3.2.2 \(\int \frac {\sin (x)}{x} \, dx\) [102]

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

[Out]

Si(x)

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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}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]/x,x]

[Out]

SinIntegral[x]

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 (x)}{x} \, dx &=\text {Si}(x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 2, normalized size = 1.00 \begin {gather*} \text {Si}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/x,x]

[Out]

SinIntegral[x]

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Maple [A]
time = 0.01, size = 3, normalized size = 1.50

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Si(x)

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Maxima [C] Result contains complex when optimal does not.
time = 1.28, size = 13, normalized size = 6.50 \begin {gather*} -\frac {1}{2} i \, {\rm Ei}\left (i \, x\right ) + \frac {1}{2} i \, {\rm Ei}\left (-i \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*Ei(I*x) + 1/2*I*Ei(-I*x)

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Fricas [A]
time = 0.57, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {Si}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin_integral(x)

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Sympy [A]
time = 0.33, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {Si}{\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/x,x)

[Out]

Si(x)

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Giac [A]
time = 0.71, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {Si}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin_integral(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/x,x)

[Out]

sinint(x)

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