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

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

[Out]

Si(y)

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Rubi [A]  time = 0.01, antiderivative size = 2, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 2, normalized size = 1.00 \[ \text {Si}(y) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[y]/y,y]

[Out]

SinIntegral[y]

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fricas [A]  time = 0.42, size = 2, normalized size = 1.00 \[ \operatorname {Si}\relax (y) \]

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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giac [A]  time = 0.98, size = 2, normalized size = 1.00 \[ \operatorname {Si}\relax (y) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin_integral(y)

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maple [A]  time = 0.00, size = 3, normalized size = 1.50 \[ \Si \relax (y ) \]

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 = 0.70, size = 13, normalized size = 6.50 \[ -\frac {1}{2} i \, {\rm Ei}\left (i \, y\right ) + \frac {1}{2} i \, {\rm Ei}\left (-i \, y\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.50 \[ \mathrm {sinint}\relax (y) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)/y,y)

[Out]

sinint(y)

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sympy [A]  time = 0.60, size = 2, normalized size = 1.00 \[ \operatorname {Si}{\relax (y )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)/y,y)

[Out]

Si(y)

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