3.773 \(\int \sin (\pi (1+2 x)) \, dx\)

Optimal. Leaf size=12 \[ \frac {\cos (2 \pi x)}{2 \pi } \]

[Out]

1/2*cos(2*Pi*x)/Pi

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Rubi [A]  time = 0.00, antiderivative size = 12, 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 = {2638} \[ \frac {\cos (2 \pi x)}{2 \pi } \]

Antiderivative was successfully verified.

[In]

Int[Sin[Pi*(1 + 2*x)],x]

[Out]

Cos[2*Pi*x]/(2*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 (1+2 x)) \, dx &=\frac {\cos (2 \pi x)}{2 \pi }\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ \frac {\cos (2 \pi x)}{2 \pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[Pi*(1 + 2*x)],x]

[Out]

Cos[2*Pi*x]/(2*Pi)

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fricas [A]  time = 0.57, size = 12, normalized size = 1.00 \[ -\frac {\cos \left (\pi + 2 \, \pi x\right )}{2 \, \pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*(1+2*x)),x, algorithm="fricas")

[Out]

-1/2*cos(pi + 2*pi*x)/pi

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giac [A]  time = 0.14, size = 10, normalized size = 0.83 \[ \frac {\cos \left (2 \, \pi x\right )}{2 \, \pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*(1+2*x)),x, algorithm="giac")

[Out]

1/2*cos(2*pi*x)/pi

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maple [A]  time = 0.04, size = 11, normalized size = 0.92 \[ \frac {\cos \left (2 \pi x \right )}{2 \pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(Pi*(1+2*x)),x)

[Out]

1/2*cos(2*Pi*x)/Pi

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maxima [A]  time = 0.32, size = 10, normalized size = 0.83 \[ \frac {\cos \left (2 \, \pi x\right )}{2 \, \pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*(1+2*x)),x, algorithm="maxima")

[Out]

1/2*cos(2*pi*x)/pi

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mupad [B]  time = 2.92, size = 13, normalized size = 1.08 \[ -\frac {\cos \left (\Pi \,\left (2\,x+1\right )\right )}{2\,\Pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(Pi*(2*x + 1)),x)

[Out]

-cos(Pi*(2*x + 1))/(2*Pi)

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sympy [A]  time = 0.79, size = 12, normalized size = 1.00 \[ - \frac {\cos {\left (\pi \left (2 x + 1\right ) \right )}}{2 \pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(pi*(1+2*x)),x)

[Out]

-cos(pi*(2*x + 1))/(2*pi)

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