3.32 \(\int -\cos (e+f x) \, dx\)

Optimal. Leaf size=11 \[ -\frac{\sin (e+f x)}{f} \]

[Out]

-(Sin[e + f*x]/f)

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Rubi [A]  time = 0.0045571, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2637} \[ -\frac{\sin (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[-Cos[e + f*x],x]

[Out]

-(Sin[e + f*x]/f)

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [B]  time = 0.0098691, size = 23, normalized size = 2.09 \[ -\frac{\sin (e) \cos (f x)}{f}-\frac{\cos (e) \sin (f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[-Cos[e + f*x],x]

[Out]

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

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Maple [A]  time = 0.011, size = 12, normalized size = 1.1 \begin{align*} -{\frac{\sin \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(f*x+e),x)

[Out]

-sin(f*x+e)/f

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Maxima [A]  time = 0.915883, size = 15, normalized size = 1.36 \begin{align*} -\frac{\sin \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(f*x+e),x, algorithm="maxima")

[Out]

-sin(f*x + e)/f

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Fricas [A]  time = 0.458646, size = 23, normalized size = 2.09 \begin{align*} -\frac{\sin \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(f*x+e),x, algorithm="fricas")

[Out]

-sin(f*x + e)/f

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Sympy [A]  time = 0.193778, size = 14, normalized size = 1.27 \begin{align*} - \begin{cases} \frac{\sin{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \cos{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(f*x+e),x)

[Out]

-Piecewise((sin(e + f*x)/f, Ne(f, 0)), (x*cos(e), True))

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Giac [A]  time = 1.16142, size = 16, normalized size = 1.45 \begin{align*} -\frac{\sin \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(f*x+e),x, algorithm="giac")

[Out]

-sin(f*x + e)/f