Optimal. Leaf size=90 \[ \frac {i a e^{n \text {ArcTan}(a x)} (i+n)}{c n}-\frac {e^{n \text {ArcTan}(a x)}}{c x}-\frac {2 i a e^{n \text {ArcTan}(a x)} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};-1+\frac {2 i}{i+a x}\right )}{c} \]
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Rubi [A]
time = 0.09, antiderivative size = 166, normalized size of antiderivative = 1.84, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5190, 105, 160,
12, 133} \begin {gather*} -\frac {2 i a (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}-\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 160
Rule 5190
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{x^2 \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^2} \, dx}{c}\\ &=-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}-\frac {\int \frac {(1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \left (-a n+a^2 x\right )}{x} \, dx}{c}\\ &=-\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}+\frac {\int \frac {a^2 n^2 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{a c n}\\ &=-\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}+\frac {(a n) \int \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c}\\ &=-\frac {a (1-i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c x}-\frac {2 a n (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 = 0.04, size = 142, normalized size = 1.58 \begin {gather*} \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left ((-2 i+n) (1+i a x) (i a x+n (i+a x))+2 a n^2 x (1-i a 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) x (-i+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{2} \left (a^{2} c \,x^{2}+c \right )}\, dx\]
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^{4} + x^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^2\,\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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