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

Optimal. Leaf size=16 \[ a x-\frac {a \cos (e+f x)}{f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2718} \begin {gather*} a x-\frac {a \cos (e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 27, normalized size = 1.69 \begin {gather*} a x-\frac {a \cos (e) \cos (f x)}{f}+\frac {a \sin (e) \sin (f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + a*Sin[e + f*x],x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 17, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 17, normalized size = 1.06 \begin {gather*} a x - \frac {a \cos \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - a*cos(f*x + e)/f

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 19, normalized size = 1.19 \begin {gather*} \frac {a f x - a \cos \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*f*x - a*cos(f*x + e))/f

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 19, normalized size = 1.19 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]
time = 0.46, size = 17, normalized size = 1.06 \begin {gather*} a x - \frac {a \cos \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - a*cos(f*x + e)/f

________________________________________________________________________________________

Mupad [B]
time = 6.72, size = 25, normalized size = 1.56 \begin {gather*} a\,x-\frac {2\,a}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + a*sin(e + f*x),x)

[Out]

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

________________________________________________________________________________________