Optimal. Leaf size=65 \[ \frac {i e^{n \text {ArcTan}(a x)}}{c n}-\frac {2 i e^{n \text {ArcTan}(a x)} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};e^{2 i \text {ArcTan}(a x)}\right )}{c n} \]
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Rubi [A]
time = 0.07, antiderivative size = 122, normalized size of antiderivative = 1.88, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5190, 98, 133}
\begin {gather*} \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {2 i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};\frac {i a x+1}{1-i a x}\right )}{c n} \end {gather*}
Antiderivative was successfully verified.
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Rule 98
Rule 133
Rule 5190
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )} \, dx &=\frac {\int \frac {(1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c}\\ &=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}+\frac {\int \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c}\\ &=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {2 (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, _2F_1\left (1,1+\frac {i n}{2};2+\frac {i n}{2};\frac {1-i a x}{1+i a x}\right )}{c (2+i n)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 120, normalized size = 1.85 \begin {gather*} \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left ((2+i n) (-i+a x)+2 (n-i a n x) \, _2F_1\left (1,1+\frac {i n}{2};2+\frac {i n}{2};\frac {i+a x}{i-a x}\right )\right )}{c n (-2 i+n) (-i+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x \left (a^{2} c \,x^{2}+c \right )}\, dx\]
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*} \frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{3} + x}\, dx}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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