Integrand size = 18, antiderivative size = 162 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=-\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f} \]
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Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5156, 4967, 2449, 2352, 2497} \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{f}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f} \]
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Rule 2352
Rule 2449
Rule 2497
Rule 4967
Rule 5156
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {b \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{f}+\frac {b \text {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )}{\left (\frac {i f}{d}+\frac {d e-c f}{d}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {(i b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i (c+d x)}\right )}{f} \\ & = -\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(336\) vs. \(2(162)=324\).
Time = 0.33 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.07 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\frac {a \log (e+f x)+b \left (\left (\cot ^{-1}(c+d x)+\arctan (c+d x)\right ) \log (e+f x)+\arctan (c+d x) \left (\log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-\log \left (\sin \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )\right )\right )+\frac {1}{2} \left (\frac {1}{4} i (\pi -2 \arctan (c+d x))^2+i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )^2-(\pi -2 \arctan (c+d x)) \log \left (1+e^{-2 i \arctan (c+d x)}\right )-2 \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )}\right )+(\pi -2 \arctan (c+d x)) \log \left (\frac {2}{\sqrt {1+(c+d x)^2}}\right )+2 \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right ) \log \left (2 \sin \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )\right )+i \operatorname {PolyLog}\left (2,-e^{-2 i \arctan (c+d x)}\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )}\right )\right )\right )}{f} \]
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Time = 0.75 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.22
method | result | size |
parts | \(\frac {a \ln \left (f x +e \right )}{f}+\frac {b \left (\frac {d \ln \left (f \left (d x +c \right )-c f +d e \right ) \operatorname {arccot}\left (d x +c \right )}{f}+d \left (-\frac {i \ln \left (f \left (d x +c \right )-c f +d e \right ) \left (\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}\right )\right )}{d}\) | \(197\) |
derivativedivides | \(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}-b d \left (-\frac {\ln \left (c f -d e -f \left (d x +c \right )\right ) \operatorname {arccot}\left (d x +c \right )}{f}-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{d}\) | \(211\) |
default | \(\frac {\frac {a d \ln \left (c f -d e -f \left (d x +c \right )\right )}{f}-b d \left (-\frac {\ln \left (c f -d e -f \left (d x +c \right )\right ) \operatorname {arccot}\left (d x +c \right )}{f}-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{d}\) | \(211\) |
risch | \(-\frac {i b \operatorname {dilog}\left (\frac {i c f -i d e +\left (-i d x -i c +1\right ) f -f}{i c f -i d e -f}\right )}{2 f}-\frac {i b \ln \left (-i d x -i c +1\right ) \ln \left (\frac {i c f -i d e +\left (-i d x -i c +1\right ) f -f}{i c f -i d e -f}\right )}{2 f}+\frac {\ln \left (i c f -i d e +\left (-i d x -i c +1\right ) f -f \right ) b \pi }{2 f}+\frac {a \ln \left (i c f -i d e +\left (-i d x -i c +1\right ) f -f \right )}{f}+\frac {i b \operatorname {dilog}\left (\frac {-i c f +i d e +\left (i d x +i c +1\right ) f -f}{-i c f +i d e -f}\right )}{2 f}+\frac {i b \ln \left (i d x +i c +1\right ) \ln \left (\frac {-i c f +i d e +\left (i d x +i c +1\right ) f -f}{-i c f +i d e -f}\right )}{2 f}\) | \(302\) |
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\[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\int { \frac {b \operatorname {arccot}\left (d x + c\right ) + a}{f x + e} \,d x } \]
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Timed out. \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\int { \frac {b \operatorname {arccot}\left (d x + c\right ) + a}{f x + e} \,d x } \]
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\[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\int { \frac {b \operatorname {arccot}\left (d x + c\right ) + a}{f x + e} \,d x } \]
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Timed out. \[ \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx=\int \frac {a+b\,\mathrm {acot}\left (c+d\,x\right )}{e+f\,x} \,d x \]
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