((A - B)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x])^2) - ((A + 2*B)*Cos[e + f*x])/(3*a*f*(a + a*Sin[e + f*x]))
3.3.75.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.47 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {-6 A \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-6 A -6 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 A -2 B}{3 f \,a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(60\) |
risch | \(-\frac {2 \left (-A +3 i A \,{\mathrm e}^{i \left (f x +e \right )}+3 i B \,{\mathrm e}^{i \left (f x +e \right )}+3 B \,{\mathrm e}^{2 i \left (f x +e \right )}-2 B \right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(68\) |
derivativedivides | \(\frac {-\frac {2 A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (-2 B +2 A \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{2} f}\) | \(70\) |
default | \(\frac {-\frac {2 A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (-2 B +2 A \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{2} f}\) | \(70\) |
norman | \(\frac {-\frac {4 A +2 B}{3 a f}-\frac {2 A \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (5 A +B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {\left (2 A +2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {2 \left (A +B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(139\) |
1/3*(-6*A*tan(1/2*f*x+1/2*e)^2+(-6*A-6*B)*tan(1/2*f*x+1/2*e)-4*A-2*B)/f/a^ 2/(tan(1/2*f*x+1/2*e)+1)^3
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.80 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\frac {{\left (A + 2 \, B\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, A + B\right )} \cos \left (f x + e\right ) + {\left ({\left (A + 2 \, B\right )} \cos \left (f x + e\right ) - A + B\right )} \sin \left (f x + e\right ) + A - B}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
1/3*((A + 2*B)*cos(f*x + e)^2 + (2*A + B)*cos(f*x + e) + ((A + 2*B)*cos(f* x + e) - A + B)*sin(f*x + e) + A - B)/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f* x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (56) = 112\).
Time = 1.12 (sec) , antiderivative size = 372, normalized size of antiderivative = 5.72 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 A \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {6 A \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {4 A}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {6 B \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {2 B}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\left (e \right )}\right )}{\left (a \sin {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Piecewise((-6*A*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**3 + 9*a**2 *f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*A*tan(e /2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**3 + 9*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 4*A/(3*a**2*f*tan(e/2 + f*x/2)**3 + 9*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*B*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**3 + 9*a**2*f*tan(e/2 + f* x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 2*B/(3*a**2*f*tan(e/2 + f*x/2)**3 + 9*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a **2*f), Ne(f, 0)), (x*(A + B*sin(e))/(a*sin(e) + a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (61) = 122\).
Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.29 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (\frac {A {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {B {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \]
-2/3*(A*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e ) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + B*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f *x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e )^3/(cos(f*x + e) + 1)^3))/f
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (3 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A + B\right )}}{3 \, a^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} \]
-2/3*(3*A*tan(1/2*f*x + 1/2*e)^2 + 3*A*tan(1/2*f*x + 1/2*e) + 3*B*tan(1/2* f*x + 1/2*e) + 2*A + B)/(a^2*f*(tan(1/2*f*x + 1/2*e) + 1)^3)
Time = 13.84 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,A}{2}+\frac {B}{2}-\frac {A\,\cos \left (e+f\,x\right )}{2}+\frac {B\,\cos \left (e+f\,x\right )}{2}+\frac {3\,A\,\sin \left (e+f\,x\right )}{2}+\frac {3\,B\,\sin \left (e+f\,x\right )}{2}\right )}{3\,a^2\,f\,\left (\frac {3\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}-\frac {\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{2}\right )} \]