3.6 \(\int \frac{\text{Si}(b x)}{x} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},-i b x)+\frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},i b x) \]

[Out]

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

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Rubi [A]  time = 0.0200348, antiderivative size = 43, 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 = {6501} \[ \frac{1}{2} b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac{1}{2} b x \, _3F_3(1,1,1;2,2,2;i b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6501

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

Rubi steps

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

Mathematica [A]  time = 0.0042725, size = 43, normalized size = 1. \[ \frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},-i b x)+\frac{1}{2} b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},i b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.069, size = 20, normalized size = 0.5 \begin{align*} bx{\mbox{$_2$F$_3$}({\frac{1}{2}},{\frac{1}{2}};\,{\frac{3}{2}},{\frac{3}{2}},{\frac{3}{2}};\,-{\frac{{b}^{2}{x}^{2}}{4}})} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*hypergeom([1/2,1/2],[3/2,3/2,3/2],-1/4*b^2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(b*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin_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(Si(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 Si}\left (b x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(b*x)/x, x)