3.115 \(\int \frac {\cos (b x) \text {Ci}(b x)}{x^2} \, dx\)

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

[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)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {\cos (b x) \text {CosIntegral}(b x)}{x^2} \, dx \]

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*}

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Mathematica [A]  time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {\cos (b x) \text {Ci}(b x)}{x^2} \, dx \]

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]

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fricas [A]  time = 2.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x\right ) \operatorname {Ci}\left (b x\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*cos(b*x)/x^2, x)

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maple [A]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\Ci \left (b x \right ) \cos \left (b x \right )}{x^{2}}\, dx \]

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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*cos(b*x)/x^2, x)

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

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)

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

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)

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