3.280 \(\int \sin (\pi x) \, dx\)

Optimal. Leaf size=9 \[ -\frac {\cos (\pi x)}{\pi } \]

[Out]

-cos(Pi*x)/Pi

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2638} \[ -\frac {\cos (\pi x)}{\pi } \]

Antiderivative was successfully verified.

[In]

Int[Sin[Pi*x],x]

[Out]

-(Cos[Pi*x]/Pi)

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 (\pi x) \, dx &=-\frac {\cos (\pi x)}{\pi }\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ -\frac {\cos (\pi x)}{\pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[Pi*x],x]

[Out]

-(Cos[Pi*x]/Pi)

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fricas [A]  time = 0.41, size = 9, normalized size = 1.00 \[ -\frac {\cos \left (\pi x\right )}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*x),x, algorithm="fricas")

[Out]

-cos(pi*x)/pi

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giac [A]  time = 0.93, size = 9, normalized size = 1.00 \[ -\frac {\cos \left (\pi x\right )}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*x),x, algorithm="giac")

[Out]

-cos(pi*x)/pi

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maple [A]  time = 0.01, size = 10, normalized size = 1.11 \[ -\frac {\cos \left (\pi x \right )}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(Pi*x),x)

[Out]

-cos(Pi*x)/Pi

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maxima [A]  time = 0.50, size = 9, normalized size = 1.00 \[ -\frac {\cos \left (\pi x\right )}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*x),x, algorithm="maxima")

[Out]

-cos(pi*x)/pi

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mupad [B]  time = 0.02, size = 9, normalized size = 1.00 \[ -\frac {\cos \left (\Pi \,x\right )}{\Pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(Pi*x),x)

[Out]

-cos(Pi*x)/Pi

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sympy [A]  time = 0.06, size = 7, normalized size = 0.78 \[ - \frac {\cos {\left (\pi x \right )}}{\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*x),x)

[Out]

-cos(pi*x)/pi

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