3.56 \(\int \frac{1}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0116354, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2648} \[ -\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(-1),x]

[Out]

-(Cos[c + d*x]/(d*(a + a*Sin[c + d*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]

Rubi steps

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

Mathematica [B]  time = 0.0414675, size = 48, normalized size = 2.09 \[ \frac{2 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d (a \sin (c+d x)+a)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.022, size = 22, normalized size = 1. \begin{align*} -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(1/(a+a*sin(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.07772, size = 36, normalized size = 1.57 \begin{align*} -\frac{2}{{\left (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(1/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.46038, size = 108, normalized size = 4.7 \begin{align*} -\frac{\cos \left (d x + c\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(1/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.72599, size = 27, normalized size = 1.17 \begin{align*} \begin{cases} - \frac{2}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.27038, size = 28, normalized size = 1.22 \begin{align*} -\frac{2}{a d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*d*(tan(1/2*d*x + 1/2*c) + 1))