3.73 \(\int \text{CosIntegral}(b x) \, dx\)

Optimal. Leaf size=16 \[ x \text{CosIntegral}(b x)-\frac{\sin (b x)}{b} \]

[Out]

x*CosIntegral[b*x] - Sin[b*x]/b

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Rubi [A]  time = 0.0044011, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6500} \[ x \text{CosIntegral}(b x)-\frac{\sin (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[CosIntegral[b*x],x]

[Out]

x*CosIntegral[b*x] - Sin[b*x]/b

Rule 6500

Int[CosIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*CosIntegral[a + b*x])/b, x] - Simp[Sin[a + b
*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0014589, size = 16, normalized size = 1. \[ x \text{CosIntegral}(b x)-\frac{\sin (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[CosIntegral[b*x],x]

[Out]

x*CosIntegral[b*x] - Sin[b*x]/b

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Maple [A]  time = 0.045, size = 19, normalized size = 1.2 \begin{align*}{\frac{bx{\it Ci} \left ( bx \right ) -\sin \left ( bx \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x),x)

[Out]

1/b*(b*x*Ci(b*x)-sin(b*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Ci}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x),x, algorithm="maxima")

[Out]

integrate(Ci(b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{Ci}\left (b x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x),x, algorithm="fricas")

[Out]

integral(cos_integral(b*x), x)

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Sympy [B]  time = 1.68395, size = 31, normalized size = 1.94 \begin{align*} - x \log{\left (b x \right )} + \frac{x \log{\left (b^{2} x^{2} \right )}}{2} + x \operatorname{Ci}{\left (b x \right )} - \frac{\sin{\left (b x \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x),x)

[Out]

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

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Giac [A]  time = 1.19137, size = 22, normalized size = 1.38 \begin{align*} x \operatorname{Ci}\left (b x\right ) - \frac{\sin \left (b x\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x),x, algorithm="giac")

[Out]

x*cos_integral(b*x) - sin(b*x)/b