Optimal. Leaf size=207 \[ -\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{2 \left (1+a^2\right ) x^2}-\frac {2 b^2 (2 a-n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \, _2F_1\left (2,1+\frac {i n}{2};2+\frac {i n}{2};\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i-a) (i+a)^3 (2 i-n)} \]
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Rubi [A]
time = 0.09, 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 = {5203, 98, 133}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 98
Rule 133
Rule 5203
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 = 0.05, size = 173, normalized size = 0.84 \begin {gather*} -\frac {i (1+i a+i b x)^{-\frac {i n}{2}} (-i (i+a+b x))^{1+\frac {i n}{2}} \left ((i+a)^2 (-2 i+n) (-i+a+b x)^2+4 b^2 (-2 a+n) x^2 \, _2F_1\left (2,1+\frac {i n}{2};2+\frac {i n}{2};\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{2 (-i+a) (i+a)^3 (-2 i+n) x^2 (-i+a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {e^{n \operatorname {atan}{\left (a + b x \right )}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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