3.74 \(\int \frac{\text{Chi}(b x)}{x} \, dx\)

Optimal. Leaf size=52 \[ -\frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},-b x)+\frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},b x)+\frac{1}{2} \log ^2(b x)+\gamma \log (x) \]

[Out]

-(b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2 + (b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x])/
2 + EulerGamma*Log[x] + Log[b*x]^2/2

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Rubi [A]  time = 0.0221243, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6531} \[ -\frac{1}{2} b x \, _3F_3(1,1,1;2,2,2;-b x)+\frac{1}{2} b x \, _3F_3(1,1,1;2,2,2;b x)+\frac{1}{2} \log ^2(b x)+\gamma \log (x) \]

Antiderivative was successfully verified.

[In]

Int[CoshIntegral[b*x]/x,x]

[Out]

-(b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2 + (b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x])/
2 + EulerGamma*Log[x] + Log[b*x]^2/2

Rule 6531

Int[CoshIntegral[(b_.)*(x_)]/(x_), x_Symbol] :> -Simp[(b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2,
x] + (Simp[(1*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x])/2, x] + Simp[EulerGamma*Log[x], x] + Simp[(1*
Log[b*x]^2)/2, x]) /; FreeQ[b, x]

Rubi steps

\begin{align*} \int \frac{\text{Chi}(b x)}{x} \, dx &=-\frac{1}{2} b x \, _3F_3(1,1,1;2,2,2;-b x)+\frac{1}{2} b x \, _3F_3(1,1,1;2,2,2;b x)+\gamma \log (x)+\frac{1}{2} \log ^2(b x)\\ \end{align*}

Mathematica [A]  time = 0.0045848, size = 52, normalized size = 1. \[ -\frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},-b x)+\frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},b x)+\frac{1}{2} \log ^2(b x)+\gamma \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[CoshIntegral[b*x]/x,x]

[Out]

-(b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2 + (b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x])/
2 + EulerGamma*Log[x] + Log[b*x]^2/2

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Chi} \left ( bx \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Chi(b*x)/x,x)

[Out]

int(Chi(b*x)/x,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(b*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh_integral(b*x)/x, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)/x,x)

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(b*x)/x, x)