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

Optimal. Leaf size=10 \[ \frac{1}{2} \text{CosIntegral}(b x)^2 \]

[Out]

CosIntegral[b*x]^2/2

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Rubi [A]  time = 0.0197507, antiderivative size = 10, normalized size of antiderivative = 1., 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 = 0.0031494, size = 10, normalized size = 1. \[ \frac{1}{2} \text{CosIntegral}(b x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

CosIntegral[b*x]^2/2

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Maple [A]  time = 0.046, size = 9, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Ci} \left ( bx \right ) \right ) ^{2}}{2}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (b x\right ) \operatorname{Ci}\left (b x\right )}{x}, x\right ) \end{align*}

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)

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Sympy [A]  time = 1.06805, size = 7, normalized size = 0.7 \begin{align*} \frac{\operatorname{Ci}^{2}{\left (b x \right )}}{2} \end{align*}

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

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Giac [A]  time = 1.24342, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{2} \, \operatorname{Ci}\left (b x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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