3.103 \(\int \frac {\cos (x)}{x} \, dx\)

Optimal. Leaf size=2 \[ \operatorname {CosIntegral}(x) \]

[Out]

Ci(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 = {3302} \[ \text {CosIntegral}(x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/x,x]

[Out]

CosIntegral[x]

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

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \[ \operatorname {CosIntegral}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/x,x]

[Out]

CosIntegral[x]

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fricas [B]  time = 0.45, size = 11, normalized size = 5.50 \[ \frac {1}{2} \, \operatorname {Ci}\left (-x\right ) + \frac {1}{2} \, \operatorname {Ci}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.30, size = 2, normalized size = 1.00 \[ \operatorname {Ci}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos_integral(x)

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maple [A]  time = 0.02, size = 3, normalized size = 1.50 \[ \Ci \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/x,x)

[Out]

Ci(x)

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maxima [C]  time = 0.57, size = 13, normalized size = 6.50 \[ \frac {1}{2} \, {\rm Ei}\left (i \, x\right ) + \frac {1}{2} \, {\rm Ei}\left (-i \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*Ei(I*x) + 1/2*Ei(-I*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/x,x)

[Out]

cosint(x)

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sympy [B]  time = 1.00, size = 12, normalized size = 6.00 \[ - \log {\relax (x )} + \frac {\log {\left (x^{2} \right )}}{2} + \operatorname {Ci}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/x,x)

[Out]

-log(x) + log(x**2)/2 + Ci(x)

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