\(\int \frac {1}{a+a \sin (x)} \, dx\) [7]

Optimal result

Integrand size = 8, antiderivative size = 12 \[ \int \frac {1}{a+a \sin (x)} \, dx=-\frac {\cos (x)}{a+a \sin (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2727} \[ \int \frac {1}{a+a \sin (x)} \, dx=-\frac {\cos (x)}{a \sin (x)+a} \]

[In]

Int[(a + a*Sin[x])^(-1),x]

[Out]

-(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (x)}{a+a \sin (x)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(12)=24\).

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.42 \[ \int \frac {1}{a+a \sin (x)} \, dx=\frac {2 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{a+a \sin (x)} \]

[In]

Integrate[(a + a*Sin[x])^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(14\)
norman	\(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(14\)
parallelrisch	\(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(14\)
risch	\(-\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}\)	\(16\)

[In]

int(1/(a+a*sin(x)),x,method=_RETURNVERBOSE)

[Out]

-2/a/(tan(1/2*x)+1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {1}{a+a \sin (x)} \, dx=-\frac {\cos \left (x\right ) - \sin \left (x\right ) + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \]

[In]

integrate(1/(a+a*sin(x)),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{a+a \sin (x)} \, dx=- \frac {2}{a \tan {\left (\frac {x}{2} \right )} + a} \]

[In]

integrate(1/(a+a*sin(x)),x)

[Out]

-2/(a*tan(x/2) + a)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {1}{a+a \sin (x)} \, dx=-\frac {2}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1}} \]

[In]

integrate(1/(a+a*sin(x)),x, algorithm="maxima")

[Out]

-2/(a + a*sin(x)/(cos(x) + 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1}{a+a \sin (x)} \, dx=-\frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]

[In]

integrate(1/(a+a*sin(x)),x, algorithm="giac")

[Out]

-2/(a*(tan(1/2*x) + 1))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1}{a+a \sin (x)} \, dx=-\frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]

[In]

int(1/(a + a*sin(x)),x)

[Out]

-2/(a*(tan(x/2) + 1))