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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0045919, antiderivative size = 12, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0050726, size = 12, normalized size = 1. \[ \frac{\cos (2 \pi x)}{2 \pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[Pi*(1 + 2*x)],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 11, normalized size = 0.9 \begin{align*}{\frac{\cos \left ( 2\,\pi \,x \right ) }{2\,\pi }} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 0.951001, size = 14, normalized size = 1.17 \begin{align*} \frac{\cos \left (2 \, \pi x\right )}{2 \, \pi } \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.92326, size = 35, normalized size = 2.92 \begin{align*} -\frac{\cos \left (\pi + 2 \, \pi x\right )}{2 \, \pi } \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 1.04472, size = 12, normalized size = 1. \begin{align*} - \frac{\cos{\left (\pi \left (2 x + 1\right ) \right )}}{2 \pi } \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.08439, size = 14, normalized size = 1.17 \begin{align*} \frac{\cos \left (2 \, \pi x\right )}{2 \, \pi } \end{align*}

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