Integrand size = 8, antiderivative size = 61 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=-\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\gamma \log (x)+\frac {1}{2} \log ^2(b x) \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6637} \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=-\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;-i b x)+\frac {1}{2} i b x \, _3F_3(1,1,1;2,2,2;i b x)+\frac {1}{2} \log ^2(b x)+\gamma \log (x) \]
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Rule 6637
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\frac {1}{2} \left (-i b x \, _3F_3(1,1,1;2,2,2;-i b x)+i b x \, _3F_3(1,1,1;2,2,2;i b x)+\log (x) (2 \gamma +2 \operatorname {CosIntegral}(b x)+\Gamma (0,-i b x)+\Gamma (0,i b x)-\log (x)+\log (-i b x)+\log (i b x))\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(51)=102\).
Time = 0.45 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.59
method | result | size |
meijerg | \(\frac {\sqrt {\pi }\, \left (-\frac {b^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, 1\right ], \left [\frac {3}{2}, 2, 2, 2\right ], -\frac {b^{2} x^{2}}{4}\right )}{2 \sqrt {\pi }}+\frac {-2 \gamma \left (-\gamma -2 \ln \left (2\right )\right )-4 \ln \left (x \right ) \left (-\gamma -2 \ln \left (2\right )\right )+4 \ln \left (2\right ) \left (-\gamma -2 \ln \left (2\right )\right )-4 \ln \left (b \right ) \left (-\gamma -2 \ln \left (2\right )\right )+8 \ln \left (x \right ) \ln \left (b \right )-8 \ln \left (x \right ) \ln \left (2\right )-\frac {\pi ^{2}}{3}+\left (-\gamma -2 \ln \left (2\right )\right )^{2}+\gamma ^{2}+4 \ln \left (x \right )^{2}+4 \ln \left (b \right )^{2}+4 \ln \left (2\right )^{2}-4 \ln \left (2\right ) \gamma +4 \ln \left (b \right ) \gamma -8 \ln \left (2\right ) \ln \left (b \right )+4 \ln \left (x \right ) \gamma }{2 \sqrt {\pi }}\right )}{4}\) | \(158\) |
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\[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x} \,d x } \]
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Time = 0.58 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=- \frac {b^{2} x^{2} {{}_{3}F_{4}\left (\begin {matrix} 1, 1, 1 \\ \frac {3}{2}, 2, 2, 2 \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8} + \frac {\log {\left (b^{2} x^{2} \right )}^{2}}{8} + \frac {\gamma \log {\left (b^{2} x^{2} \right )}}{2} \]
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\[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x} \,d x } \]
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\[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {CosIntegral}(b x)}{x} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )}{x} \,d x \]
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