\(\int \frac {1}{3+3 \sin (e+f x)} \, dx\) [457]

Optimal result

Integrand size = 12, antiderivative size = 23 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=-\frac {\cos (e+f x)}{f (3+3 \sin (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2727} \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=-\frac {\cos (e+f x)}{f (a \sin (e+f x)+a)} \]

[In]

Int[(a + a*Sin[e + f*x])^(-1),x]

[Out]

-(Cos[e + f*x]/(f*(a + a*Sin[e + f*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 (e+f x)}{f (a+a \sin (e+f x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=\frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f (3+3 \sin (e+f x))} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(-1),x]

[Out]

(2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*(3 + 3*Sin[e + f*x]))

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {2}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(22\)
default	\(-\frac {2}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(22\)
risch	\(-\frac {2}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\)	\(23\)
norman	\(\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(31\)
parallelrisch	\(\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\)	\(31\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=-\frac {\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=\begin {cases} - \frac {2}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} & \text {for}\: f \neq 0 \\\frac {x}{a \sin {\left (e \right )} + a} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=-\frac {2}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=-\frac {2}{a f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 6.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{3+3 \sin (e+f x)} \, dx=-\frac {2}{a\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \]

[In]

int(1/(a + a*sin(e + f*x)),x)

[Out]

-2/(a*f*(tan(e/2 + (f*x)/2) + 1))