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

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 94, normalized size = 1.54 \[ \frac {1}{2} \left (-i b x \, _3F_3(1,1,1;2,2,2;-i b x)+i b x \, _3F_3(1,1,1;2,2,2;i b x)+\log (x) (2 \text {Ci}(b x)+\log (-i b x)+\log (i b x)+\Gamma (0,-i b x)+\Gamma (0,i b x)-\log (x)+2 \gamma )\right ) \]

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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fricas [F]  time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x\right )}{x}, x\right ) \]

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

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)

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maple [B]  time = 0.06, size = 158, normalized size = 2.59 \[ \frac {\sqrt {\pi }\, \left (-\frac {b^{2} x^{2} \hypergeom \left (\left [1, 1, 1\right ], \left [\frac {3}{2}, 2, 2, 2\right ], -\frac {b^{2} x^{2}}{4}\right )}{2 \sqrt {\pi }}+\frac {4 \ln \relax (x )^{2}+4 \ln \relax (2)^{2}+4 \ln \relax (b )^{2}-\frac {\pi ^{2}}{3}-4 \ln \relax (x ) \left (-\gamma -2 \ln \relax (2)\right )+4 \ln \relax (2) \left (-\gamma -2 \ln \relax (2)\right )-4 \ln \relax (b ) \left (-\gamma -2 \ln \relax (2)\right )-2 \gamma \left (-\gamma -2 \ln \relax (2)\right )-4 \ln \relax (2) \gamma +4 \ln \relax (x ) \gamma +\gamma ^{2}+4 \ln \relax (b ) \gamma -8 \ln \relax (x ) \ln \relax (2)+8 \ln \relax (x ) \ln \relax (b )-8 \ln \relax (2) \ln \relax (b )+\left (-\gamma -2 \ln \relax (2)\right )^{2}}{2 \sqrt {\pi }}\right )}{4} \]

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

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

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

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

[Out]

Exception raised: AttributeError

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