3.119 \(\int \frac {C(b x)}{x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{2} b \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{x} \]

[Out]

1/2*b*Ci(1/2*b^2*Pi*x^2)-FresnelC(b*x)/x

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6427, 3376} \[ \frac {1}{2} b \text {CosIntegral}\left (\frac {1}{2} \pi b^2 x^2\right )-\frac {\text {FresnelC}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[FresnelC[b*x]/x^2,x]

[Out]

(b*CosIntegral[(b^2*Pi*x^2)/2])/2 - FresnelC[b*x]/x

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 6427

Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*FresnelC[b*x])/(d*(m + 1)), x] -
Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Cos[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {C(b x)}{x^2} \, dx &=-\frac {C(b x)}{x}+b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=\frac {1}{2} b \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{x}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.00 \[ \frac {1}{2} b \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[b*x]/x^2,x]

[Out]

(b*CosIntegral[(b^2*Pi*x^2)/2])/2 - FresnelC[b*x]/x

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)/x^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)/x^2, x)

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maple [A]  time = 0.01, size = 28, normalized size = 1.04 \[ b \left (-\frac {\FresnelC \left (b x \right )}{b x}+\frac {\Ci \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)/x^2,x)

[Out]

b*(-FresnelC(b*x)/b/x+1/2*Ci(1/2*b^2*Pi*x^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)/x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)/x^2,x)

[Out]

int(FresnelC(b*x)/x^2, x)

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sympy [B]  time = 1.06, size = 53, normalized size = 1.96 \[ - \frac {\pi ^{2} b^{5} x^{4} \Gamma \left (\frac {5}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2, \frac {9}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{128 \Gamma \left (\frac {9}{4}\right )} + \frac {b \log {\left (b^{4} x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x**2,x)

[Out]

-pi**2*b**5*x**4*gamma(5/4)*hyper((1, 1, 5/4), (3/2, 2, 2, 9/4), -pi**2*b**4*x**4/16)/(128*gamma(9/4)) + b*log
(b**4*x**4)/4

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