Integrand size = 10, antiderivative size = 38 \[ \int (a+b \arctan (c+d x)) \, dx=a x+\frac {b (c+d x) \arctan (c+d x)}{d}-\frac {b \log \left (1+(c+d x)^2\right )}{2 d} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5147, 4930, 266} \[ \int (a+b \arctan (c+d x)) \, dx=a x+\frac {b (c+d x) \arctan (c+d x)}{d}-\frac {b \log \left ((c+d x)^2+1\right )}{2 d} \]
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Rule 266
Rule 4930
Rule 5147
Rubi steps \begin{align*} \text {integral}& = a x+b \int \arctan (c+d x) \, dx \\ & = a x+\frac {b \text {Subst}(\int \arctan (x) \, dx,x,c+d x)}{d} \\ & = a x+\frac {b (c+d x) \arctan (c+d x)}{d}-\frac {b \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = a x+\frac {b (c+d x) \arctan (c+d x)}{d}-\frac {b \log \left (1+(c+d x)^2\right )}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int (a+b \arctan (c+d x)) \, dx=a x+b x \arctan (c+d x)-\frac {b \left (-2 c \arctan (c+d x)+\log \left (1+c^2+2 c d x+d^2 x^2\right )\right )}{2 d} \]
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Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
default | \(a x +\frac {b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(35\) |
parts | \(a x +\frac {b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(35\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(40\) |
parallelrisch | \(-\frac {b \left (-2 \arctan \left (d x +c \right ) x \,d^{2}-2 c \arctan \left (d x +c \right ) d +\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) d \right )}{2 d^{2}}+a x\) | \(54\) |
risch | \(a x -\frac {i b x \ln \left (1+i \left (d x +c \right )\right )}{2}+\frac {i b x \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {b c \arctan \left (d x +c \right )}{d}-\frac {b \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}\) | \(73\) |
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Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int (a+b \arctan (c+d x)) \, dx=\frac {2 \, a d x + 2 \, {\left (b d x + b c\right )} \arctan \left (d x + c\right ) - b \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{2 \, d} \]
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Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int (a+b \arctan (c+d x)) \, dx=a x + b \left (\begin {cases} \frac {c \operatorname {atan}{\left (c + d x \right )}}{d} + x \operatorname {atan}{\left (c + d x \right )} - \frac {\log {\left (c^{2} + 2 c d x + d^{2} x^{2} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\x \operatorname {atan}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int (a+b \arctan (c+d x)) \, dx=a x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int (a+b \arctan (c+d x)) \, dx=a x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \]
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Time = 1.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int (a+b \arctan (c+d x)) \, dx=a\,x+b\,x\,\mathrm {atan}\left (c+d\,x\right )-\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )}{2\,d}+\frac {b\,c\,\mathrm {atan}\left (c+d\,x\right )}{d} \]
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