\(\int \operatorname {CosIntegral}(a+b x) \, dx\) [89]

Optimal result

Integrand size = 6, antiderivative size = 27 \[ \int \operatorname {CosIntegral}(a+b x) \, dx=\frac {(a+b x) \operatorname {CosIntegral}(a+b x)}{b}-\frac {\sin (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6635} \[ \int \operatorname {CosIntegral}(a+b x) \, dx=\frac {(a+b x) \operatorname {CosIntegral}(a+b x)}{b}-\frac {\sin (a+b x)}{b} \]

[In]

Int[CosIntegral[a + b*x],x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \operatorname {CosIntegral}(a+b x) \, dx=\frac {a \operatorname {CosIntegral}(a+b x)}{b}+x \operatorname {CosIntegral}(a+b x)-\frac {\cos (b x) \sin (a)}{b}-\frac {\cos (a) \sin (b x)}{b} \]

[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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x +a \right ) \left (b x +a \right )-\sin \left (b x +a \right )}{b}\)	\(26\)
default	\(\frac {\operatorname {Ci}\left (b x +a \right ) \left (b x +a \right )-\sin \left (b x +a \right )}{b}\)	\(26\)
parts	\(x \,\operatorname {Ci}\left (b x +a \right )-\frac {-a \,\operatorname {Ci}\left (b x +a \right )+\sin \left (b x +a \right )}{b}\)	\(31\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \operatorname {CosIntegral}(a+b x) \, dx=\frac {{\left (\pi b x + \pi a\right )} \operatorname {C}\left (b x + a\right ) - \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi b} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \operatorname {CosIntegral}(a+b x) \, dx=\int \operatorname {Ci}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(Ci(a + b*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \operatorname {CosIntegral}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {C}\left (b x + a\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi }}{b} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \operatorname {CosIntegral}(a+b x) \, dx=\int { \operatorname {C}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integrate(fresnel_cos(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \operatorname {CosIntegral}(a+b x) \, dx=x\,\mathrm {cosint}\left (a+b\,x\right )-\frac {\sin \left (a+b\,x\right )-a\,\mathrm {cosint}\left (a+b\,x\right )}{b} \]

[In]

int(cosint(a + b*x),x)

[Out]

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