3.89 \(\int \text{CosIntegral}(a+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac{(a+b x) \text{CosIntegral}(a+b x)}{b}-\frac{\sin (a+b x)}{b} \]

[Out]

((a + b*x)*CosIntegral[a + b*x])/b - Sin[a + b*x]/b

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Rubi [A]  time = 0.0056449, antiderivative size = 27, normalized size of antiderivative = 1., 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 = {6500} \[ \frac{(a+b x) \text{CosIntegral}(a+b x)}{b}-\frac{\sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[CosIntegral[a + b*x],x]

[Out]

((a + b*x)*CosIntegral[a + b*x])/b - Sin[a + 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}(a+b x) \, dx &=\frac{(a+b x) \text{Ci}(a+b x)}{b}-\frac{\sin (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0363525, size = 42, normalized size = 1.56 \[ x \text{CosIntegral}(a+b x)+\frac{a \text{CosIntegral}(a+b x)}{b}-\frac{\sin (a) \cos (b x)}{b}-\frac{\cos (a) \sin (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[CosIntegral[a + b*x],x]

[Out]

(a*CosIntegral[a + b*x])/b + x*CosIntegral[a + b*x] - (Cos[b*x]*Sin[a])/b - (Cos[a]*Sin[b*x])/b

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Maple [A]  time = 0.043, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ){\it Ci} \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos_integral(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Ci}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x+a),x)

[Out]

Integral(Ci(a + b*x), x)

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Giac [B]  time = 1.13353, size = 93, normalized size = 3.44 \begin{align*} x \operatorname{Ci}\left (b x + a\right ) + \frac{a \cos \left (a\right )^{2} \operatorname{Ci}\left (b x + a\right ) + a \cos \left (a\right )^{2} \operatorname{Ci}\left (-b x - a\right ) + a \operatorname{Ci}\left (b x + a\right ) \sin \left (a\right )^{2} + a \operatorname{Ci}\left (-b x - a\right ) \sin \left (a\right )^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*cos_integral(b*x + a) + 1/2*(a*cos(a)^2*cos_integral(b*x + a) + a*cos(a)^2*cos_integral(-b*x - a) + a*cos_in
tegral(b*x + a)*sin(a)^2 + a*cos_integral(-b*x - a)*sin(a)^2)/b