Integrand size = 19, antiderivative size = 48 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=-\frac {1}{2} b e x+\frac {b e \arctan (c+d x)}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5151, 12, 4946, 327, 209} \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {e (c+d x)^2 (a+b \arctan (c+d x))}{2 d}+\frac {b e \arctan (c+d x)}{2 d}-\frac {b e x}{2} \]
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Rule 12
Rule 209
Rule 327
Rule 4946
Rule 5151
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int e x (a+b \arctan (x)) \, dx,x,c+d x)}{d} \\ & = \frac {e \text {Subst}(\int x (a+b \arctan (x)) \, dx,x,c+d x)}{d} \\ & = \frac {e (c+d x)^2 (a+b \arctan (c+d x))}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,c+d x\right )}{2 d} \\ & = -\frac {1}{2} b e x+\frac {e (c+d x)^2 (a+b \arctan (c+d x))}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{2 d} \\ & = -\frac {1}{2} b e x+\frac {b e \arctan (c+d x)}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {e \left (b (-d x+\arctan (c+d x))+(c+d x)^2 (a+b \arctan (c+d x))\right )}{2 d} \]
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Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+b e \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(51\) |
default | \(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+b e \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(51\) |
parts | \(e a \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {b e \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(52\) |
parallelrisch | \(\frac {d^{3} e b \arctan \left (d x +c \right ) x^{2}+x^{2} a \,d^{3} e +2 c b e \arctan \left (d x +c \right ) x \,d^{2}+2 x a c \,d^{2} e +\arctan \left (d x +c \right ) b \,c^{2} d e -x b \,d^{2} e -5 a \,c^{2} d e +e b \arctan \left (d x +c \right ) d +2 b c d e -d e a}{2 d^{2}}\) | \(105\) |
risch | \(-\frac {i e b \left (d \,x^{2}+2 c x \right ) \ln \left (1+i \left (d x +c \right )\right )}{4}+\frac {i e d b \,x^{2} \ln \left (1-i \left (d x +c \right )\right )}{4}+\frac {i e b c x \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {a d e \,x^{2}}{2}+\frac {e \arctan \left (d x +c \right ) b \,c^{2}}{2 d}+a c e x -\frac {b e x}{2}+\frac {b e \arctan \left (d x +c \right )}{2 d}\) | \(113\) |
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.25 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {a d^{2} e x^{2} + {\left (2 \, a c - b\right )} d e x + {\left (b d^{2} e x^{2} + 2 \, b c d e x + {\left (b c^{2} + b\right )} e\right )} \arctan \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (41) = 82\).
Time = 6.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.98 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {atan}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {atan}{\left (c + d x \right )} + \frac {b d e x^{2} \operatorname {atan}{\left (c + d x \right )}}{2} - \frac {b e x}{2} + \frac {b e \operatorname {atan}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {atan}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (42) = 84\).
Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.50 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (d x + c\right ) - d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b c e}{2 \, d} \]
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\[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\int { {\left (d e x + c e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )} \,d x } \]
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Time = 1.60 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.52 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=a\,c\,e\,x-\frac {b\,e\,x}{2}+\frac {b\,e\,\mathrm {atan}\left (c+d\,x\right )}{2\,d}+\frac {a\,d\,e\,x^2}{2}+\frac {b\,c^2\,e\,\mathrm {atan}\left (c+d\,x\right )}{2\,d}+b\,c\,e\,x\,\mathrm {atan}\left (c+d\,x\right )+\frac {b\,d\,e\,x^2\,\mathrm {atan}\left (c+d\,x\right )}{2} \]
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