### 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 = 0.0037205, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, 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*}

Mathematica [A]  time = 0.0036049, size = 9, normalized size = 1. $-\frac{\cos (\pi x)}{\pi }$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sin[Pi*x],x]

[Out]

-(Cos[Pi*x]/Pi)

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Maple [A]  time = 0.003, size = 10, normalized size = 1.1 \begin{align*} -{\frac{\cos \left ( \pi \,x \right ) }{\pi }} \end{align*}

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 = 0.967854, size = 12, normalized size = 1.33 \begin{align*} -\frac{\cos \left (\pi x\right )}{\pi } \end{align*}

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 = 2.02815, size = 20, normalized size = 2.22 \begin{align*} -\frac{\cos \left (\pi x\right )}{\pi } \end{align*}

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 = 0.060021, size = 7, normalized size = 0.78 \begin{align*} - \frac{\cos{\left (\pi x \right )}}{\pi } \end{align*}

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 = 1.04805, size = 12, normalized size = 1.33 \begin{align*} -\frac{\cos \left (\pi x\right )}{\pi } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(pi*x)/pi