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

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

[Out]

-(Cos[a + b*x]/b)

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Rubi [A]  time = 0.0035687, antiderivative size = 11, normalized size of antiderivative = 1., 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 = 0.0092239, size = 22, normalized size = 2. \[ \frac{\sin (a) \sin (b x)}{b}-\frac{\cos (a) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 12, normalized size = 1.1 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x+a)/b

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Maxima [A]  time = 0.940373, size = 15, normalized size = 1.36 \begin{align*} -\frac{\cos \left (b x + a\right )}{b} \end{align*}

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

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Fricas [A]  time = 1.75709, size = 23, normalized size = 2.09 \begin{align*} -\frac{\cos \left (b x + a\right )}{b} \end{align*}

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

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Sympy [A]  time = 0.125937, size = 14, normalized size = 1.27 \begin{align*} \begin{cases} - \frac{\cos{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \sin{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

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))

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Giac [A]  time = 1.07024, size = 15, normalized size = 1.36 \begin{align*} -\frac{\cos \left (b x + a\right )}{b} \end{align*}

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