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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ x \text {Ci}(b x)-\frac {\sin (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[CosIntegral[b*x],x]

[Out]

x*CosIntegral[b*x] - Sin[b*x]/b

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Ci}\left (b x\right ), x\right ) \]

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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giac [A]  time = 0.23, size = 16, normalized size = 1.00 \[ x \operatorname {Ci}\left (b x\right ) - \frac {\sin \left (b x\right )}{b} \]

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

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maple [A]  time = 0.01, size = 19, normalized size = 1.19 \[ \frac {b x \Ci \left (b x \right )-\sin \left (b x \right )}{b} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Ci}\left (b x\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ x\,\mathrm {cosint}\left (b\,x\right )-\frac {\sin \left (b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosint(b*x),x)

[Out]

x*cosint(b*x) - sin(b*x)/b

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sympy [B]  time = 1.24, size = 31, normalized size = 1.94 \[ - x \log {\left (b x \right )} + \frac {x \log {\left (b^{2} x^{2} \right )}}{2} + x \operatorname {Ci}{\left (b x \right )} - \frac {\sin {\left (b x \right )}}{b} \]

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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