3.182 \(\int \frac{\sin (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=28 \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac{x}{a} \]

[Out]

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

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Rubi [A]  time = 0.0380084, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2735, 2648} \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

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

Mathematica [B]  time = 0.108201, size = 72, normalized size = 2.57 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left ((c+d x-2) \sin \left (\frac{1}{2} (c+d x)\right )+(c+d x) \cos \left (\frac{1}{2} (c+d x)\right )\right )}{a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 41, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}+2\,{\frac{1}{da \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42433, size = 68, normalized size = 2.43 \begin{align*} \frac{2 \,{\left (\frac{\arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{1}{a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.74005, size = 142, normalized size = 5.07 \begin{align*} \frac{d x +{\left (d x + 1\right )} \cos \left (d x + c\right ) +{\left (d x - 1\right )} \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + (d*x + 1)*cos(d*x + c) + (d*x - 1)*sin(d*x + c) + 1)/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)

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Sympy [A]  time = 1.54983, size = 90, normalized size = 3.21 \begin{align*} \begin{cases} \frac{d x \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{d x}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12511, size = 43, normalized size = 1.54 \begin{align*} \frac{\frac{d x + c}{a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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