Integrand size = 23, antiderivative size = 65 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {(c-d) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}-\frac {(c+2 d) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )} \] Output:
-1/3*(c-d)*cos(f*x+e)/f/(a+a*sin(f*x+e))^2-1/3*(c+2*d)*cos(f*x+e)/f/(a^2+a ^2*sin(f*x+e))
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {\cos (e+f x) (2 c+d+(c+2 d) \sin (e+f x))}{3 a^2 f (1+\sin (e+f x))^2} \] Input:
Integrate[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]
Output:
-1/3*(Cos[e + f*x]*(2*c + d + (c + 2*d)*Sin[e + f*x]))/(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.
\(\displaystyle \int \frac {c+d \sin (e+f x)}{(a \sin (e+f x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {c+d \sin (e+f x)}{(a \sin (e+f x)+a)^2}dx\) |
\(\Big \downarrow \) 3229 |
\(\displaystyle \frac {(c+2 d) \int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {(c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(c+2 d) \int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {(c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle -\frac {(c+2 d) \cos (e+f x)}{3 a f (a \sin (e+f x)+a)}-\frac {(c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\) |
Input:
Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]
Output:
-1/3*((c - d)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x])^2) - ((c + 2*d)*Cos[e + f*x])/(3*a*f*(a + a*Sin[e + f*x]))
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 = 0.52 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c +\left (-6 c -6 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 c -2 d}{3 f \,a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(60\) |
risch | \(-\frac {2 \left (-c +3 i c \,{\mathrm e}^{i \left (f x +e \right )}+3 i d \,{\mathrm e}^{i \left (f x +e \right )}+3 d \,{\mathrm e}^{2 i \left (f x +e \right )}-2 d \right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(68\) |
derivativedivides | \(\frac {-\frac {-2 c +2 d}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 c -2 d \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 c}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} f}\) | \(70\) |
default | \(\frac {-\frac {-2 c +2 d}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 c -2 d \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 c}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} f}\) | \(70\) |
norman | \(\frac {-\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{a f}+\frac {\left (-2 c -2 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {-4 c -2 d}{3 a f}+\frac {2 \left (-c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{a f}+\frac {2 \left (-5 c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{3 a f}}{a \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(144\) |
Input:
int((c+d*sin(f*x+e))/(a+sin(f*x+e)*a)^2,x,method=_RETURNVERBOSE)
Output:
1/3*(-6*tan(1/2*f*x+1/2*e)^2*c+(-6*c-6*d)*tan(1/2*f*x+1/2*e)-4*c-2*d)/f/a^ 2/(tan(1/2*f*x+1/2*e)+1)^3
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.80 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\frac {{\left (c + 2 \, d\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c + d\right )} \cos \left (f x + e\right ) + {\left ({\left (c + 2 \, d\right )} \cos \left (f x + e\right ) - c + d\right )} \sin \left (f x + e\right ) + c - d}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
Output:
1/3*((c + 2*d)*cos(f*x + e)^2 + (2*c + d)*cos(f*x + e) + ((c + 2*d)*cos(f* x + e) - c + d)*sin(f*x + e) + c - d)/(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 = 1.25 (sec) , antiderivative size = 372, normalized size of antiderivative = 5.72 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 c \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {6 c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {4 c}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {6 d \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {2 d}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (c + d \sin {\left (e \right )}\right )}{\left (a \sin {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))**2,x)
Output:
Piecewise((-6*c*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*c*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*c/(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*d*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*d/(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*(c + d*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 = 0.04 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.29 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (\frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
Output:
-2/3*(c*(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) + d*(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 = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, c + d\right )}}{3 \, a^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^2,x, algorithm="giac")
Output:
-2/3*(3*c*tan(1/2*f*x + 1/2*e)^2 + 3*c*tan(1/2*f*x + 1/2*e) + 3*d*tan(1/2* f*x + 1/2*e) + 2*c + d)/(a^2*f*(tan(1/2*f*x + 1/2*e) + 1)^3)
Time = 16.56 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,c}{2}+\frac {d}{2}-\frac {c\,\cos \left (e+f\,x\right )}{2}+\frac {d\,\cos \left (e+f\,x\right )}{2}+\frac {3\,c\,\sin \left (e+f\,x\right )}{2}+\frac {3\,d\,\sin \left (e+f\,x\right )}{2}\right )}{3\,a^2\,f\,\left (\frac {3\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}-\frac {\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{2}\right )} \] Input:
int((c + d*sin(e + f*x))/(a + a*sin(e + f*x))^2,x)
Output:
-(2*cos(e/2 + (f*x)/2)*((5*c)/2 + d/2 - (c*cos(e + f*x))/2 + (d*cos(e + f* x))/2 + (3*c*sin(e + f*x))/2 + (3*d*sin(e + f*x))/2))/(3*a^2*f*((3*2^(1/2) *cos(e/2 - pi/4 + (f*x)/2))/2 - (2^(1/2)*cos((3*e)/2 + pi/4 + (3*f*x)/2))/ 2))
Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d -\frac {2 c}{3}-\frac {2 d}{3}}{a^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \] Input:
int((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^2,x)
Output:
(2*(tan((e + f*x)/2)**3*c - 3*tan((e + f*x)/2)*d - c - d))/(3*a**2*f*(tan( (e + f*x)/2)**3 + 3*tan((e + f*x)/2)**2 + 3*tan((e + f*x)/2) + 1))