3.2.7 \(\int \sin (a+b x) \, dx\) [107]

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

[Out]

-cos(b*x+a)/b

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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 = {2718} \begin {gather*} -\frac {\cos (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x],x]

[Out]

-(Cos[a + b*x]/b)

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

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

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 = 0.02, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(-\frac {\cos \left (b x +a \right )}{b}\)	\(12\)
default	\(-\frac {\cos \left (b x +a \right )}{b}\)	\(12\)
risch	\(-\frac {\cos \left (b x +a \right )}{b}\)	\(12\)
norman	\(\frac {2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\)	\(32\)
meijerg	\(\frac {\sin \left (a \right ) \sin \left (b x \right )}{b}+\frac {\cos \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x+a)/b

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Maxima [A]
time = 0.93, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (b x + a\right )}{b} \end {gather*}

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 = 0.81, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (b x + a\right )}{b} \end {gather*}

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 = 0.05, size = 14, normalized size = 1.27 \begin {gather*} \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 {gather*}

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 = 1.15, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (b x + a\right )}{b} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x),x)

[Out]

-cos(a + b*x)/b

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