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

Optimal. Leaf size=10 \[ \operatorname {CosIntegral}(x)-\frac {\sin (x)}{x} \]

[Out]

Ci(x)-sin(x)/x

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3297, 3302} \[ \text {CosIntegral}(x)-\frac {\sin (x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

CosIntegral[x] - Sin[x]/x

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \operatorname {CosIntegral}(x)-\frac {\sin (x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

CosIntegral[x] - Sin[x]/x

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fricas [A]  time = 0.43, size = 20, normalized size = 2.00 \[ \frac {x \operatorname {Ci}\left (-x\right ) + x \operatorname {Ci}\relax (x) - 2 \, \sin \relax (x)}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x*cos_integral(-x) + x*cos_integral(x) - 2*sin(x))/x

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giac [A]  time = 1.28, size = 13, normalized size = 1.30 \[ \frac {x \operatorname {Ci}\relax (x) - \sin \relax (x)}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*cos_integral(x) - sin(x))/x

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maple [A]  time = 0.01, size = 11, normalized size = 1.10 \[ \Ci \relax (x )-\frac {\sin \relax (x )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ci(x)-sin(x)/x

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maxima [C]  time = 0.55, size = 15, normalized size = 1.50 \[ \frac {1}{2} \, \Gamma \left (-1, i \, x\right ) + \frac {1}{2} \, \Gamma \left (-1, -i \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*gamma(-1, I*x) + 1/2*gamma(-1, -I*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.10 \[ \mathrm {cosint}\relax (x)-\frac {\sin \relax (x)}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosint(x) - sin(x)/x

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sympy [B]  time = 1.45, size = 17, normalized size = 1.70 \[ - \log {\relax (x )} + \frac {\log {\left (x^{2} \right )}}{2} + \operatorname {Ci}{\relax (x )} - \frac {\sin {\relax (x )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x**2)/2 + Ci(x) - sin(x)/x

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