Optimal. Leaf size=61 \[ -\frac{1}{2} i b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},-i b x)+\frac{1}{2} i b x \text{HypergeometricPFQ}(\{1,1,1\},\{2,2,2\},i b x)+\frac{1}{2} \log ^2(b x)+\gamma \log (x) \]
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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.
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Rule 6502
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.
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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.
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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.
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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.
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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.
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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.
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