3.74 \(\int \frac{\text{CosIntegral}(b x)}{x} \, dx\)

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

[Out]

(-I/2)*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

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Rubi [A]  time = 0.02331, antiderivative size = 61, normalized size of antiderivative = 1., 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 = {6502} \[ -\frac{1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac{1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\frac{1}{2} \log ^2(b x)+\gamma \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-I/2)*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 6502

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

Mathematica [A]  time = 0.039013, size = 94, normalized size = 1.54 \[ \frac{1}{2} (-i b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},-i b x)+i b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},i b x)+\log (x) (\text{Gamma}(0,-i b x)+\text{Gamma}(0,i b x)+2 \text{CosIntegral}(b x)+\log (-i b x)+\log (i b x)-\log (x)+2 \gamma )) \]

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]  time = 0.091, size = 158, normalized size = 2.6 \begin{align*}{\frac{\sqrt{\pi }}{4} \left ({\frac{1}{2\,\sqrt{\pi }} \left ( -{\frac{{\pi }^{2}}{3}}+4\,\ln \left ( x \right ) \gamma -4\,\ln \left ( 2 \right ) \gamma +4\,\ln \left ( b \right ) \gamma + \left ( -\gamma -2\,\ln \left ( 2 \right ) \right ) ^{2}+4\, \left ( \ln \left ( 2 \right ) \right ) ^{2}+4\, \left ( \ln \left ( b \right ) \right ) ^{2}+4\, \left ( \ln \left ( x \right ) \right ) ^{2}-2\,\gamma \, \left ( -\gamma -2\,\ln \left ( 2 \right ) \right ) -4\,\ln \left ( b \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 ( x \right ) \left ( -\gamma -2\,\ln \left ( 2 \right ) \right ) +{\gamma }^{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 ) \right ) }-{\frac{{b}^{2}{x}^{2}}{2\,\sqrt{\pi }}{\mbox{$_3$F$_4$}(1,1,1;\,{\frac{3}{2}},2,2,2;\,-{\frac{{b}^{2}{x}^{2}}{4}})}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Ci}\left (b x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos_integral(b*x)/x, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)/x, x)