Integrand size = 14, antiderivative size = 220 \[ \int e^{n \arctan (a+b x)} x^2 \, dx=-\frac {(4 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}+\frac {2^{-\frac {i n}{2}} \left (2-6 a^2-6 a n-n^2\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{3 b^3 (2 i-n)} \]
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Time = 0.10 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5203, 92, 81, 71} \[ \int e^{n \arctan (a+b x)} x^2 \, dx=\frac {2^{-\frac {i n}{2}} \left (-6 a^2-6 a n-n^2+2\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}+1,\frac {i n}{2},\frac {i n}{2}+2,\frac {1}{2} (-i a-i b x+1)\right )}{3 b^3 (-n+2 i)}-\frac {(4 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{3 b^2} \]
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Rule 71
Rule 81
Rule 92
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int x^2 (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx \\ & = \frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}+\frac {\int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (-1-a^2-b (4 a+n) x\right ) \, dx}{3 b^2} \\ & = -\frac {(4 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}-\frac {\left (2-6 a^2-6 a n-n^2\right ) \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx}{6 b^2} \\ & = -\frac {(4 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}+\frac {2^{-\frac {i n}{2}} \left (2-6 a^2-6 a n-n^2\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{3 b^3 (2 i-n)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73 \[ \int e^{n \arctan (a+b x)} x^2 \, dx=\frac {(-i (i+a+b x))^{1+\frac {i n}{2}} \left (-\left ((4 a+n) (1+i a+i b x)^{1-\frac {i n}{2}}\right )+2 b x (1+i a+i b x)^{1-\frac {i n}{2}}+\frac {2^{1-\frac {i n}{2}} \left (-2+6 a^2+6 a n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )}{-2 i+n}\right )}{6 b^3} \]
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\[\int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{2}d x\]
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\[ \int e^{n \arctan (a+b x)} x^2 \, dx=\int { x^{2} e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
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\[ \int e^{n \arctan (a+b x)} x^2 \, dx=\int x^{2} e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \]
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\[ \int e^{n \arctan (a+b x)} x^2 \, dx=\int { x^{2} e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]
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Timed out. \[ \int e^{n \arctan (a+b x)} x^2 \, dx=\text {Timed out} \]
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Timed out. \[ \int e^{n \arctan (a+b x)} x^2 \, dx=\int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
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