3.456 \(\int \frac {c+d \sin (e+f x)}{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=35 \[ \frac {d x}{a}-\frac {(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)} \]

[Out]

d*x/a-(c-d)*cos(f*x+e)/f/(a+a*sin(f*x+e))

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2735, 2648} \[ \frac {d x}{a}-\frac {(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x]),x]

[Out]

(d*x)/a - ((c - d)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))

Rule 2648

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]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rubi steps

\begin {align*} \int \frac {c+d \sin (e+f x)}{a+a \sin (e+f x)} \, dx &=\frac {d x}{a}-(-c+d) \int \frac {1}{a+a \sin (e+f x)} \, dx\\ &=\frac {d x}{a}-\frac {(c-d) \cos (e+f x)}{f (a+a \sin (e+f x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.17, size = 79, normalized size = 2.26 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right ) (2 c+d (e+f x-2))+d (e+f x) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{a f (\sin (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x]),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*(e + f*x)*Cos[(e + f*x)/2] + (2*c + d*(-2 + e + f*x))*Sin[(e + f*x)/
2]))/(a*f*(1 + Sin[e + f*x]))

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 66, normalized size = 1.89 \[ \frac {d f x + {\left (d f x - c + d\right )} \cos \left (f x + e\right ) + {\left (d f x + c - d\right )} \sin \left (f x + e\right ) - c + d}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

(d*f*x + (d*f*x - c + d)*cos(f*x + e) + (d*f*x + c - d)*sin(f*x + e) - c + d)/(a*f*cos(f*x + e) + a*f*sin(f*x
+ e) + a*f)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 40, normalized size = 1.14 \[ \frac {\frac {{\left (f x + e\right )} d}{a} - \frac {2 \, {\left (c - d\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

((f*x + e)*d/a - 2*(c - d)/(a*(tan(1/2*f*x + 1/2*e) + 1)))/f

________________________________________________________________________________________

maple [A]  time = 0.15, size = 65, normalized size = 1.86 \[ \frac {2 d \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 c}{a f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {2 d}{a f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))/(a+a*sin(f*x+e)),x)

[Out]

2/a/f*d*arctan(tan(1/2*f*x+1/2*e))-2/a/f/(tan(1/2*f*x+1/2*e)+1)*c+2/a/f/(tan(1/2*f*x+1/2*e)+1)*d

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 78, normalized size = 2.23 \[ \frac {2 \, {\left (d {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {c}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

2*(d*(arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a + 1/(a + a*sin(f*x + e)/(cos(f*x + e) + 1))) - c/(a + a*sin(f*
x + e)/(cos(f*x + e) + 1)))/f

________________________________________________________________________________________

mupad [B]  time = 6.83, size = 35, normalized size = 1.00 \[ \frac {d\,x}{a}-\frac {2\,c-2\,d}{a\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*sin(e + f*x))/(a + a*sin(e + f*x)),x)

[Out]

(d*x)/a - (2*c - 2*d)/(a*f*(tan(e/2 + (f*x)/2) + 1))

________________________________________________________________________________________

sympy [A]  time = 1.82, size = 109, normalized size = 3.11 \[ \begin {cases} - \frac {2 c}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {d f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {d f x}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2 d}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} & \text {for}\: f \neq 0 \\\frac {x \left (c + d \sin {\relax (e )}\right )}{a \sin {\relax (e )} + a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e)),x)

[Out]

Piecewise((-2*c/(a*f*tan(e/2 + f*x/2) + a*f) + d*f*x*tan(e/2 + f*x/2)/(a*f*tan(e/2 + f*x/2) + a*f) + d*f*x/(a*
f*tan(e/2 + f*x/2) + a*f) + 2*d/(a*f*tan(e/2 + f*x/2) + a*f), Ne(f, 0)), (x*(c + d*sin(e))/(a*sin(e) + a), Tru
e))

________________________________________________________________________________________