3.21 \(\int \text {Si}(a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac {(a+b x) \text {Si}(a+b x)}{b}+\frac {\cos (a+b x)}{b} \]

[Out]

cos(b*x+a)/b+(b*x+a)*Si(b*x+a)/b

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Rubi [A]  time = 0.01, antiderivative size = 26, 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 = {6499} \[ \frac {(a+b x) \text {Si}(a+b x)}{b}+\frac {\cos (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinIntegral[a + b*x],x]

[Out]

Cos[a + b*x]/b + ((a + b*x)*SinIntegral[a + b*x])/b

Rule 6499

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

Rubi steps

\begin {align*} \int \text {Si}(a+b x) \, dx &=\frac {\cos (a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 41, normalized size = 1.58 \[ x \text {Si}(a+b x)+\frac {a \text {Si}(a+b x)}{b}-\frac {\sin (a) \sin (b x)}{b}+\frac {\cos (a) \cos (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinIntegral[a + b*x],x]

[Out]

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

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Si}\left (b x + a\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin_integral(b*x + a), x)

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giac [C]  time = 0.61, size = 303, normalized size = 11.65 \[ x \operatorname {Si}\left (b x + a\right ) + \frac {{\left (a \Im \left (\operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} - a \Im \left (\operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, a \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + a \Im \left (\operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} - a \Im \left (\operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, a \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + a \Im \left (\operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} - a \Im \left (\operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, a \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + a \Im \left (\operatorname {Ci}\left (b x + a\right ) \right ) - a \Im \left (\operatorname {Ci}\left (-b x - a\right ) \right ) + 2 \, a \operatorname {Si}\left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, b x\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 2\right )} b}{2 \, {\left (b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} + b^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin_integral(b*x + a) + 1/2*(a*imag_part(cos_integral(b*x + a))*tan(1/2*b*x)^2*tan(1/2*a)^2 - a*imag_part(co
s_integral(-b*x - a))*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*a*sin_integral(b*x + a)*tan(1/2*b*x)^2*tan(1/2*a)^2 + a*
imag_part(cos_integral(b*x + a))*tan(1/2*b*x)^2 - a*imag_part(cos_integral(-b*x - a))*tan(1/2*b*x)^2 + 2*a*sin
_integral(b*x + a)*tan(1/2*b*x)^2 + a*imag_part(cos_integral(b*x + a))*tan(1/2*a)^2 - a*imag_part(cos_integral
(-b*x - a))*tan(1/2*a)^2 + 2*a*sin_integral(b*x + a)*tan(1/2*a)^2 + 2*tan(1/2*b*x)^2*tan(1/2*a)^2 + a*imag_par
t(cos_integral(b*x + a)) - a*imag_part(cos_integral(-b*x - a)) + 2*a*sin_integral(b*x + a) - 2*tan(1/2*b*x)^2
- 8*tan(1/2*b*x)*tan(1/2*a) - 2*tan(1/2*a)^2 + 2)*b/(b^2*tan(1/2*b*x)^2*tan(1/2*a)^2 + b^2*tan(1/2*b*x)^2 + b^
2*tan(1/2*a)^2 + b^2)

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maple [A]  time = 0.01, size = 24, normalized size = 0.92 \[ \frac {\left (b x +a \right ) \Si \left (b x +a \right )+\cos \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*((b*x+a)*Si(b*x+a)+cos(b*x+a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinint(a + b*x),x)

[Out]

x*sinint(a + b*x) + (cos(a + b*x) + a*sinint(a + b*x))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(Si(a + b*x), x)

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