3.1.74 \(\int \frac {\text {CosIntegral}(b x)}{x} \, dx\) [74]

Optimal. Leaf size=61 \[ -\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\gamma \log (x)+\frac {1}{2} \log ^2(b x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 61, 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 = {6637} \begin {gather*} -\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\frac {1}{2} \log ^2(b x)+\gamma \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6637

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 94, normalized size = 1.54 \begin {gather*} \frac {1}{2} \left (-i b x \, _3F_3(1,1,1;2,2,2;-i b x)+i b x \, _3F_3(1,1,1;2,2,2;i b x)+\log (x) (2 \gamma +2 \text {CosIntegral}(b x)+\Gamma (0,-i b x)+\Gamma (0,i b x)-\log (x)+\log (-i b x)+\log (i b x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-I)*b*x] + I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*
b*x] + Log[x]*(2*EulerGamma + 2*CosIntegral[b*x] + Gamma[0, (-I)*b*x] + Gamma[0, I*b*x] - Log[x] + Log[(-I)*b*
x] + Log[I*b*x]))/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(51)=102\).
time = 0.30, size = 158, normalized size = 2.59

method	result	size
meijerg	\(\frac {\sqrt {\pi }\, \left (-\frac {b^{2} x^{2} \hypergeom \left (\left [1, 1, 1\right ], \left [\frac {3}{2}, 2, 2, 2\right ], -\frac {b^{2} x^{2}}{4}\right )}{2 \sqrt {\pi }}+\frac {-2 \gamma \left (-\gamma -2 \ln \left (2\right )\right )-4 \ln \left (x \right ) \left (-\gamma -2 \ln \left (2\right )\right )+4 \ln \left (2\right ) \left (-\gamma -2 \ln \left (2\right )\right )-4 \ln \left (b \right ) \left (-\gamma -2 \ln \left (2\right )\right )-\frac {\pi ^{2}}{3}+\left (-\gamma -2 \ln \left (2\right )\right )^{2}+4 \ln \left (b \right )^{2}+4 \ln \left (2\right )^{2}+4 \ln \left (x \right )^{2}-8 \ln \left (x \right ) \ln \left (2\right )+8 \ln \left (x \right ) \ln \left (b \right )-8 \ln \left (2\right ) \ln \left (b \right )+\gamma ^{2}+4 \ln \left (b \right ) \gamma +4 \ln \left (x \right ) \gamma -4 \ln \left (2\right ) \gamma }{2 \sqrt {\pi }}\right )}{4}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Pi^(1/2)*(-1/2/Pi^(1/2)*b^2*x^2*hypergeom([1,1,1],[3/2,2,2,2],-1/4*b^2*x^2)+1/2*(-2*gamma*(-gamma-2*ln(2))
-4*ln(x)*(-gamma-2*ln(2))+4*ln(2)*(-gamma-2*ln(2))-4*ln(b)*(-gamma-2*ln(2))-1/3*Pi^2+(-gamma-2*ln(2))^2+4*ln(b
)^2+4*ln(2)^2+4*ln(x)^2-8*ln(x)*ln(2)+8*ln(x)*ln(b)-8*ln(2)*ln(b)+gamma^2+4*ln(b)*gamma+4*ln(x)*gamma-4*ln(2)*
gamma)/Pi^(1/2))

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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(fresnel_cos(b*x)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(fresnel_cos(b*x)/x, x)

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Sympy [A]
time = 0.55, size = 44, normalized size = 0.72 \begin {gather*} - \frac {b^{2} x^{2} {{}_{3}F_{4}\left (\begin {matrix} 1, 1, 1 \\ \frac {3}{2}, 2, 2, 2 \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8} + \frac {\log {\left (b^{2} x^{2} \right )}^{2}}{8} + \frac {\gamma \log {\left (b^{2} x^{2} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**2*x**2*hyper((1, 1, 1), (3/2, 2, 2, 2), -b**2*x**2/4)/8 + log(b**2*x**2)**2/8 + EulerGamma*log(b**2*x**2)/
2

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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(fresnel_cos(b*x)/x,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(cosint(b*x)/x, x)

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