\(\int (3+3 \sin (e+f x)) \, dx\) [429]

Optimal result

Integrand size = 10, antiderivative size = 15 \[ \int (3+3 \sin (e+f x)) \, dx=3 x-\frac {3 \cos (e+f x)}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2718} \[ \int (3+3 \sin (e+f x)) \, dx=a x-\frac {a \cos (e+f x)}{f} \]

[In]

Int[a + a*Sin[e + f*x],x]

[Out]

a*x - (a*Cos[e + f*x])/f

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int (3+3 \sin (e+f x)) \, dx=3 x-\frac {3 \cos (e) \cos (f x)}{f}+\frac {3 \sin (e) \sin (f x)}{f} \]

[In]

Integrate[3 + 3*Sin[e + f*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13

method	result	size
default	\(a x -\frac {a \cos \left (f x +e \right )}{f}\)	\(17\)
risch	\(a x -\frac {a \cos \left (f x +e \right )}{f}\)	\(17\)
parts	\(a x -\frac {a \cos \left (f x +e \right )}{f}\)	\(17\)
parallelrisch	\(\frac {a \left (-\cos \left (f x +e \right )-1\right )}{f}+a x\)	\(20\)
derivativedivides	\(\frac {a \left (f x +e \right )-a \cos \left (f x +e \right )}{f}\)	\(22\)
norman	\(\frac {a x +a x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int (3+3 \sin (e+f x)) \, dx=\frac {a f x - a \cos \left (f x + e\right )}{f} \]

[In]

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

[Out]

(a*f*x - a*cos(f*x + e))/f

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int (3+3 \sin (e+f x)) \, dx=a x + a \left (\begin {cases} - \frac {\cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \sin {\left (e \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int (3+3 \sin (e+f x)) \, dx=a x - \frac {a \cos \left (f x + e\right )}{f} \]

[In]

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

[Out]

a*x - a*cos(f*x + e)/f

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int (3+3 \sin (e+f x)) \, dx=a x - \frac {a \cos \left (f x + e\right )}{f} \]

[In]

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

[Out]

a*x - a*cos(f*x + e)/f

Mupad [B] (verification not implemented)

Time = 6.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int (3+3 \sin (e+f x)) \, dx=a\,x-\frac {2\,a}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

int(a + a*sin(e + f*x),x)

[Out]

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