3.9 \(\int \frac {S(b x)}{x} \, dx\)

Optimal. Leaf size=73 \[ \frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {1}{2} i b^2 \pi x^2\right ) \]

[Out]

1/2*I*b*x*HypergeometricPFQ([1/2, 1/2],[3/2, 3/2],-1/2*I*b^2*Pi*x^2)-1/2*I*b*x*HypergeometricPFQ([1/2, 1/2],[3
/2, 3/2],1/2*I*b^2*Pi*x^2)

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Rubi [A]  time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6424, 6358, 6360} \[ \frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {1}{2} i b^2 \pi x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I/2)*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-I/2)*b^2*Pi*x^2] - (I/2)*b*x*HypergeometricPFQ[{1/2, 1/2
}, {3/2, 3/2}, (I/2)*b^2*Pi*x^2]

Rule 6358

Int[Erf[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt
[Pi], x] /; FreeQ[b, x]

Rule 6360

Int[Erfi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2])/Sqrt[P
i], x] /; FreeQ[b, x]

Rule 6424

Int[FresnelS[(b_.)*(x_)]/(x_), x_Symbol] :> Dist[(1 + I)/4, Int[Erf[(Sqrt[Pi]*(1 + I)*b*x)/2]/x, x], x] + Dist
[(1 - I)/4, Int[Erf[(Sqrt[Pi]*(1 - I)*b*x)/2]/x, x], x] /; FreeQ[b, x]

Rubi steps

\begin {align*} \int \frac {S(b x)}{x} \, dx &=\left (-\frac {1}{4}-\frac {i}{4}\right ) \int \frac {\text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )}{x} \, dx+\left (\frac {1}{4}+\frac {i}{4}\right ) \int \frac {\text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )}{x} \, dx\\ &=\frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {1}{2} i b^2 \pi x^2\right )\\ \end {align*}

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Mathematica [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {S(b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[FresnelS[b*x]/x, x]

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnels(b*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)/x, x)

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maple [A]  time = 0.06, size = 29, normalized size = 0.40 \[ \frac {\pi \,x^{3} b^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {7}{4}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*Pi*x^3*b^3*hypergeom([3/4,3/4],[3/2,7/4,7/4],-1/16*x^4*Pi^2*b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(FresnelS(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(fresnels(b*x)/x,x)

[Out]

Exception raised: AttributeError

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