Integrand size = 14, antiderivative size = 39 \[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-\frac {i c}{x}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {i c}{x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4944, 4940, 2438} \[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-\frac {i c}{x}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {i c}{x}\right ) \]
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Rule 2438
Rule 4940
Rule 4944
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \arctan (c x)}{x} \, dx,x,\frac {1}{x}\right ) \\ & = a \log (x)-\frac {1}{2} (i b) \text {Subst}\left (\int \frac {\log (1-i c x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} (i b) \text {Subst}\left (\int \frac {\log (1+i c x)}{x} \, dx,x,\frac {1}{x}\right ) \\ & = a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-\frac {i c}{x}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {i c}{x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-\frac {i c}{x}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {i c}{x}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31 ) = 62\).
Time = 1.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.23
method | result | size |
parts | \(a \ln \left (x \right )+b \left (-\ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}+\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}-\frac {i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}+\frac {i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}\right )\) | \(87\) |
derivativedivides | \(-a \ln \left (\frac {c}{x}\right )-b \left (\ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )+\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}\right )\) | \(92\) |
default | \(-a \ln \left (\frac {c}{x}\right )-b \left (\ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )+\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}\right )\) | \(92\) |
risch | \(\text {Expression too large to display}\) | \(669\) |
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\[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=\int { \frac {b \arctan \left (\frac {c}{x}\right ) + a}{x} \,d x } \]
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\[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=\int \frac {a + b \operatorname {atan}{\left (\frac {c}{x} \right )}}{x}\, dx \]
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\[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=\int { \frac {b \arctan \left (\frac {c}{x}\right ) + a}{x} \,d x } \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.85 \[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=-\frac {{\left (2 \, b c^{4} \arctan \left (\frac {c}{x}\right ) - \frac {i \, b c^{6} \log \left (\frac {i \, c}{x} + 1\right )}{x^{2}} + \frac {i \, b c^{6} \log \left (-\frac {i \, c}{x} + 1\right )}{x^{2}} + 2 \, a c^{4} + \frac {2 \, b c^{5}}{x}\right )} x^{2}}{4 \, c^{5}} \]
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Time = 0.37 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{x} \, dx=a\,\ln \left (x\right )+\frac {b\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-\frac {c\,1{}\mathrm {i}}{x}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+\frac {c\,1{}\mathrm {i}}{x}\right )\right )\,1{}\mathrm {i}}{2} \]
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