3.108 \(\int \cos (a+b x) \, dx\)

Optimal. Leaf size=10 \[ \frac{\sin (a+b x)}{b} \]

[Out]

Sin[a + b*x]/b

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Rubi [A]  time = 0.00802069, antiderivative size = 10, 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 \[ \frac{\sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[Cos[a + b*x],x]

[Out]

Sin[a + b*x]/b

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Rubi in Sympy [A]  time = 0.612479, size = 7, normalized size = 0.7 \[ \frac{\sin{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(b*x+a),x)

[Out]

sin(a + b*x)/b

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Mathematica [B]  time = 0.0086357, size = 21, normalized size = 2.1 \[ \frac{\sin (a) \cos (b x)}{b}+\frac{\cos (a) \sin (b x)}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 11, normalized size = 1.1 \[{\frac{\sin \left ( bx+a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(b*x+a),x)

[Out]

sin(b*x+a)/b

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Maxima [A]  time = 1.3898, size = 14, normalized size = 1.4 \[ \frac{\sin \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(b*x + a),x, algorithm="maxima")

[Out]

sin(b*x + a)/b

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Fricas [A]  time = 0.208757, size = 14, normalized size = 1.4 \[ \frac{\sin \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(b*x + a),x, algorithm="fricas")

[Out]

sin(b*x + a)/b

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Sympy [A]  time = 0.139349, size = 12, normalized size = 1.2 \[ \begin{cases} \frac{\sin{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cos{\left (a \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20459, size = 14, normalized size = 1.4 \[ \frac{\sin \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(b*x + a),x, algorithm="giac")

[Out]

sin(b*x + a)/b