\(\int \frac {\sin (x)}{a+a \sin (x)} \, dx\) [6]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {x}{a}+\frac {\cos (x)}{a+a \sin (x)} \]

[Out]

x/a+cos(x)/(a+a*sin(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2814, 2727} \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {x}{a}+\frac {\cos (x)}{a \sin (x)+a} \]

[In]

Int[Sin[x]/(a + a*Sin[x]),x]

[Out]

x/a + Cos[x]/(a + a*Sin[x])

Rule 2727

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 2814

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(17)=34\).

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.47 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (x \cos \left (\frac {x}{2}\right )+(-2+x) \sin \left (\frac {x}{2}\right )\right )}{a (1+\sin (x))} \]

[In]

Integrate[Sin[x]/(a + a*Sin[x]),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29

method	result	size
risch	\(\frac {x}{a}+\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}\)	\(22\)
parallelrisch	\(\frac {2+\tan \left (\frac {x}{2}\right ) x +x}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(22\)
default	\(\frac {\frac {4}{2 \tan \left (\frac {x}{2}\right )+2}+2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}\)	\(24\)
norman	\(\frac {\frac {x}{a}+\frac {2}{a}+\frac {x \tan \left (\frac {x}{2}\right )}{a}+\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(73\)

[In]

int(sin(x)/(a+a*sin(x)),x,method=_RETURNVERBOSE)

[Out]

x/a+2/(exp(I*x)+I)/a

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {{\left (x + 1\right )} \cos \left (x\right ) + {\left (x - 1\right )} \sin \left (x\right ) + x + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {x \tan {\left (\frac {x}{2} \right )}}{a \tan {\left (\frac {x}{2} \right )} + a} + \frac {x}{a \tan {\left (\frac {x}{2} \right )} + a} + \frac {2}{a \tan {\left (\frac {x}{2} \right )} + a} \]

[In]

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

[Out]

x*tan(x/2)/(a*tan(x/2) + a) + x/(a*tan(x/2) + a) + 2/(a*tan(x/2) + a)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} + \frac {2}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {x}{a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]

[In]

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

[Out]

x/a + 2/(a*(tan(1/2*x) + 1))

Mupad [B] (verification not implemented)

Time = 5.93 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\sin (x)}{a+a \sin (x)} \, dx=\frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}+\frac {x}{a} \]

[In]

int(sin(x)/(a + a*sin(x)),x)

[Out]

2/(a*(tan(x/2) + 1)) + x/a