Optimal. Leaf size=147 \[ \frac {2^{-\frac {i n}{2}} (2 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{b^2 (-n+2 i)}+\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5095, 80, 69} \[ \frac {2^{-\frac {i n}{2}} (2 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{b^2 (-n+2 i)}+\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 80
Rule 5095
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a+b x)} x \, dx &=\int x (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx\\ &=\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 b^2}-\frac {(2 a+n) \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx}{2 b}\\ &=\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 b^2}+\frac {2^{-\frac {i n}{2}} (2 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} \, _2F_1\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 )}{b^2 (2 i-n)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 128, normalized size = 0.87 \[ \frac {i (-i (a+b x+i))^{1+\frac {i n}{2}} \left (\frac {2^{1-\frac {i n}{2}} (2 a+n) \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;-\frac {1}{2} i (a+b x+i)\right )}{-2-i n}+(a+b x-i) (i a+i b x+1)^{-\frac {i n}{2}}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x e^{\left (n \arctan \left (b x + a\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (b x +a \right )} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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