3.2.15 \(\int \frac {\cos (b x) \text {CosIntegral}(b x)}{x^2} \, dx\) [115]

Optimal. Leaf size=51 \[ -\frac {\cos ^2(b x)}{x}-\frac {\cos (b x) \text {CosIntegral}(b x)}{x}-b \text {Si}(2 b x)-b \text {Int}\left (\frac {\text {CosIntegral}(b x) \sin (b x)}{x},x\right ) \]

[Out]

-b*CannotIntegrate(Ci(b*x)*sin(b*x)/x,x)-Ci(b*x)*cos(b*x)/x-cos(b*x)^2/x-b*Si(2*b*x)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cos (b x) \text {CosIntegral}(b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Cos[b*x]*CosIntegral[b*x])/x^2,x]

[Out]

-(Cos[b*x]^2/x) - (Cos[b*x]*CosIntegral[b*x])/x - b*SinIntegral[2*b*x] - b*Defer[Int][(CosIntegral[b*x]*Sin[b*
x])/x, x]

Rubi steps

\begin {align*} \int \frac {\cos (b x) \text {Ci}(b x)}{x^2} \, dx &=-\frac {\cos (b x) \text {Ci}(b x)}{x}+b \int \frac {\cos ^2(b x)}{b x^2} \, dx-b \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{x}-b \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx+\int \frac {\cos ^2(b x)}{x^2} \, dx\\ &=-\frac {\cos ^2(b x)}{x}-\frac {\cos (b x) \text {Ci}(b x)}{x}-b \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx+(2 b) \int -\frac {\sin (2 b x)}{2 x} \, dx\\ &=-\frac {\cos ^2(b x)}{x}-\frac {\cos (b x) \text {Ci}(b x)}{x}-b \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx-b \int \frac {\sin (2 b x)}{x} \, dx\\ &=-\frac {\cos ^2(b x)}{x}-\frac {\cos (b x) \text {Ci}(b x)}{x}-b \text {Si}(2 b x)-b \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos (b x) \text {CosIntegral}(b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^2,x]

[Out]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^2, x]

________________________________________________________________________________________

Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\cosineIntegral \left (b x \right ) \cos \left (b x \right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x)*cos(b*x)/x^2,x)

[Out]

int(Ci(b*x)*cos(b*x)/x^2,x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(b*x)*cos(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(cos(b*x)*fresnel_cos(b*x)/x^2, x)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(b*x)*cos(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(cos(b*x)*fresnel_cos(b*x)/x^2, x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*cos(b*x)/x**2,x)

[Out]

Integral(cos(b*x)*Ci(b*x)/x**2, x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_cos(b*x)*cos(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(cos(b*x)*fresnel_cos(b*x)/x^2, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosint(b*x)*cos(b*x))/x^2,x)

[Out]

int((cosint(b*x)*cos(b*x))/x^2, x)

________________________________________________________________________________________