3.1.32 \(\int -\cos (e+f x) \, dx\) [32]

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

[Out]

-sin(f*x+e)/f

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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 = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2717} \begin {gather*} -\frac {\sin (e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-Cos[e + f*x],x]

[Out]

-(Sin[e + f*x]/f)

Rule 2717

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(11)=22\).
time = 0.01, size = 23, normalized size = 2.09 \begin {gather*} -\frac {\cos (f x) \sin (e)}{f}-\frac {\cos (e) \sin (f x)}{f} \end {gather*}

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 = 0.12, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(-\frac {\sin \left (f x +e \right )}{f}\)	\(12\)
default	\(-\frac {\sin \left (f x +e \right )}{f}\)	\(12\)
risch	\(-\frac {\sin \left (f x +e \right )}{f}\)	\(12\)
norman	\(-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\)	\(30\)
meijerg	\(-\frac {\cos \left (e \right ) \sin \left (f x \right )}{f}+\frac {\sin \left (e \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (f x \right )}{\sqrt {\pi }}\right )}{f}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(f*x+e),x,method=_RETURNVERBOSE)

[Out]

-sin(f*x+e)/f

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Maxima [A]
time = 0.30, size = 12, normalized size = 1.09 \begin {gather*} -\frac {\sin \left (f x + e\right )}{f} \end {gather*}

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 = 3.94, size = 12, normalized size = 1.09 \begin {gather*} -\frac {\sin \left (f x + e\right )}{f} \end {gather*}

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 = 0.04, size = 14, normalized size = 1.27 \begin {gather*} - \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 {gather*}

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 = 0.41, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\sin \left (f x + e\right )}{f} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(e + f*x),x)

[Out]

-sin(e + f*x)/f

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