\(\int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx\) [129]

Optimal result

Integrand size = 13, antiderivative size = 33 \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b} \]

[Out]

-1/2*Si(2*b*x+2*a)/b+Ci(b*x+a)*sin(b*x+a)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6647, 4491, 12, 3380} \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b} \]

[In]

Int[Cos[a + b*x]*CosIntegral[a + b*x],x]

[Out]

(CosIntegral[a + b*x]*Sin[a + b*x])/b - SinIntegral[2*a + 2*b*x]/(2*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 6647

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a + b*x]*(CosIntegral[c + d
*x]/b), x] - Dist[d/b, Int[Sin[a + b*x]*(Cos[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx \\ & = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx \\ & = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{a+b x} \, dx \\ & = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 (a+b x))}{2 b} \]

[In]

Integrate[Cos[a + b*x]*CosIntegral[a + b*x],x]

[Out]

(CosIntegral[a + b*x]*Sin[a + b*x])/b - SinIntegral[2*(a + b*x)]/(2*b)

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x +a \right ) \sin \left (b x +a \right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}}{b}\)	\(30\)
default	\(\frac {\operatorname {Ci}\left (b x +a \right ) \sin \left (b x +a \right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}}{b}\)	\(30\)

[In]

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

[Out]

1/b*(Ci(b*x+a)*sin(b*x+a)-1/2*Si(2*b*x+2*a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (31) = 62\).

Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.82 \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {2 \, b \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right ) - \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right )}{2 \, b^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(cos(a + b*x)*Ci(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(cos(b*x + a)*fresnel_cos(b*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(cos(b*x + a)*fresnel_cos(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(cosint(a + b*x)*cos(a + b*x),x)

[Out]

int(cosint(a + b*x)*cos(a + b*x), x)