3.116 \(\int \frac {\cos (b x) \text {Ci}(b x)}{x} \, dx\)

Optimal. Leaf size=10 \[ \frac {\text {Ci}(b x)^2}{2} \]

[Out]

1/2*Ci(b*x)^2

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6686} \[ \frac {1}{2} \text {CosIntegral}(b x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

CosIntegral[b*x]^2/2

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\text {Ci}(b x)^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

CosIntegral[b*x]^2/2

________________________________________________________________________________________

fricas [F]  time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x\right ) \operatorname {Ci}\left (b x\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*cos(b*x)/x, x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 9, normalized size = 0.90 \[ \frac {\Ci \left (b x \right )^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*Ci(b*x)^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*cos(b*x)/x, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.10 \[ \frac {{\mathrm {cosint}\left (b\,x\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosint(b*x)^2/2

________________________________________________________________________________________

sympy [A]  time = 0.71, size = 7, normalized size = 0.70 \[ \frac {\operatorname {Ci}^{2}{\left (b x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ci(b*x)**2/2

________________________________________________________________________________________