Optimal. Leaf size=207 \[ -\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 \left (a^2+1\right ) x^2}-\frac {2 b^2 (2 a-n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{-1-\frac {i n}{2}} \, _2F_1\left (2,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{(-a+i) (a+i)^3 (-n+2 i)} \]
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Rubi [A] time = 0.11, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5095, 96, 131} \[ -\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 \left (a^2+1\right ) x^2}-\frac {2 b^2 (2 a-n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{-1-\frac {i n}{2}} \, _2F_1\left (2,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{(-a+i) (a+i)^3 (-n+2 i)} \]
Antiderivative was successfully verified.
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Rule 96
Rule 131
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {(1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}}}{x^3} \, 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 \left (1+a^2\right ) x^2}-\frac {(b (2 a-n)) \int \frac {(1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}}}{x^2} \, dx}{2 \left (1+a^2\right )}\\ &=-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 \left (1+a^2\right ) x^2}+\frac {2 b^2 (2 a-n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \, _2F_1\left (2,1+\frac {i n}{2};2+\frac {i n}{2};\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i+a)^2 \left (1+a^2\right ) (2 i-n)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 173, normalized size = 0.84 \[ -\frac {i (i a+i b x+1)^{-\frac {i n}{2}} (-i (a+b x+i))^{1+\frac {i n}{2}} \left (4 b^2 x^2 (n-2 a) \, _2F_1\left (2,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )+(a+i)^2 (n-2 i) (a+b x-i)^2\right )}{2 (a-i) (a+i)^3 (n-2 i) x^2 (a+b x-i)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (n \arctan \left (b x + a\right )\right )}}{x^{3}}, x\right ) \]
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 = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (b x +a \right )}}{x^{3}}\, dx \]
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 \[ \int \frac {e^{\left (n \arctan \left (b x + a\right )\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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