3.3.80 \(\int \sin (\pi x) \, dx\) [280]

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 = {2718} \begin {gather*} -\frac {\cos (\pi x)}{\pi } \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[Pi*x],x]

[Out]

-(Cos[Pi*x]/Pi)

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

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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} -\frac {\cos (\pi x)}{\pi } \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[Pi*x],x]

[Out]

-(Cos[Pi*x]/Pi)

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Maple [A]
time = 0.01, size = 10, normalized size = 1.11

method	result	size
derivativedivides	\(-\frac {\cos \left (\pi x \right )}{\pi }\)	\(10\)
default	\(-\frac {\cos \left (\pi x \right )}{\pi }\)	\(10\)
risch	\(-\frac {\cos \left (\pi x \right )}{\pi }\)	\(10\)
norman	\(-\frac {2}{\pi \left (1+\tan ^{2}\left (\frac {\pi x}{2}\right )\right )}\)	\(17\)
meijerg	\(\frac {\frac {1}{\sqrt {\pi }}-\frac {\cos \left (\pi x \right )}{\sqrt {\pi }}}{\sqrt {\pi }}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(Pi*x),x,method=_RETURNVERBOSE)

[Out]

-cos(Pi*x)/Pi

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Maxima [A]
time = 4.18, size = 9, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (\pi x\right )}{\pi } \end {gather*}

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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Fricas [A]
time = 0.49, size = 9, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (\pi x\right )}{\pi } \end {gather*}

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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Sympy [A]
time = 0.02, size = 7, normalized size = 0.78 \begin {gather*} - \frac {\cos {\left (\pi x \right )}}{\pi } \end {gather*}

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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Giac [A]
time = 1.70, size = 9, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (\pi x\right )}{\pi } \end {gather*}

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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Mupad [B]
time = 0.02, size = 9, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (\Pi \,x\right )}{\Pi } \end {gather*}

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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