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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 27, 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 = {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*}

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Mathematica [A]  time = 0.04, size = 42, normalized size = 1.56 \[ x \text {Ci}(a+b x)+\frac {a \text {Ci}(a+b x)}{b}-\frac {\sin (a) \cos (b x)}{b}-\frac {\cos (a) \sin (b x)}{b} \]

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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fricas [F]  time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Ci}\left (b x + a\right ), x\right ) \]

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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giac [B]  time = 0.21, size = 85, normalized size = 3.15 \[ x \operatorname {Ci}\left (b x + a\right ) + \frac {a \cos \relax (a)^{2} \operatorname {Ci}\left (b x + a\right ) + a \cos \relax (a)^{2} \operatorname {Ci}\left (-b x - a\right ) + a \operatorname {Ci}\left (b x + a\right ) \sin \relax (a)^{2} + a \operatorname {Ci}\left (-b x - a\right ) \sin \relax (a)^{2} - 2 \, \cos \relax (a) \sin \left (b x\right ) - 2 \, \cos \left (b x\right ) \sin \relax (a)}{2 \, b} \]

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 - 2*cos(a)*sin(b*x) - 2*cos(b*x)*sin(a))/b

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maple [A]  time = 0.00, size = 26, normalized size = 0.96 \[ \frac {\left (b x +a \right ) \Ci \left (b x +a \right )-\sin \left (b x +a \right )}{b} \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ x\,\mathrm {cosint}\left (a+b\,x\right )-\frac {\sin \left (a+b\,x\right )-a\,\mathrm {cosint}\left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosint(a + b*x),x)

[Out]

x*cosint(a + b*x) - (sin(a + b*x) - a*cosint(a + b*x))/b

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {Ci}{\left (a + b x \right )}\, dx \]

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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