Integrand size = 18, antiderivative size = 40 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=-\frac {c \arctan (a x)}{x}+a^2 c x \arctan (a x)+a c \log (x)-a c \log \left (1+a^2 x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5070, 4946, 272, 36, 29, 31, 4930, 266} \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=a^2 c x \arctan (a x)-a c \log \left (a^2 x^2+1\right )-\frac {c \arctan (a x)}{x}+a c \log (x) \]
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 272
Rule 4930
Rule 4946
Rule 5070
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\arctan (a x)}{x^2} \, dx+\left (a^2 c\right ) \int \arctan (a x) \, dx \\ & = -\frac {c \arctan (a x)}{x}+a^2 c x \arctan (a x)+(a c) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (a^3 c\right ) \int \frac {x}{1+a^2 x^2} \, dx \\ & = -\frac {c \arctan (a x)}{x}+a^2 c x \arctan (a x)-\frac {1}{2} a c \log \left (1+a^2 x^2\right )+\frac {1}{2} (a c) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {c \arctan (a x)}{x}+a^2 c x \arctan (a x)-\frac {1}{2} a c \log \left (1+a^2 x^2\right )+\frac {1}{2} (a c) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^3 c\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {c \arctan (a x)}{x}+a^2 c x \arctan (a x)+a c \log (x)-a c \log \left (1+a^2 x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=-\frac {c \arctan (a x)}{x}+a^2 c x \arctan (a x)+a c \log (x)-a c \log \left (1+a^2 x^2\right ) \]
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Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02
method | result | size |
parts | \(a^{2} c x \arctan \left (a x \right )-\frac {c \arctan \left (a x \right )}{x}-c a \left (\ln \left (a^{2} x^{2}+1\right )-\ln \left (x \right )\right )\) | \(41\) |
derivativedivides | \(a \left (a c x \arctan \left (a x \right )-\frac {c \arctan \left (a x \right )}{a x}-c \left (-\ln \left (a x \right )+\ln \left (a^{2} x^{2}+1\right )\right )\right )\) | \(45\) |
default | \(a \left (a c x \arctan \left (a x \right )-\frac {c \arctan \left (a x \right )}{a x}-c \left (-\ln \left (a x \right )+\ln \left (a^{2} x^{2}+1\right )\right )\right )\) | \(45\) |
parallelrisch | \(\frac {a^{2} c \,x^{2} \arctan \left (a x \right )+c a \ln \left (x \right ) x -c a \ln \left (a^{2} x^{2}+1\right ) x -c \arctan \left (a x \right )}{x}\) | \(46\) |
risch | \(-\frac {i c \left (a^{2} x^{2}-1\right ) \ln \left (i a x +1\right )}{2 x}+\frac {i c \left (a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a \ln \left (x \right ) x +2 i a \ln \left (-2 a^{2} x^{2}-2\right ) x -\ln \left (-i a x +1\right )\right )}{2 x}\) | \(82\) |
meijerg | \(\frac {a c \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}+\frac {a c \left (4 \ln \left (x \right )+4 \ln \left (a \right )-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}\) | \(92\) |
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Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=-\frac {a c x \log \left (a^{2} x^{2} + 1\right ) - a c x \log \left (x\right ) - {\left (a^{2} c x^{2} - c\right )} \arctan \left (a x\right )}{x} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=\begin {cases} a^{2} c x \operatorname {atan}{\left (a x \right )} + a c \log {\left (x \right )} - a c \log {\left (x^{2} + \frac {1}{a^{2}} \right )} - \frac {c \operatorname {atan}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=-{\left (c \log \left (a^{2} x^{2} + 1\right ) - c \log \left (x\right )\right )} a + {\left (a^{2} c x - \frac {c}{x}\right )} \arctan \left (a x\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )}{x^{2}} \,d x } \]
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Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)}{x^2} \, dx=a^2\,c\,x\,\mathrm {atan}\left (a\,x\right )-\frac {c\,\mathrm {atan}\left (a\,x\right )}{x}-c\,\left (a\,\ln \left (a^2\,x^2+1\right )-a\,\ln \left (x\right )\right ) \]
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