3.1.11 \(\int \frac {\sin (y)}{y} \, dy\) [11]

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 = {3380} \begin {gather*} \text {Si}(y) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[y]/y,y]

[Out]

SinIntegral[y]

Rule 3380

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 = 0.01, size = 2, normalized size = 1.00 \begin {gather*} \text {Si}(y) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[y]/y,y]

[Out]

SinIntegral[y]

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Maple [A]
time = 0.02, size = 3, normalized size = 1.50

method	result	size
default	\(\sinIntegral \left (y \right )\)	\(3\)
meijerg	\(\sinIntegral \left (y \right )\)	\(3\)
risch	\(-\frac {\pi \,\mathrm {csgn}\left (y \right )}{2}+\sinIntegral \left (y \right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)/y,y,method=_RETURNVERBOSE)

[Out]

Si(y)

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Maxima [C] Result contains complex when optimal does not.
time = 1.97, size = 13, normalized size = 6.50 \begin {gather*} -\frac {1}{2} i \, {\rm Ei}\left (i \, y\right ) + \frac {1}{2} i \, {\rm Ei}\left (-i \, y\right ) \end {gather*}

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 = 0.69, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {Si}\left (y\right ) \end {gather*}

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 = 0.31, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {Si}{\left (y \right )} \end {gather*}

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 = 0.43, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {Si}\left (y\right ) \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.50 \begin {gather*} \mathrm {sinint}\left (y\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)/y,y)

[Out]

sinint(y)

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