Integrand size = 12, antiderivative size = 102 \[ \int (a+b \arctan (c+d x))^2 \, dx=\frac {i (a+b \arctan (c+d x))^2}{d}+\frac {(c+d x) (a+b \arctan (c+d x))^2}{d}+\frac {2 b (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d} \]
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Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5147, 4930, 5040, 4964, 2449, 2352} \[ \int (a+b \arctan (c+d x))^2 \, dx=\frac {(c+d x) (a+b \arctan (c+d x))^2}{d}+\frac {i (a+b \arctan (c+d x))^2}{d}+\frac {2 b \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))}{d}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{d} \]
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Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 5040
Rule 5147
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) (a+b \arctan (c+d x))^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {i (a+b \arctan (c+d x))^2}{d}+\frac {(c+d x) (a+b \arctan (c+d x))^2}{d}+\frac {(2 b) \text {Subst}\left (\int \frac {a+b \arctan (x)}{i-x} \, dx,x,c+d x\right )}{d} \\ & = \frac {i (a+b \arctan (c+d x))^2}{d}+\frac {(c+d x) (a+b \arctan (c+d x))^2}{d}+\frac {2 b (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {i (a+b \arctan (c+d x))^2}{d}+\frac {(c+d x) (a+b \arctan (c+d x))^2}{d}+\frac {2 b (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d} \\ & = \frac {i (a+b \arctan (c+d x))^2}{d}+\frac {(c+d x) (a+b \arctan (c+d x))^2}{d}+\frac {2 b (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07 \[ \int (a+b \arctan (c+d x))^2 \, dx=\frac {b^2 (-i+c+d x) \arctan (c+d x)^2+2 b \arctan (c+d x) \left (a c+a d x+b \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+a \left (a c+a d x+2 b \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )-i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )}{d} \]
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Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.36
method | result | size |
parts | \(a^{2} x +\frac {b^{2} \left (\arctan \left (d x +c \right )^{2} \left (d x +c +i\right )+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )-2 i \arctan \left (d x +c \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right )\right )}{d}+\frac {2 a b \left (\left (d x +c \right ) \arctan \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}\) | \(139\) |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}-i \arctan \left (d x +c \right )^{2} b^{2}+\arctan \left (d x +c \right )^{2} b^{2} \left (d x +c \right )-i \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 a b \left (d x +c \right ) \arctan \left (d x +c \right )-a b \ln \left (1+\left (d x +c \right )^{2}\right )}{d}\) | \(146\) |
default | \(\frac {\left (d x +c \right ) a^{2}-i \arctan \left (d x +c \right )^{2} b^{2}+\arctan \left (d x +c \right )^{2} b^{2} \left (d x +c \right )-i \operatorname {polylog}\left (2, -\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 \arctan \left (d x +c \right ) \ln \left (1+\frac {\left (1+i \left (d x +c \right )\right )^{2}}{1+\left (d x +c \right )^{2}}\right ) b^{2}+2 a b \left (d x +c \right ) \arctan \left (d x +c \right )-a b \ln \left (1+\left (d x +c \right )^{2}\right )}{d}\) | \(146\) |
risch | \(-\frac {b^{2} \left (d x +c -i\right ) \ln \left (1+i \left (d x +c \right )\right )^{2}}{4 d}-\frac {\ln \left (-i d x -i c +1\right )^{2} b^{2} c}{4 d}-\frac {\ln \left (-i d x -i c +1\right ) a b}{d}+\left (\frac {b^{2} x \ln \left (1-i \left (d x +c \right )\right )}{2}-\frac {i b \left (2 a x d -b \ln \left (1-i \left (d x +c \right )\right )+i \ln \left (1-i \left (d x +c \right )\right ) b c \right )}{2 d}\right ) \ln \left (1+i \left (d x +c \right )\right )-\frac {a b \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}+\frac {a^{2} c}{d}+a^{2} x +\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {1}{2} i d x -\frac {1}{2} i c \right )}{d}-\frac {i \ln \left (-i d x -i c +1\right )^{2} b^{2}}{4 d}+\frac {i \ln \left (-i d x -i c +1\right ) a b c}{d}-\frac {i a b c \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}+\frac {i a^{2}}{d}-\frac {\ln \left (-i d x -i c +1\right )^{2} b^{2} x}{4}+i \ln \left (-i d x -i c +1\right ) a b x +\frac {a b c \arctan \left (d x +c \right )}{d}-\frac {i a b \arctan \left (d x +c \right )}{d}+\frac {i b^{2} \ln \left (\frac {1}{2} i d x +\frac {1}{2} i c +\frac {1}{2}\right ) \ln \left (\frac {1}{2}-\frac {1}{2} i d x -\frac {1}{2} i c \right )}{d}-\frac {i b^{2} \ln \left (\frac {1}{2} i d x +\frac {1}{2} i c +\frac {1}{2}\right ) \ln \left (-i d x -i c +1\right )}{d}\) | \(415\) |
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\[ \int (a+b \arctan (c+d x))^2 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (a+b \arctan (c+d x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{2}\, dx \]
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\[ \int (a+b \arctan (c+d x))^2 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (a+b \arctan (c+d x))^2 \, dx=\int { {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (a+b \arctan (c+d x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \]
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