Optimal. Leaf size=51 \[ \frac {x^{1+m} F_1\left (1+m;2-\frac {i n}{2},2+\frac {i n}{2};2+m;i a x,-i a x\right )}{c^2 (1+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5190, 138}
\begin {gather*} \frac {x^{m+1} F_1\left (m+1;2-\frac {i n}{2},\frac {i n}{2}+2;m+2;i a x,-i a x\right )}{c^2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 5190
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac {\int x^m (1-i a x)^{-2+\frac {i n}{2}} (1+i a x)^{-2-\frac {i n}{2}} \, dx}{c^2}\\ &=\frac {x^{1+m} F_1\left (1+m;2-\frac {i n}{2},2+\frac {i n}{2};2+m;i a x,-i a x\right )}{c^2 (1+m)}\\ \end {align*}
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Mathematica [F]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \text {ArcTan}(a x)} x^m}{\left (c+a^2 c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{2}}\, dx\]
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 {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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 {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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