3.1.6 \(\int \frac {\text {Si}(b x)}{x} \, dx\) [6]

Optimal. Leaf size=43 \[ \frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;i b x) \]

[Out]

1/2*b*x*hypergeom([1, 1, 1],[2, 2, 2],-I*b*x)+1/2*b*x*hypergeom([1, 1, 1],[2, 2, 2],I*b*x)

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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 = {6636} \begin {gather*} \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 {gather*}

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 6636

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

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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.28, size = 20, normalized size = 0.47

method	result	size
meijerg	\(b x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}, \frac {3}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Si(b*x)/x,x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin_integral(b*x)/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin_integral(b*x)/x, x)

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Sympy [A]
time = 0.30, size = 22, normalized size = 0.51 \begin {gather*} b x {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*hyper((1/2, 1/2), (3/2, 3/2, 3/2), -b**2*x**2/4)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin_integral(b*x)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {sinint}\left (b\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinint(b*x)/x, x)

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