3.1.73 \(\int \text {CosIntegral}(b x) \, dx\) [73]

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

[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 = {6635} \begin {gather*} x \text {CosIntegral}(b x)-\frac {\sin (b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[CosIntegral[b*x],x]

[Out]

x*CosIntegral[b*x] - Sin[b*x]/b

Rule 6635

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 \begin {gather*} x \text {CosIntegral}(b x)-\frac {\sin (b x)}{b} \end {gather*}

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 = 0.17, size = 19, normalized size = 1.19

method	result	size
derivativedivides	\(\frac {\cosineIntegral \left (b x \right ) b x -\sin \left (b x \right )}{b}\)	\(19\)
default	\(\frac {\cosineIntegral \left (b x \right ) b x -\sin \left (b x \right )}{b}\)	\(19\)
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {2 b x}{\sqrt {\pi }}-\frac {2 b x \gamma }{\sqrt {\pi }}-\frac {2 b x \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 b x \ln \left (\frac {b x}{2}\right )}{\sqrt {\pi }}-\frac {2 \sin \left (b x \right )}{\sqrt {\pi }}+\frac {2 b x \cosineIntegral \left (b x \right )}{\sqrt {\pi }}+\frac {\left (2 \gamma -2+2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x b}{\sqrt {\pi }}\right )}{2 b}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 27, normalized size = 1.69 \begin {gather*} \frac {b x \operatorname {C}\left (b x\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{\pi }}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*fresnel_cos(b*x) - sin(1/2*pi*b^2*x^2)/pi)/b

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Fricas [A]
time = 0.36, size = 28, normalized size = 1.75 \begin {gather*} \frac {\pi b x \operatorname {C}\left (b x\right ) - \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{\pi b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(pi*b*x*fresnel_cos(b*x) - sin(1/2*pi*b^2*x^2))/(pi*b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.75, size = 31, normalized size = 1.94 \begin {gather*} - 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 {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnel_cos(b*x), x)

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

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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