3.105 \(\int \frac {\sin ^2(x)}{x} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log (x)}{2}-\frac {1}{2} \operatorname {CosIntegral}(2 x) \]

[Out]

-1/2*Ci(2*x)+1/2*ln(x)

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Rubi [A]  time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3312, 3302} \[ \frac {\log (x)}{2}-\frac {1}{2} \text {CosIntegral}(2 x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/x,x]

[Out]

-CosIntegral[2*x]/2 + Log[x]/2

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin {align*} \int \frac {\sin ^2(x)}{x} \, dx &=\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \int \frac {\cos (2 x)}{x} \, dx\\ &=-\frac {\text {Ci}(2 x)}{2}+\frac {\log (x)}{2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\log (x)}{2}-\frac {1}{2} \operatorname {CosIntegral}(2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/x,x]

[Out]

-1/2*CosIntegral[2*x] + Log[x]/2

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fricas [A]  time = 0.43, size = 17, normalized size = 1.13 \[ -\frac {1}{4} \, \operatorname {Ci}\left (2 \, x\right ) - \frac {1}{4} \, \operatorname {Ci}\left (-2 \, x\right ) + \frac {1}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/x,x, algorithm="fricas")

[Out]

-1/4*cos_integral(2*x) - 1/4*cos_integral(-2*x) + 1/2*log(x)

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giac [A]  time = 1.13, size = 11, normalized size = 0.73 \[ -\frac {1}{2} \, \operatorname {Ci}\left (2 \, x\right ) + \frac {1}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/x,x, algorithm="giac")

[Out]

-1/2*cos_integral(2*x) + 1/2*log(x)

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maple [A]  time = 0.02, size = 12, normalized size = 0.80 \[ -\frac {\Ci \left (2 x \right )}{2}+\frac {\ln \relax (x )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/x,x)

[Out]

-1/2*Ci(2*x)+1/2*ln(x)

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maxima [C]  time = 0.56, size = 17, normalized size = 1.13 \[ -\frac {1}{4} \, {\rm Ei}\left (2 i \, x\right ) - \frac {1}{4} \, {\rm Ei}\left (-2 i \, x\right ) + \frac {1}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/x,x, algorithm="maxima")

[Out]

-1/4*Ei(2*I*x) - 1/4*Ei(-2*I*x) + 1/2*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \frac {\ln \relax (x)}{2}-\frac {\mathrm {cosint}\left (2\,x\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/x,x)

[Out]

log(x)/2 - cosint(2*x)/2

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sympy [A]  time = 1.08, size = 10, normalized size = 0.67 \[ \frac {\log {\relax (x )}}{2} - \frac {\operatorname {Ci}{\left (2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/x,x)

[Out]

log(x)/2 - Ci(2*x)/2

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