3.107 \(\int \sin (a+b x) \, dx\)

Optimal. Leaf size=11 \[ -\frac {\cos (a+b x)}{b} \]

[Out]

-cos(b*x+a)/b

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2638} \[ -\frac {\cos (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x],x]

[Out]

-(Cos[a + b*x]/b)

Rule 2638

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 2.00 \[ \frac {\sin (a) \sin (b x)}{b}-\frac {\cos (a) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 11, normalized size = 1.00 \[ -\frac {\cos \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x + a)/b

________________________________________________________________________________________

giac [A]  time = 0.96, size = 11, normalized size = 1.00 \[ -\frac {\cos \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x + a)/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 12, normalized size = 1.09 \[ -\frac {\cos \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a),x)

[Out]

-cos(b*x+a)/b

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 11, normalized size = 1.00 \[ -\frac {\cos \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x + a)/b

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 11, normalized size = 1.00 \[ -\frac {\cos \left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x),x)

[Out]

-cos(a + b*x)/b

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 14, normalized size = 1.27 \[ \begin {cases} - \frac {\cos {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \sin {\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a),x)

[Out]

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

________________________________________________________________________________________