Integrand size = 16, antiderivative size = 138 \[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=-\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e} \]
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Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e} \]
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Rule 2352
Rule 2449
Rule 2497
Rule 4966
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {(b c) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{e}-\frac {(b c) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{e} \\ & = -\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}+\frac {(i b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{e} \\ & = -\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\frac {2 a \log (d+e x)+i b \log (1-i c x) \log \left (\frac {c (d+e x)}{c d-i e}\right )-i b \log (1+i c x) \log \left (\frac {c (d+e x)}{c d+i e}\right )+i b \operatorname {PolyLog}\left (2,\frac {e (1-i c x)}{i c d+e}\right )-i b \operatorname {PolyLog}\left (2,-\frac {e (-i+c x)}{c d+i e}\right )}{2 e} \]
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Time = 3.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.13
method | result | size |
parts | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \left (\frac {c \ln \left (c e x +c d \right ) \arctan \left (c x \right )}{e}-c \left (-\frac {i \ln \left (c e x +c d \right ) \left (\ln \left (\frac {-c e x +i e}{c d +i e}\right )-\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )\right )}{c}\) | \(156\) |
derivativedivides | \(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (\frac {\ln \left (c e x +c d \right ) \arctan \left (c x \right )}{e}-\frac {i \ln \left (c e x +c d \right ) \left (-\ln \left (\frac {-c e x +i e}{c d +i e}\right )+\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}+\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{c}\) | \(157\) |
default | \(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (\frac {\ln \left (c e x +c d \right ) \arctan \left (c x \right )}{e}-\frac {i \ln \left (c e x +c d \right ) \left (-\ln \left (\frac {-c e x +i e}{c d +i e}\right )+\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}+\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{c}\) | \(157\) |
risch | \(\frac {i b \operatorname {dilog}\left (\frac {-i d c +e \left (-i c x +1\right )-e}{-i d c -e}\right )}{2 e}+\frac {i b \ln \left (-i c x +1\right ) \ln \left (\frac {-i d c +e \left (-i c x +1\right )-e}{-i d c -e}\right )}{2 e}+\frac {a \ln \left (i d c -e \left (-i c x +1\right )+e \right )}{e}-\frac {i b \operatorname {dilog}\left (\frac {i d c +e \left (i c x +1\right )-e}{i d c -e}\right )}{2 e}-\frac {i b \ln \left (i c x +1\right ) \ln \left (\frac {i d c +e \left (i c x +1\right )-e}{i d c -e}\right )}{2 e}\) | \(193\) |
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\[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x + d} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{d + e x}\, dx \]
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\[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x + d} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{d+e\,x} \,d x \]
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