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

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

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Mathematica [A]  time = 0.01, size = 43, normalized size = 1.00 \[ \frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} b x \, _3F_3(1,1,1;2,2,2;i b x) \]

Antiderivative was successfully verified.

[In]

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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fricas [F]  time = 1.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Si}\left (b x\right )}{x}, x\right ) \]

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

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)

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maple [A]  time = 0.04, size = 20, normalized size = 0.47 \[ b x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}, \frac {3}{2}\right ], -\frac {b^{2} x^{2}}{4}\right ) \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Si}\left (b x\right )}{x}\,{d x} \]

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

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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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

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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