3.11 \(\int \frac{\sin (y)}{y} \, dy\)

Optimal. Leaf size=2 \[ \text{Si}(y) \]

[Out]

SinIntegral[y]

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Rubi [A]  time = 0.0111784, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3299} \[ \text{Si}(y) \]

Antiderivative was successfully verified.

[In]

Int[Sin[y]/y,y]

[Out]

SinIntegral[y]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\sin (y)}{y} \, dy &=\text{Si}(y)\\ \end{align*}

Mathematica [A]  time = 0.0132604, size = 2, normalized size = 1. \[ \text{Si}(y) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[y]/y,y]

[Out]

SinIntegral[y]

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Maple [A]  time = 0.001, size = 3, normalized size = 1.5 \begin{align*}{\it Si} \left ( y \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)/y,y)

[Out]

Si(y)

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Maxima [C]  time = 1.05461, size = 18, normalized size = 9. \begin{align*} -\frac{1}{2} i \,{\rm Ei}\left (i \, y\right ) + \frac{1}{2} i \,{\rm Ei}\left (-i \, y\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)/y,y, algorithm="maxima")

[Out]

-1/2*I*Ei(I*y) + 1/2*I*Ei(-I*y)

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Fricas [A]  time = 1.85999, size = 23, normalized size = 11.5 \begin{align*} \operatorname{Si}\left (y\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)/y,y, algorithm="fricas")

[Out]

sin_integral(y)

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Sympy [A]  time = 0.539575, size = 2, normalized size = 1. \begin{align*} \operatorname{Si}{\left (y \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)/y,y)

[Out]

Si(y)

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Giac [A]  time = 1.11709, size = 3, normalized size = 1.5 \begin{align*} \operatorname{Si}\left (y\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)/y,y, algorithm="giac")

[Out]

sin_integral(y)