3.457 \(\int \frac{1}{a+a \sin (e+f x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0124415, 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} \[ -\frac{\cos (e+f x)}{f (a \sin (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(-1),x]

[Out]

-(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]

Rubi steps

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

Mathematica [B]  time = 0.0423332, size = 48, normalized size = 2.09 \[ \frac{2 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f (a \sin (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.023, size = 22, normalized size = 1. \begin{align*} -2\,{\frac{1}{af \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e)),x)

[Out]

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

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Maxima [A]  time = 1.06959, size = 36, normalized size = 1.57 \begin{align*} -\frac{2}{{\left (a + \frac{a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.49856, size = 108, normalized size = 4.7 \begin{align*} -\frac{\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.867783, size = 27, normalized size = 1.17 \begin{align*} \begin{cases} - \frac{2}{a f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f} & \text{for}\: f \neq 0 \\\frac{x}{a \sin{\left (e \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.41652, size = 30, normalized size = 1.3 \begin{align*} -\frac{2}{a f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*f*(tan(1/2*f*x + 1/2*e) + 1))