Integrand size = 12, antiderivative size = 76 \[ \int x (a+b \arctan (c x))^2 \, dx=-\frac {a b x}{c}-\frac {b^2 x \arctan (c x)}{c}+\frac {(a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} x^2 (a+b \arctan (c x))^2+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^2} \]
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Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4946, 5036, 4930, 266, 5004} \[ \int x (a+b \arctan (c x))^2 \, dx=\frac {(a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} x^2 (a+b \arctan (c x))^2-\frac {a b x}{c}-\frac {b^2 x \arctan (c x)}{c}+\frac {b^2 \log \left (c^2 x^2+1\right )}{2 c^2} \]
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Rule 266
Rule 4930
Rule 4946
Rule 5004
Rule 5036
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \arctan (c x))^2-(b c) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{2} x^2 (a+b \arctan (c x))^2-\frac {b \int (a+b \arctan (c x)) \, dx}{c}+\frac {b \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c} \\ & = -\frac {a b x}{c}+\frac {(a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} x^2 (a+b \arctan (c x))^2-\frac {b^2 \int \arctan (c x) \, dx}{c} \\ & = -\frac {a b x}{c}-\frac {b^2 x \arctan (c x)}{c}+\frac {(a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} x^2 (a+b \arctan (c x))^2+b^2 \int \frac {x}{1+c^2 x^2} \, dx \\ & = -\frac {a b x}{c}-\frac {b^2 x \arctan (c x)}{c}+\frac {(a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} x^2 (a+b \arctan (c x))^2+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^2} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int x (a+b \arctan (c x))^2 \, dx=\frac {a c x (-2 b+a c x)+2 b \left (a-b c x+a c^2 x^2\right ) \arctan (c x)+b^2 \left (1+c^2 x^2\right ) \arctan (c x)^2+b^2 \log \left (1+c^2 x^2\right )}{2 c^2} \]
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Time = 0.97 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{c^{2}}+x^{2} a b \arctan \left (c x \right )-\frac {a b x}{c}+\frac {a b \arctan \left (c x \right )}{c^{2}}\) | \(88\) |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )+2 a b \left (\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {c x}{2}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\) | \(91\) |
default | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )+2 a b \left (\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {c x}{2}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\) | \(91\) |
parallelrisch | \(\frac {b^{2} \arctan \left (c x \right )^{2} x^{2} c^{2}+2 a b \arctan \left (c x \right ) x^{2} c^{2}+c^{2} x^{2} a^{2}-2 b^{2} \arctan \left (c x \right ) x c -2 a b c x +b^{2} \arctan \left (c x \right )^{2}+b^{2} \ln \left (c^{2} x^{2}+1\right )+2 a b \arctan \left (c x \right )-a^{2}}{2 c^{2}}\) | \(101\) |
risch | \(-\frac {b^{2} \left (c^{2} x^{2}+1\right ) \ln \left (i c x +1\right )^{2}}{8 c^{2}}-\frac {i b \left (2 c^{2} x^{2} a +i b \,c^{2} x^{2} \ln \left (-i c x +1\right )-2 x b c +i b \ln \left (-i c x +1\right )\right ) \ln \left (i c x +1\right )}{4 c^{2}}+\frac {i a b \,x^{2} \ln \left (-i c x +1\right )}{2}-\frac {b^{2} x^{2} \ln \left (-i c x +1\right )^{2}}{8}-\frac {i b^{2} x \ln \left (-i c x +1\right )}{2 c}+\frac {a^{2} x^{2}}{2}-\frac {b^{2} \ln \left (-i c x +1\right )^{2}}{8 c^{2}}-\frac {a b x}{c}+\frac {a b \arctan \left (c x \right )}{c^{2}}+\frac {b^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}\) | \(203\) |
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Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09 \[ \int x (a+b \arctan (c x))^2 \, dx=\frac {a^{2} c^{2} x^{2} - 2 \, a b c x + {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x\right )^{2} + b^{2} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (a b c^{2} x^{2} - b^{2} c x + a b\right )} \arctan \left (c x\right )}{2 \, c^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.41 \[ \int x (a+b \arctan (c x))^2 \, dx=\begin {cases} \frac {a^{2} x^{2}}{2} + a b x^{2} \operatorname {atan}{\left (c x \right )} - \frac {a b x}{c} + \frac {a b \operatorname {atan}{\left (c x \right )}}{c^{2}} + \frac {b^{2} x^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2} - \frac {b^{2} x \operatorname {atan}{\left (c x \right )}}{c} + \frac {b^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{2}} + \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{2}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.37 \[ \int x (a+b \arctan (c x))^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \arctan \left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b - \frac {1}{2} \, {\left (2 \, c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac {\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} \]
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\[ \int x (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
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Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16 \[ \int x (a+b \arctan (c x))^2 \, dx=\frac {\frac {b^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+\frac {b^2\,\ln \left (c^2\,x^2+1\right )}{2}-c\,\left (x\,\mathrm {atan}\left (c\,x\right )\,b^2+a\,x\,b\right )+a\,b\,\mathrm {atan}\left (c\,x\right )}{c^2}+\frac {a^2\,x^2}{2}+\frac {b^2\,x^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+a\,b\,x^2\,\mathrm {atan}\left (c\,x\right ) \]
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