Optimal. Leaf size=191 \[ \frac {2 i (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \, _2F_1\left (1,\frac {i n}{2};\frac {i n}{2}+1;\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{n}-\frac {i 2^{1-\frac {i n}{2}} (-i a-i b x+1)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;\frac {1}{2} (-i a-i b x+1)\right )}{n} \]
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Rubi [A] time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5095, 105, 69, 131} \[ \frac {2 i (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \, _2F_1\left (1,\frac {i n}{2};\frac {i n}{2}+1;\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{n}-\frac {i 2^{1-\frac {i n}{2}} (-i a-i b x+1)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;\frac {1}{2} (-i a-i b x+1)\right )}{n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 105
Rule 131
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac {(1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}}}{x} \, dx\\ &=-\left ((-1+i a) \int \frac {(1-i a-i b x)^{-1+\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}}}{x} \, dx\right )-(i b) \int (1-i a-i b x)^{-1+\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx\\ &=\frac {2 i (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, _2F_1\left (1,\frac {i n}{2};1+\frac {i n}{2};\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{n}-\frac {i 2^{1-\frac {i n}{2}} (1-i a-i b x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};1+\frac {i n}{2};\frac {1}{2} (1-i a-i b x)\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 170, normalized size = 0.89 \[ \frac {2 i (i a+i b x+1)^{-\frac {i n}{2}} (-i (a+b x+i))^{\frac {i n}{2}} \left (\, _2F_1\left (1,\frac {i n}{2};\frac {i n}{2}+1;\frac {a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )-2^{-\frac {i n}{2}} (i a+i b x+1)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;-\frac {1}{2} i (a+b x+i)\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (n \arctan \left (b x + a\right )\right )}}{x}, x\right ) \]
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 = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (b x +a \right )}}{x}\, dx \]
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 \[ \int \frac {e^{\left (n \arctan \left (b x + a\right )\right )}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )}}{x} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atan}{\left (a + b x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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