Integrand size = 16, antiderivative size = 244 \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=-\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac {i d \log (1-i a-i b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}+\frac {i d \log (1+i a+i b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}+\frac {i d \operatorname {PolyLog}\left (2,\frac {c (i-a-b x)}{i c-a c+b d}\right )}{2 c^2}-\frac {i d \operatorname {PolyLog}\left (2,\frac {c (i+a+b x)}{(i+a) c-b d}\right )}{2 c^2} \]
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Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5159, 2456, 2436, 2332, 2441, 2440, 2438} \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=\frac {i d \operatorname {PolyLog}\left (2,\frac {c (-a-b x+i)}{-a c+i c+b d}\right )}{2 c^2}-\frac {i d \operatorname {PolyLog}\left (2,\frac {c (a+b x+i)}{(a+i) c-b d}\right )}{2 c^2}+\frac {i d \log (i a+i b x+1) \log \left (\frac {b (c x+d)}{b d+(-a+i) c}\right )}{2 c^2}-\frac {i d \log (-i a-i b x+1) \log \left (-\frac {b (c x+d)}{-b d+(a+i) c}\right )}{2 c^2}-\frac {(i a+i b x+1) \log (i a+i b x+1)}{2 b c}-\frac {(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c} \]
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Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 5159
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} i \int \frac {\log (1-i a-i b x)}{c+\frac {d}{x}} \, dx-\frac {1}{2} i \int \frac {\log (1+i a+i b x)}{c+\frac {d}{x}} \, dx \\ & = \frac {1}{2} i \int \left (\frac {\log (1-i a-i b x)}{c}-\frac {d \log (1-i a-i b x)}{c (d+c x)}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\log (1+i a+i b x)}{c}-\frac {d \log (1+i a+i b x)}{c (d+c x)}\right ) \, dx \\ & = \frac {i \int \log (1-i a-i b x) \, dx}{2 c}-\frac {i \int \log (1+i a+i b x) \, dx}{2 c}-\frac {(i d) \int \frac {\log (1-i a-i b x)}{d+c x} \, dx}{2 c}+\frac {(i d) \int \frac {\log (1+i a+i b x)}{d+c x} \, dx}{2 c} \\ & = -\frac {i d \log (1-i a-i b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}+\frac {i d \log (1+i a+i b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}-\frac {\text {Subst}(\int \log (x) \, dx,x,1-i a-i b x)}{2 b c}-\frac {\text {Subst}(\int \log (x) \, dx,x,1+i a+i b x)}{2 b c}+\frac {(b d) \int \frac {\log \left (-\frac {i b (d+c x)}{-((1-i a) c)-i b d}\right )}{1-i a-i b x} \, dx}{2 c^2}+\frac {(b d) \int \frac {\log \left (\frac {i b (d+c x)}{-((1+i a) c)+i b d}\right )}{1+i a+i b x} \, dx}{2 c^2} \\ & = -\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac {i d \log (1-i a-i b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}+\frac {i d \log (1+i a+i b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}+\frac {(i d) \text {Subst}\left (\int \frac {\log \left (1+\frac {c x}{-((1-i a) c)-i b d}\right )}{x} \, dx,x,1-i a-i b x\right )}{2 c^2}-\frac {(i d) \text {Subst}\left (\int \frac {\log \left (1+\frac {c x}{-((1+i a) c)+i b d}\right )}{x} \, dx,x,1+i a+i b x\right )}{2 c^2} \\ & = -\frac {(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac {(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac {i d \log (1-i a-i b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}+\frac {i d \log (1+i a+i b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}+\frac {i d \operatorname {PolyLog}\left (2,\frac {c (i-a-b x)}{(i-a) c+b d}\right )}{2 c^2}-\frac {i d \operatorname {PolyLog}\left (2,\frac {c (i+a+b x)}{(i+a) c-b d}\right )}{2 c^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(771\) vs. \(2(244)=488\).
Time = 8.24 (sec) , antiderivative size = 771, normalized size of antiderivative = 3.16 \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=\frac {-2 a^2 c^2 \arctan (a+b x)+2 a b c d \arctan (a+b x)+i a b c d \pi \arctan (a+b x)-i b^2 d^2 \pi \arctan (a+b x)-2 a b c^2 x \arctan (a+b x)+2 b^2 c d x \arctan (a+b x)+2 i a b c d \arctan \left (a-\frac {b d}{c}\right ) \arctan (a+b x)-2 i b^2 d^2 \arctan \left (a-\frac {b d}{c}\right ) \arctan (a+b x)-b c d \arctan (a+b x)^2+i a b c d \arctan (a+b x)^2-i b^2 d^2 \arctan (a+b x)^2+b c d \sqrt {1+a^2-\frac {2 a b d}{c}+\frac {b^2 d^2}{c^2}} e^{-i \arctan \left (a-\frac {b d}{c}\right )} \arctan (a+b x)^2+a b c d \pi \log \left (1+e^{-2 i \arctan (a+b x)}\right )-b^2 d^2 \pi \log \left (1+e^{-2 i \arctan (a+b x)}\right )-2 a b c d \arctan (a+b x) \log \left (1+e^{2 i \arctan (a+b x)}\right )+2 b^2 d^2 \arctan (a+b x) \log \left (1+e^{2 i \arctan (a+b x)}\right )-2 a b c d \arctan \left (a-\frac {b d}{c}\right ) \log \left (1-e^{2 i \left (-\arctan \left (a-\frac {b d}{c}\right )+\arctan (a+b x)\right )}\right )+2 b^2 d^2 \arctan \left (a-\frac {b d}{c}\right ) \log \left (1-e^{2 i \left (-\arctan \left (a-\frac {b d}{c}\right )+\arctan (a+b x)\right )}\right )+2 a b c d \arctan (a+b x) \log \left (1-e^{2 i \left (-\arctan \left (a-\frac {b d}{c}\right )+\arctan (a+b x)\right )}\right )-2 b^2 d^2 \arctan (a+b x) \log \left (1-e^{2 i \left (-\arctan \left (a-\frac {b d}{c}\right )+\arctan (a+b x)\right )}\right )-2 a c^2 \log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )+2 b c d \log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )-a b c d \pi \log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )+b^2 d^2 \pi \log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )+2 a b c d \arctan \left (a-\frac {b d}{c}\right ) \log \left (\sin \left (\arctan \left (\frac {-a c+b d}{c}\right )+\arctan (a+b x)\right )\right )-2 b^2 d^2 \arctan \left (a-\frac {b d}{c}\right ) \log \left (\sin \left (\arctan \left (\frac {-a c+b d}{c}\right )+\arctan (a+b x)\right )\right )+i b d (a c-b d) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a+b x)}\right )+i b d (-a c+b d) \operatorname {PolyLog}\left (2,e^{2 i \left (-\arctan \left (a-\frac {b d}{c}\right )+\arctan (a+b x)\right )}\right )}{2 b c^2 (-a c+b d)} \]
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Time = 0.34 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\arctan \left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}+\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )+c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (-\frac {i \ln \left (a c -b d -c \left (b x +a \right )\right ) \left (\ln \left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\ln \left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}-\frac {i \left (\operatorname {dilog}\left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\operatorname {dilog}\left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}\right )}{c}}{b}\) | \(295\) |
default | \(\frac {\frac {\arctan \left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\arctan \left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}+\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )+c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (-\frac {i \ln \left (a c -b d -c \left (b x +a \right )\right ) \left (\ln \left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\ln \left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}-\frac {i \left (\operatorname {dilog}\left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\operatorname {dilog}\left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}\right )}{c}}{b}\) | \(295\) |
parts | \(\frac {\arctan \left (b x +a \right ) x}{c}-\frac {\arctan \left (b x +a \right ) d \ln \left (c x +d \right )}{c^{2}}-\frac {b \left (\frac {\ln \left (a^{2} c^{2}-2 a b c d +2 a b c \left (c x +d \right )+b^{2} d^{2}-2 b^{2} d \left (c x +d \right )+b^{2} \left (c x +d \right )^{2}+c^{2}\right )}{2 b^{2}}-\frac {a \arctan \left (\frac {2 a b c -2 b^{2} d +2 b^{2} \left (c x +d \right )}{2 b c}\right )}{b^{2}}-d \left (-\frac {i \ln \left (c x +d \right ) \left (\ln \left (\frac {i c -a c +b d -b \left (c x +d \right )}{-a c +b d +i c}\right )-\ln \left (\frac {i c +a c -b d +b \left (c x +d \right )}{a c -b d +i c}\right )\right )}{2 b c}-\frac {i \left (\operatorname {dilog}\left (\frac {i c -a c +b d -b \left (c x +d \right )}{-a c +b d +i c}\right )-\operatorname {dilog}\left (\frac {i c +a c -b d +b \left (c x +d \right )}{a c -b d +i c}\right )\right )}{2 b c}\right )\right )}{c}\) | \(313\) |
risch | \(\frac {i \ln \left (-b x i-i a +1\right ) a}{2 b c}+\frac {i d \ln \left (b x i+i a +1\right ) \ln \left (\frac {-i a c +i b d +\left (b x i+i a +1\right ) c -c}{-i a c +i b d -c}\right )}{2 c^{2}}-\frac {i \ln \left (b x i+i a +1\right ) a}{2 b c}+\frac {i \ln \left (-b x i-i a +1\right ) x}{2 c}-\frac {\ln \left (-b x i-i a +1\right )}{2 b c}+\frac {1}{b c}-\frac {i d \ln \left (-b x i-i a +1\right ) \ln \left (\frac {i a c -i b d +\left (-b x i-i a +1\right ) c -c}{i a c -i b d -c}\right )}{2 c^{2}}+\frac {i d \operatorname {dilog}\left (\frac {-i a c +i b d +\left (b x i+i a +1\right ) c -c}{-i a c +i b d -c}\right )}{2 c^{2}}-\frac {i \ln \left (b x i+i a +1\right ) x}{2 c}-\frac {i d \operatorname {dilog}\left (\frac {i a c -i b d +\left (-b x i-i a +1\right ) c -c}{i a c -i b d -c}\right )}{2 c^{2}}-\frac {\ln \left (b x i+i a +1\right )}{2 b c}\) | \(363\) |
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\[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{c + \frac {d}{x}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=\text {Timed out} \]
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Time = 0.36 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.16 \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=-\frac {b d \arctan \left (b x + a\right ) \log \left (-\frac {b^{2} c^{2} x^{2} + 2 \, b^{2} c d x + b^{2} d^{2}}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}\right ) + i \, b d {\rm Li}_2\left (-\frac {i \, b c x + {\left (i \, a - 1\right )} c}{{\left (-i \, a + 1\right )} c + i \, b d}\right ) - i \, b d {\rm Li}_2\left (-\frac {i \, b c x + {\left (i \, a + 1\right )} c}{{\left (-i \, a - 1\right )} c + i \, b d}\right ) - 2 \, {\left (b c x + a c\right )} \arctan \left (b x + a\right ) - {\left (b d \arctan \left (-\frac {b c^{2} x + b c d}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}, \frac {a b c d - b^{2} d^{2} + {\left (a b c^{2} - b^{2} c d\right )} x}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}\right ) - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b c^{2}} \]
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\[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{c + \frac {d}{x}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \]
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